Duality Principles and Numerical Procedures for a Large Class of Non-Convex Models in the Calculus of Variations

1. Introduction

In this section we establish a dual formulation for a large class of models in non-convex optimization. It is worth highlighting the main duality principle is applied to double well models similar as those found in the phase transition theory.

Such results are based on the works of J.J. Telega and W.R. Bielski [1,2,3,4] and on a D.C. optimization approach developed in Toland [5]. About the other references, details on the Sobolev spaces involved are found in [6]. Related results on convex analysis and duality theory are addressed in [7,8,9,10,11,12,13].

Similar models on the superconductivity physics may be found in [14,15,16].

At this point we recall that the duality principles are important since the related dual variational formulations are either convex (in fact concave) or have a large region of convexity around their critical points. These features are relevant considering that, from a concerning strict convexity, the standard Newton, Newton type and similar methods are in general convergent. Moreover, the dual variational formulations are also relevant since in some situations, it is possible to assure the global optimality of some critical points which satisfy certain specific constraints theoretically established.

Among the main results here developed, we highlight the duality principles for the quasi-convex formulations in the context of the vectorial calculus of variations. An important example in non-linear elasticity is addressed along the text in details.

Also, for the applications in physics in the final sections, we believe to have found a path to connect the quantum approach with a more classical one in a unified framework.

Indeed, we have presented a path to model a great variety of chemical reactions through such a connection between the atomic and classical worlds.

Finally, in this text we adopt the standard Einstein convention of summing up repeated indices, unless otherwise indicated.

In order to clarify the notation, here we introduce the definition of topological dual space.

Definition 1

(Topological dual spaces). Let U be a Banach space. We shall define its dual topological space, as the set of all linear continuous functionals defined on U. We suppose such a dual space of U, may be represented by another Banach space $U$, through a bilinear form $\langle ·,·\rangle U:U\times U*\to \mathbb{R}$ (here we are referring to standard representations of dual spaces of Sobolev and Lebesgue spaces). Thus, given $f:U\to \mathbb{R}$ linear and continuous, we assume the existence of a unique $u*\in U*$ such that

$$\begin{array}{c}f\left(u\right)=\langle u,u*\rangle U,\forall u\in U.\end{array}$$

(1)

The norm of f , denoted by $\parallel f\parallel$, is defined as

$$\begin{array}{c}\parallel f\parallel U*=\underset{u\in U}{sup}\left\{\right|\langle u,u*\rangle U|:\parallel u\parallel U\le \}\equiv \parallel u*\parallel U*.\end{array}$$

(2)

At this point we start to describe the primal and dual variational formulations.

2. A General Duality Principle Non-Convex Optimization

In this section we present a duality principle applicable to a model in phase transition.

This case corresponds to the vectorial one in the calculus of variations.

Let $\Omega \subset \mathbb{R}n$ be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider a functional $J:V\to \mathbb{R}$ where

$$J\left(u\right)=F(\nabla u,\dots ,\nabla uN)+G(u,\dots ,uN)-\langle ui,hi\rangle L,$$

and where

$$F(\nabla u,\dots ,\nabla uN)=\int \Omega f(\nabla u,\dots ,\nabla uN)dx$$

$f:\mathbb{R}N\times n\to \mathbb{R}$ is a three times Fréchet differentiable function not necessarily convex. Moreover,

$$V=\{u=(u,\dots ,uN)\in W,p(\Omega ;\mathbb{R}N):u=u\mathrm{on}\partial \Omega \},$$

$h=(h,\dots ,hN)\in L(\Omega ;\mathbb{R}N),$ and $<p<+\infty .$

We assume there exists $\alpha \in \mathbb{R}$ such that

$$\alpha =\underset{u\in V}{inf}J\left(u\right).$$

Furthermore, suppose G is Fréchet differentiable but not necessarily convex. A global optimum point may not be attained for J so that the problem of finding a global minimum for J may not be a solution.

Anyway, one question remains, how the minimizing sequences behave close the infimum of J.

We intend to use duality theory to approximately solve such a global optimization problem.

Define $V=W,(\Omega ;\mathbb{R}N)$ and

$$V\left(u\right)=\{\varphi \in V:\mathrm{supp}\varphi \subset B\left(u\right)\},$$

where

$$B\left(u\right)=\{x\in \Omega :f**(\nabla u\left(x\right))<f(\nabla u\left(x\right))\}.$$

Moreover, $Y=Y*=L(\Omega ;\mathbb{R}N\times n)$, $Y=Y*=L(\Omega ;\mathbb{R}N\times n)$, $Y=Y*=L(\Omega ;\mathbb{R}N),$ so that at this point we define, $F:V\times V\to \mathbb{R}$, $G:V\to \mathbb{R}$, $G:V\to \mathbb{R}$, $G:V\to \mathbb{R}$ and $G:V\to \mathbb{R},$ by

$$\begin{array}{ccc}\hfill F(u,\varphi )& =& F(\nabla u+\nabla \varphi ,\dots ,\nabla uN+\nabla \varphi N)+{\displaystyle K}\int \Omega \nabla uj·\nabla ujdx\hfill \\ & & +{\displaystyle K22}\int \Omega \nabla \varphi j·\nabla \varphi jdx\hfill \end{array}$$

(3)

and

$$G(u,\dots ,un)=G(u,\dots ,uN)+{\displaystyle K12}\int \Omega ujujdx-\langle ui,fi\rangle L,$$

$$G(\nabla u,\dots ,\nabla uN)={\displaystyle K12}\int \Omega \nabla uj·\nabla ujdx,$$

$$G(\nabla \varphi ,\dots ,\nabla \varphi N)={\displaystyle K22}\int \Omega \nabla \varphi j·\nabla \varphi jdx,$$

and

$$G(u,\dots ,uN)={\displaystyle K12}\int \Omega ujujdx.$$

Define now $J:V\times V\to \mathbb{R}$,

$$J(u,\varphi )=F(\nabla u+\nabla \varphi )+G\left(u\right)-\langle ui,hi\rangle L.$$

Observe that

$$\begin{array}{ccc}\hfill J(u,\varphi )& =& F(u,\varphi )+G\left(u\right)-G(\nabla u)-G(\nabla \varphi )-G\left(u\right)\hfill \\ & \le & F(u,\varphi )+G\left(u\right)-\langle \nabla u,z*\rangle L-\langle \nabla \varphi ,z*\rangle L-\langle u,z*\rangle L\hfill \\ & & +\underset{v\in Y}{sup}\{\langle v,z*\rangle L-G\left(v\right)\}\hfill \\ & & +\underset{v\in Y}{sup}\{\langle v,z*\rangle L-G\left(v\right)\}\hfill \\ & & +\underset{u\in V}{sup}\{\langle u,z*\rangle L-G\left(u\right)\}\hfill \\ & =& F(u,\varphi )+G\left(u\right)-\langle \nabla u,z*\rangle L-\langle \nabla \varphi ,z*\rangle L-\langle u,z*\rangle L\hfill \\ & & +G*(z*)+G*(z*)+G*(z*)\hfill \\ & =& J*(u,\varphi ,z*),\hfill \end{array}$$

(4)

$\forall u\in V,\varphi \in V\left(u\right),z*=(z*,z*,z*)\in Y*=Y*\times Y*\times Y*$.

From the general results in [5], we may infer that

$$\begin{array}{ccc}\hfill \underset{(u,\varphi )\in V\times V\left(u\right)}{inf}J(u,\varphi )& =& \underset{(u,\varphi ,z*)\in V\times V\left(u\right)\times Y*}{inf}J*(u,\varphi ,z*).\hfill \end{array}$$

(5)

On the other hand

$$\underset{u\in V}{inf}J\left(u\right)\ge \underset{(u,\varphi )\in V\times V\left(u\right)}{inf}J(u,\varphi ).$$

From these last two results we may obtain

$$\underset{u\in V}{inf}J\left(u\right)\ge \underset{(u,\varphi ,z*)\in V\times V\left(u\right)\times Y*}{inf}J*(u,\varphi ,z*).$$

Moreover, from standards results on convex analysis, we may have

$$\begin{array}{ccc}\hfill \underset{u\in V}{inf}J*(u,\varphi ,z*)& =& \underset{u\in V}{inf}\{F(u,\varphi )+G\left(u\right)\hfill \\ & & -\langle \nabla u,z*\rangle L-\langle \nabla \varphi ,z*\rangle L-\langle u,z*\rangle L\hfill \\ & & +G*(z*)+G*(z*)+G*(z*)\}\hfill \\ & =& \underset{(v*,v*)\in C*}{sup}\{-F*(v*+z*,\varphi )-G*(v*+z*)-\langle \nabla \varphi ,z*\rangle L\hfill \\ & & +G*(z*)+G*(z*)+G*(z*)\},\hfill \end{array}$$

(6)

where

$$C*=\{v*=(v*,v*)\in Y*\times Y*:-\mathrm{div}(v*)i+(v*)i=\mathbf{0},\forall i\in \{,\dots ,N\}\},$$

$$F*(v*+z*,\varphi )=\underset{u\in V}{sup}\{\langle u,-\mathrm{div}(z*+v*)\rangle L-F(u,\varphi )\},$$

and

$$G*(v*+z*)=\underset{u\in V}{sup}\{\langle u,v*+z*\rangle L-G\left(u\right)\}.$$

Thus, defining

$$J*(\varphi ,z*,v*)=F*(v*+z*,\varphi )-G*(v*+z*)-\langle \nabla \varphi ,z*\rangle L+G*(z*)+G*(z*)+G*(z*),$$

we have got

$$\begin{array}{ccc}\hfill \underset{u\in V}{inf}J\left(u\right)& \ge & \underset{(u,\varphi )\in V\times V}{inf}J(u,\varphi )\hfill \\ & =& \underset{(u,\varphi ,z*)\in V\times V\left(u\right)\times Y*}{inf}J*(u,\varphi ,z*)\hfill \\ & =& \underset{z*\in Y*}{inf}\left\{\underset{\varphi \in V}{inf}\left\{\underset{v*\in C*}{sup}J*(\varphi ,z*,v*)\right\}\right\}.\hfill \end{array}$$

(7)

Finally, observe that

$$\begin{array}{ccc}& & \underset{u\in V}{inf}J\left(u\right)\hfill \\ & \ge & \underset{z*\in Y*}{inf}\left\{\underset{\varphi \in V\left(u\right)}{inf}\left\{\underset{v*\in C*}{sup}J*(\varphi ,z*,v*)\right\}\right\}\hfill \\ & \ge & \underset{v*\in C*}{sup}\left\{\underset{(z*,\varphi )\in Y*\times V\left(u\right)}{inf}J*(\varphi ,z*,v*)\right\}.\hfill \end{array}$$

(8)

This last variational formulation corresponds to a concave relaxed formulation in $v$ concerning the original primal formulation.

3. Another Duality Principle for a Simpler Related Model in Phase Transition with a Respective Numerical Example

In this section we present another duality principle for a related model in phase transition.

Let $\Omega =[,]\subset \mathbb{R}$ and consider a functional $J:V\to \mathbb{R}$ where

$$J\left(u\right)={\displaystyle 12}\int \Omega ((u\prime )-)dx+{\displaystyle 12}\int \Omega udx-\langle u,f\rangle L,$$

and where

$$V=\{u\in W,\left(\Omega \right):u\left(\right)=\mathrm{and}u\left(\right)=/\}$$

and $f\in L\left(\Omega \right).$

A global optimum point is not attained for J so that the problem of finding a global minimum for J has no solution.

Anyway, one question remains, how the minimizing sequences behave close the infimum of J.

We intend to use duality theory to approximately solve such a global optimization problem.

Denoting $V=W,\left(\Omega \right)$, at this point we define, $F:V\to \mathbb{R}$ and $F:V\times V\to \mathbb{R}$ by

$$F\left(u\right)={\displaystyle 12}\int \Omega ((u\prime )-)dx,$$

and

$$F(u,\varphi )={\displaystyle 12}\int \Omega ((u\prime +\varphi \prime )-)dx.$$

Observe that

$$F\left(u\right)\ge \underset{\varphi \in V}{inf}F(u,\varphi ),\forall u\in V.$$

In order to restrict the action of $\varphi$ on the region where the primal functional is non-convex, we redefine a not relabeled

$$V=\left\{\varphi \in W,\left(\Omega \right):(\varphi \prime )-\le ,\mathrm{in}\Omega \right\}$$

and define also

$$F:V\times V\to \mathbb{R},$$

$$F:V\times V\to \mathbb{R}$$

and

$$G:V\times V\to \mathbb{R}$$

by

$$F(u,\varphi )={\displaystyle 12}\int \Omega ((u\prime +\varphi \prime )-)dx+{\displaystyle 12}\int \Omega udx-\langle u,f\rangle L,$$

$$\begin{array}{ccc}\hfill F(u,\varphi )& =& F(u,\varphi )+{\displaystyle K}\int \Omega (u\prime )dx\hfill \\ & & +{\displaystyle K12}\int \Omega (\varphi \prime )dx\hfill \end{array}$$

(9)

and

$$\begin{array}{ccc}\hfill G(u,\varphi )& =& {\displaystyle K}\int \Omega (u\prime )dx\hfill \\ & & +{\displaystyle K12}\int \Omega (\varphi \prime )dx\hfill \end{array}$$

(10)

Denoting $Y=Y*=L\left(\Omega \right)$ we also define the polar functional $G*:Y*\times Y*\to \mathbb{R}$ by

$$G*(v*,v*)=\underset{(u,\varphi )\in V\times V}{sup}\{\langle u,v*\rangle L+\langle \varphi ,v*\rangle L-G(u,\varphi )\}.$$

Observe that

$$\underset{u\in U}{inf}J\left(u\right)\ge \underset{((u,\varphi ),(v*,v*))\in V\times V\times [Y*]}{inf}\{G*(v*,v*)-\langle u,v*\rangle L-\langle \varphi ,v*\rangle L+F(u,\varphi )\}.$$

With such results in mind, we define a relaxed primal dual variational formulation for the primal problem, represented by $J*:V\times V\times [Y*]\to \mathbb{R}$, where

$$J*(u,\varphi ,v*,v*)=G*(v*,v*)-\langle u,v*\rangle L-\langle \varphi ,v*\rangle L+F(u,\varphi ).$$

Having defined such a functional, we may obtain numerical results by solving a sequence of convex auxiliary sub-problems, through the following algorithm (in order to obtain the concerning critical points, at first we have neglected the constraint $(\varphi \prime )-\le \mathrm{in}\Omega$).

Set $K\approx 0.1$ and $K=120.0$ and $<\epsilon \ll .$

Choose $(u,\varphi )\in V\times V,$ such that $\parallel u\parallel ,\infty <$ and $\parallel \varphi \parallel ,\infty <.$

Set $n=.$

Calculate $(vn*,(v*)n)$ solution of the system of equations:

$${\displaystyle \partial J*(un,\varphi n,vn*,(v*)n)\partial v*}=\mathbf{0}$$

and

$${\displaystyle \partial J*(un,\varphi n,vn*,(v*)n)\partial v*}=\mathbf{0},$$

that is

$${\displaystyle \partial G*(vn*,(v*)n)\partial v*}-un=$$

and

$${\displaystyle \partial G*(vn*,(v*)n)\partial v*}-\varphi n=$$

so that

$$vn*={\displaystyle \partial G(un,\varphi n)\partial u}$$

and

$$(v*)n*={\displaystyle \partial G(un,\varphi n)\partial \varphi }$$

Calculate $(un+,\varphi n+)$ by solving the system of equations:

$${\displaystyle \partial J*(un+,\varphi n+,vn*,(v*)n)\partial u}=\mathbf{0}$$

and

$${\displaystyle \partial J*(un+,\varphi n+,vn*,(v*)n)\partial \varphi }=\mathbf{0}$$

that is

$$-vn*+{\displaystyle \partial F(un+,\varphi n+)\partial u}=\mathbf{0}$$

and

$$-(v*)n+{\displaystyle \partial F(un+,\varphi n+)\partial \varphi }=\mathbf{0}$$

If $max\{\parallel un-un+\parallel \infty ,\parallel \varphi n+-\varphi n\parallel \infty \}\le \epsilon$, then stop, else set $n:=n+$ and go to item 4.

At this point, we present the corresponding software in MAT-LAB, in finite differences and based on the one-dimensional version of the generalized method of lines.

Here the software.

***********************

clear all

m8=300;

d=1/m8;

K=0.1;

K1=120;

for i=1:m8

$uo(i,)=i*d/;$

vo(i,1)=i*d/10;

yo(i,1)=sin(i*d*pi)/2;

end;

k=1;

b12=1.0;

while $(b12>10-4.3)$ and $(k<230000)$

k=k+1;

for i=1:m8-1

duo(i,1)=(uo(i+1,1)-uo(i,1))/d;

dvo(i,1)=(vo(i+1,1)-vo(i,1))/d;

end;

m9=zeros(2,2);

m9(1,1)=1;

i=1;

$f=*(duo(i,)+dvo(i,))-;$

m80(1,1,i)=-f1-K;

m80(1,2,i)=-f1;

m80(2,1,i)=-f1;

m80(2,2,i)=-f1-K1;

$y11(,i)=K*(uo(i+,)-*uo(i,))/d-yo(i,);$

$y11(,i)=K*(vo(i+,)-*vo(i,))/d;$

$m12=*m80(:,:,i)-m*d;$

m50(:,:,i)=m80(:,:,i)*inv(m12);

z(:,i)=inv(m12)*y11(:,i)*$d$;

for i=2:m8-1

$f=*(duo(i,)+dvo(i,))-$;

m80(1,1,i)=-f1-K;

m80(1,2,i)=-f1;

m80(2,1,i)=-f1;

m80(2,2,i)=-f1-K1;

$y11(,i)=K*(uo(i+,)-*uo(i,)+uo(i-,))/d-yo(i,);$

$y11(,i)=K*(vo(i+,)-*vo(i,)+vo(i-,))/d;$

$m12=*m80(:,:,i)-m*d-m80(:,:,i)*m50(:,:,i-);$

m50(:,:,i)=inv(m12)*m80(:,:,i);

$z(:,i)=inv\left(m12\right)*(y11(:,i)*d+m80(:,:,i)*z(:,i-));$

end;

U(1,m8)=1/2;

U(2,m8)=0.0;

for i=1:m8-1

U(:,m8-i)=m50(:,:,m8-i)*U(:,m8-i+1)+z(:,m8-i);

end;

for i=1:m8

u(i,1)=U(1,i);

v(i,1)=U(2,i);

end;

b12=max(abs(u-uo))

uo=u;

vo=v;

u(m8/2,1)

end;

for i=1:m8

y(i)=i*d;

end;

plot(y,uo)

**************************************

For the case in which $f\left(x\right)=$, we have obtained numerical results for $K=0.1$ and $K=120$. For such a concerning solution $u$ obtained, please see Figure 1. For the case in which $f\left(x\right)=sin\left(\pi x\right)/$, we have obtained numerical results also for $K=0.1$ and $K=120$. For such a concerning solution $u$ obtained, please see Figure 2.

Remark 1.

Observe that the solutions obtained are approximate critical points. They are not, in a classical sense, the global solutions for the related optimization problems. Indeed, such solutions reflect the average behavior of weak cluster points for concerning minimizing sequences.

3.1. A general proposal for relaxation

Let $\Omega \subset \mathbb{R}n$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider a functional $J:V\to \mathbb{R}$ where

$$J\left(u\right)=F(\nabla u)+G\left(u\right)-\langle u,f\rangle L,$$

where

$$V=\left\{u\in W,(\Omega ;\mathbb{R}N):u=u\mathrm{on}\partial \Omega \right\},$$

$$u\in C(\Omega ;\mathbb{R}N),$$

$f\in L(\Omega ;\mathbb{R}N)$, $G:V\to \mathbb{R}$ is convex and Fréchet differentiable, and

$$F(\nabla u)=\int \Omega f(\nabla u)dx,$$

where $f:\mathbb{R}N\times n\to \mathbb{R}$ is also Fréchet differentiable.

Assume there exists $N^\in \mathbb{N}$ such that

$$Wh\equiv \left\{y\in \mathbb{R}N\times n:f**\left(y\right)<f\left(y\right)\right\}=\cup j=N^Wj$$

where for each $j\in \{,\dots ,N^\}$$Wj\subset \mathbb{R}N\times n$ is an open connected set such that $\partial Wj$ is regular. We also suppose

$$Wj¯\cap Wk¯=\varnothing ,\forall j\ne k.$$

Define

$$W^j=\left\{vj\in W,(\Omega ;\mathbb{R}N);\nabla vj\left(x\right)\in Wj,\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega \right\}$$

and define also

$$W=\left\{v=(v,\dots ,vN^):vj\in W^j\forall j\in \{,\dots ,N^\}\mathrm{and}\mathrm{supp}vj\cap \mathrm{supp}vk=\varnothing ,\forall j\ne k\right\}.$$

At this point we define

$$h(u\left(x\right),v\left(x\right))=\left\{\begin{array}{cc}f(\nabla u\left(x\right)+\nabla vj\left(x\right)),\hfill & \mathrm{if}\nabla u\left(x\right)\in Wj,\hfill \\ f(\nabla u(x\left)\right),\hfill & \mathrm{if}\nabla u\left(x\right)\notin Wh,\hfill \end{array}\right.$$

(11)

and

$$H\left(u\right)=\underset{v\in Wu}{inf}\int \Omega h(u,v)dx,$$

where

$$Wu=\{v\in W:\nabla u\left(x\right)+\nabla vj\left(x\right)\in Wj,\mathrm{if}\nabla u\left(x\right)\in Wj,\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\forall j\in \{,\dots ,N^\}\}.$$

Moreover, we propose the relaxed functional

$$J\left(u\right)=H\left(u\right)+G\left(u\right)-\langle u,f\rangle L.$$

Observe that clearly

$$\underset{u\in V}{inf}J\left(u\right)\le \underset{u\in V}{inf}J\left(u\right).$$

4. A Convex dual Variational Formulation for a Third Similar Model

In this section we present another duality principle for a third related model in phase transition.

Let $\Omega =[,]\subset \mathbb{R}$ and consider a functional $J:V\to \mathbb{R}$ where

$$J\left(u\right)={\displaystyle 12}\int \Omega min\{(u\prime -),(u\prime +)\}dx+{\displaystyle 12}\int \Omega udx-\langle u,f\rangle L,$$

and where

$$V=\{u\in W,\left(\Omega \right):u\left(\right)=\mathrm{and}u\left(\right)=/\}$$

and $f\in L\left(\Omega \right).$

A global optimum point is not attained for J so that the problem of finding a global minimum for J has no solution.

Anyway, one question remains, how the minimizing sequences behave close to the infimum of J.

We intend to use the duality theory to solve such a global optimization problem in an appropriate sense to be specified.

At this point we define, $F:V\to \mathbb{R}$ and $G:V\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill F\left(u\right)& =& {\displaystyle 12}\int \Omega min\{(u\prime -),(u\prime +)\}dx\hfill \\ & =& {\displaystyle 12}\int \Omega (u\prime )dx-\int \Omega |u\prime |dx+/\hfill \\ & \equiv & F(u\prime ),\hfill \end{array}$$

(12)

and

$$G\left(u\right)={\displaystyle 12}\int \Omega udx-\langle u,f\rangle L.$$

Denoting $Y=Y*=L\left(\Omega \right)$ we also define the polar functional $F*:Y*\to \mathbb{R}$ and $G*:Y*\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill F*(v*)& =& \underset{v\in Y}{sup}\{\langle v,v*\rangle L-F\left(v\right)\}\hfill \\ & =& {\displaystyle 12}\int \Omega (v*)dx+\int \Omega |v*|dx,\hfill \end{array}$$

(13)

and

$$\begin{array}{ccc}\hfill G*((v*)\prime )& =& \underset{u\in V}{sup}\{-\langle u\prime ,v*\rangle L-G\left(u\right)\}\hfill \\ & =& {\displaystyle 12}\int \Omega ((v*)\prime +f)dx-{\displaystyle 12}v*\left(\right).\hfill \end{array}$$

(14)

Observe this is the scalar case of the calculus of variations, so that from the standard results on convex analysis, we have

$$\underset{u\in V}{inf}J\left(u\right)=\underset{v*\in Y*}{max}\{-F*(v*)-G*(-(v*)\prime )\}.$$

Indeed, from the direct method of the calculus of variations, the maximum for the dual formulation is attained at some $v^*\in Y*$.

Moreover, the corresponding solution $u\in V$ is obtained from the equation

$$u={\displaystyle \partial G((v^*)\prime )\partial (v*)\prime }=(v^*)\prime +f.$$

Finally, the Euler-Lagrange equations for the dual problem stands for

$$\left\{\begin{array}{cc}(v*)\prime \prime +f\prime -v*-\mathrm{sign}(v*)=,\hfill & \mathrm{in}\Omega ,\hfill \\ (v*)\prime \left(\right)+f\left(\right)=,(v*)\prime \left(\right)+f\left(\right)=/,\hfill \end{array}\right.$$

(15)

where $\mathrm{sign}(v*\left(x\right))=$ if $v*\left(x\right)>,$$\mathrm{sign}(v*\left(x\right))=-,$ if $v*\left(x\right)<$ and

$$-\le \mathrm{sign}(v*\left(x\right))\le ,$$

if $v*\left(x\right)=.$

We have computed the solutions $v$ and corresponding solutions $u\in V$ for the cases in which $f\left(x\right)=$ and $f\left(x\right)=sin\left(\pi x\right)/.$

For the solution $u\left(x\right)$ for the case in which $f\left(x\right)=$, please see Figure 3.

For the solution $u\left(x\right)$ for the case in which $f\left(x\right)=sin\left(\pi x\right)/$, please see Figure 4.

Remark 2.

Observe that such solutions $u$ obtained are not the global solutions for the related primal optimization problems. Indeed, such solutions reflect the average behavior of weak cluster points for concerning minimizing sequences.

4.1. The algorithm through which we have obtained the numerical results

In this subsection we present the software in MATLAB through which we have obtained the last numerical results.

This algorithm is for solving the concerning Euler-Lagrange equations for the dual problem, that is, for solving the equation

$$\left\{\begin{array}{cc}(v*)\prime \prime +f\prime -v*-\mathrm{sign}(v*)=,\hfill & \mathrm{in}\Omega ,\hfill \\ (v*)\prime \left(\right)=,(v*)\prime \left(\right)=/.\hfill \end{array}\right.$$

(16)

Here the concerning software in MATLAB. We emphasize to have used the smooth approximation

$$|v*|\approx \sqrt{(v*)+e},$$

where a small value for $e$ is specified in the next lines.

*************************************

clear all

$m=800;$ (number of nodes)

$d=/m;$

$e=.00001;$

$fori=:m$

$yo(i,)=0.01;$

$y(i,)=sin(\pi *i/m)/;$

$end;$

$fori=:m-$

$dy(i,)=(y(i+,)-y(i,))/d;$

$end;$

$fork=:3000$ (we have fixed the number of iterations)

$i=;$

$h=/\sqrt{vo(i,)+e};$

$m12=+d*h+d;$

$m50\left(i\right)=/m12;$

$z\left(i\right)=m50\left(i\right)*(dy(i,)*d);$

$fori=:m-$

$h=/\sqrt{vo(i,)+e};$

$m12=+h*d+d-m50(i-);$

$m50\left(i\right)=/m12;$

$z\left(i\right)=m50\left(i\right)*(z(i-)+dy(i,)*d);$

$end;$

$v(m,)=(d/+z(m-))/(-m50(m-));$

$fori=:m-$

$v(m-i,)=m50(m-i)*v(m-i+)+z(m-i);$

$end;$

$v(m/,)$

$vo=v;$

$end;$

$fori=:m-$

$u(i,)=(v(i+,)-v(i,))/d+y(i,);$

$end;$

$fori=:m-$

$x\left(i\right)=i*d;$

$end;$

$plot(x,u(:,\left)\right)$

********************************

5. An Improvement of the Convexity Conditions for a Non-Convex Related Model Through an Approximate Primal Formulation

In this section we develop an approximate primal dual formulation suitable for a large class of variational models.

Here, the applications are for the Kirchhoff-Love plate model, which may be found in Ciarlet, [17].

At this point we start to describe the primal variational formulation.

Let $\Omega \subset \mathbb{R}$ be an open, bounded, connected set which represents the middle surface of a plate of thickness h. The boundary of $\Omega$, which is assumed to be regular (Lipschitzian), is denoted by $\partial \Omega$. The vectorial basis related to the cartesian system $\{x,x,x\}$ is denoted by $(\mathbf{a}\alpha ,\mathbf{a})$, where $\alpha =,$ (in general Greek indices stand for 1 or 2), and where $\mathbf{a}$ is the vector normal to $\Omega$, whereas $\mathbf{a}$ and $\mathbf{a}$ are orthogonal vectors parallel to $\Omega .$ Also, $\mathbf{n}$ is the outward normal to the plate surface.

The displacements will be denoted by

$$\mathbf{u}^=\{u^\alpha ,u^\}=u^\alpha \mathbf{a}\alpha +u^\mathbf{a}.$$

The Kirchhoff-Love relations are

$$\begin{array}{ccc}& & u^\alpha (x,x,x)=u\alpha (x,x)-xw(x,x),\alpha \hfill \\ & & \mathrm{and}u^(x,x,x)=w(x,x).\hfill \end{array}$$

(17)

Here $-h/\le x\le h/$ so that we have $u=(u\alpha ,w)\in U$ where

$$\begin{array}{ccc}\hfill U& =& \left\{u=(u\alpha ,w)\in W,(\Omega ;\mathbb{R})\times W,\left(\Omega \right),\right.\hfill \\ & & u\alpha =w={\displaystyle \partial w\partial \mathbf{n}}=\left.\mathrm{on}\partial \Omega \right\}\hfill \\ & =& W,(\Omega ;\mathbb{R})\times W,\left(\Omega \right).\hfill \end{array}$$

It is worth emphasizing that the boundary conditions here specified refer to a clamped plate.

We also define the operator $\Lambda :U\to Y\times Y$, where $Y=Y*=L(\Omega ;\mathbb{R}\times )$, by

$$\Lambda \left(u\right)=\left\{\gamma \right(u),\kappa (u\left)\right\},$$

$$\gamma \alpha \beta \left(u\right)={\displaystyle u\alpha ,\beta +u\beta ,\alpha }+{\displaystyle w,\alpha w,\beta },$$

$$\kappa \alpha \beta \left(u\right)=-w,\alpha \beta .$$

The constitutive relations are given by

$$N\alpha \beta \left(u\right)=H\alpha \beta \lambda \mu \gamma \lambda \mu \left(u\right),$$

(18)

$$M\alpha \beta \left(u\right)=h\alpha \beta \lambda \mu \kappa \lambda \mu \left(u\right),$$

(19)

where: $\left\{H\alpha \beta \lambda \mu \right\}$ and $\left\{h\alpha \beta \lambda \mu ={\displaystyle h12}H\alpha \beta \lambda \mu \right\}$, are symmetric positive definite fourth order tensors. From now on, we denote $\{H¯\alpha \beta \lambda \mu \}=\left\{H\alpha \beta \lambda \mu \right\}-$ and $\{h¯\alpha \beta \lambda \mu \}=\left\{h\alpha \beta \lambda \mu \right\}-$.

Furthermore $\left\{N\alpha \beta \right\}$ denote the membrane force tensor and $\left\{M\alpha \beta \right\}$ the moment one. The plate stored energy, represented by $(G\circ \Lambda ):U\to \mathbb{R}$ is expressed by

$$(G\circ \Lambda )\left(u\right)={\displaystyle 12}\int \Omega N\alpha \beta \left(u\right)\gamma \alpha \beta \left(u\right)dx+{\displaystyle 12}\int \Omega M\alpha \beta \left(u\right)\kappa \alpha \beta \left(u\right)dx$$

(20)

and the external work, represented by $F:U\to \mathbb{R}$, is given by

$$F\left(u\right)=\langle w,P\rangle L+\langle u\alpha ,P\alpha \rangle L,$$

(21)

where $P,P,P\in L\left(\Omega \right)$ are external loads in the directions $\mathbf{a}$, $\mathbf{a}$ and $\mathbf{a}$ respectively. The potential energy, denoted by $J:U\to \mathbb{R}$ is expressed by:

$$J\left(u\right)=(G\circ \Lambda )\left(u\right)-F\left(u\right)$$

Define now $J:U˜\to \mathbb{R}$ by

$$J\left(u\right)=J\left(u\right)+J\left(w\right).$$

where

$$J\left(w\right)=10\int \Omega {\displaystyle aKbwln\left(a\right)K/}dx+10\int \Omega {\displaystyle a-K(bw-/100)ln\left(a\right)K/}dx.$$

In such a case for $a=2.71$, $K=185$, $b=P/\left|P\right|$ in $\Omega$ and

$$U˜=\{u\in U:\parallel w\parallel \infty \le 0.01\mathrm{and}Pw\ge \mathrm{a}.\mathrm{e}.\mathrm{in}\Omega \},$$

we get

$$\begin{array}{ccc}\hfill {\displaystyle \partial J\left(u\right)\partial w}& =& {\displaystyle \partial J\left(u\right)\partial w}+{\displaystyle \partial J\left(u\right)\partial w}\hfill \\ & \approx & {\displaystyle \partial J\left(u\right)\partial w}+\mathcal{O}(\pm 3.0),\hfill \end{array}$$

(22)

and

$$\begin{array}{ccc}\hfill {\displaystyle \partial J\left(u\right)\partial w}& =& {\displaystyle \partial J\left(u\right)\partial w}+{\displaystyle \partial J\left(u\right)\partial w}\hfill \\ & \approx & {\displaystyle \partial J\left(u\right)\partial w}+\mathcal{O}\left(850\right).\hfill \end{array}$$

(23)

This new functional $J$ has a relevant improvement in the convexity conditions concerning the previous functional J.

Indeed, we have obtained a gain in positiveness for the second variation ${\displaystyle \partial J\left(u\right)\partial w},$ which has increased of order $\mathcal{O}(700-1000).$

Moreover the difference between the approximate and exact equation

$${\displaystyle \partial J\left(u\right)\partial w}=\mathbf{0}$$

is of order $\mathcal{O}(\pm 3.0)$ which corresponds to a small perturbation in the original equation for a load of $P=1500N/m,$ for example. Summarizing, the exact equation may be approximately solved in an appropriate sense.

5.1. A duality principle for the concerning quasi-convex envelope

In this section, denoting

$$V=\{\varphi =\varphi (x,y)\in W,(\Omega \times \Omega ;\mathbb{R}):\varphi =\mathrm{on}\Omega \times \partial \Omega \},$$

we define the functional $J:U\times V\to \mathbb{R}$, where

$$\begin{array}{ccc}\hfill J(u,\varphi )& =& G\left(\left\{w,\alpha \beta \right\}\right)+G\left(\left\{{\displaystyle 12}(u\alpha ,\beta +u\beta ,\alpha )+\varphi \alpha ,y\beta +{\displaystyle 12}w,\alpha w,\beta \right\}\right)\hfill \\ & & -\langle w,P\rangle L-\langle u\alpha ,P\alpha \rangle L.\hfill \end{array}$$

(24)

where

$$G\left(\left\{w,\alpha \beta \right\}\right)={\displaystyle 12}\int \Omega h\alpha \beta \lambda \mu w,\alpha \beta w,\lambda \mu dx$$

and,

$$\begin{array}{ccc}& & G\left(\left\{{\displaystyle 12}(u\alpha ,\beta +u\beta ,\alpha )+\varphi \alpha ,y\beta +{\displaystyle 12}w,\alpha w,\beta \right\}\right)\hfill \\ & =& {\displaystyle \left|\Omega \right|}\int \Omega \int \Omega H\alpha \beta \lambda \mu \left({\displaystyle 12}(u\alpha ,\beta +u\beta ,\alpha )+\varphi \alpha ,y\beta (x,y)+{\displaystyle 12}w,\alpha w,\beta \right)\hfill \\ & & \times \left({\displaystyle 12}(u\lambda ,\mu +u\lambda ,\mu )+\varphi \lambda ,y\mu (x,y)+{\displaystyle 12}w,\lambda w,\mu \right)dxdy\hfill \end{array}$$

We define also

$$J(\left\{u\alpha \right\},\varphi )=\underset{w\in W,\left(\Omega \right)}{inf}J(u,\varphi ),$$

and

$$J\left(\left\{u\alpha \right\}\right)=\underset{\varphi \in V}{inf}J(\left\{u\alpha \right\},\varphi ).$$

It is a well known result from the modern Calculus of Variations theory (please, see [18] for details) that

$$\underset{u\in U}{inf}J\left(u\right)=\underset{\left\{u\alpha \right\}\in W,(\Omega ;\mathbb{R})}{inf}J\left(\left\{u\alpha \right\}\right).$$

At this point we denote

$$Y=Y*=Y=Y*\equiv L(\Omega \times \Omega ;\mathbb{R})$$

and

$$Y=Y*\equiv L(\Omega \times \Omega ;\mathbb{R}).$$

Observe that

$$\begin{array}{ccc}& & J\left(u\right)\hfill \\ & =& G\left(\left\{w,\alpha \beta \right\}\right)+G\left(\left\{{\displaystyle 12}(u\alpha ,\beta +u\beta ,\alpha )+\varphi \alpha ,y\beta +{\displaystyle 12}w,\alpha w,\beta \right\}\right)\hfill \\ & & -\langle w,P\rangle L-\langle u\alpha ,P\alpha \rangle L\hfill \\ & =& G\left(\left\{w,\alpha \beta \right\}\right)-\langle w,\alpha \beta ,M\alpha \beta \rangle L+\langle w,\alpha \beta ,M\alpha \beta \rangle L\hfill \\ & & +{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega w,\alpha \left(x\right),Q\alpha (x,y)dxdy-\langle w,P\rangle L\hfill \\ & & -{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega w,\alpha \left(x\right),Q\alpha (x,y)dxdy+G\left(\left\{{\displaystyle 12}(u\alpha ,\beta +u\beta ,\alpha )+\varphi \alpha ,y\beta +{\displaystyle 12}w,\alpha w,\beta \right\}\right)\hfill \\ & & -{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega \left({\displaystyle 12}(u\alpha ,\beta +u\beta ,\alpha )+\varphi \alpha ,y\beta +{\displaystyle 12}w,\alpha w,\beta \right),v\alpha \beta *(x,y)dxdy\hfill \\ & & +{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega \left({\displaystyle 12}(u\alpha ,\beta +u\beta ,\alpha )+\varphi \alpha ,y\beta +{\displaystyle 12}w,\alpha w,\beta \right),v\alpha \beta *(x,y)dxdy-\langle u\alpha ,P\alpha \rangle L\hfill \\ & \ge & \underset{v\in Y}{inf}\{-\langle \left(v\right)\alpha \beta ,M\alpha \beta \rangle L+G\left(\left(v\right)\alpha \beta \right)\}\hfill \\ & & +\underset{w\in W,\left(\Omega \right)}{inf}\left\{\langle w,\alpha \beta ,M\alpha \beta \rangle L+{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega w,\alpha \left(x\right)Q\alpha (x,y)dxdy-\langle w,P\rangle L\right\}\hfill \\ & & +\underset{v\in Y}{inf}\left\{-{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega v\alpha \beta v\alpha \beta *dxdy+G\left(\left\{v\alpha \beta \right\}\right)\right\}\hfill \\ & & +\underset{(v,\left\{u\alpha \right\})\in Y\times W,(\Omega ;\mathbb{R})}{inf}\left\{{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega \left({\displaystyle 12}(u\alpha ,\beta +u\beta ,\alpha )+\varphi \alpha ,y\beta +{\displaystyle 12}\left(v\right)\alpha (x,y)\left(v\right)\beta (x,y)\right)\right.\hfill \\ & & \left.\times v\alpha \beta *(x,y)dxdy-\langle u\alpha ,P\alpha \rangle L+{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega \left(v\right)\alpha (x,y)Q\alpha (x,y)dxdy\right\}\hfill \\ & \ge & -G*\left(M\right)-{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega \left(v\alpha \beta *¯\right)Q\alpha Q\beta dxdy-{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega H¯\alpha \beta \lambda \mu v\alpha \beta *v\lambda \mu *dxdy,\hfill \end{array}$$

(25)

$\forall u\in U,(M,Q)\in C*$, $v=\left\{v\alpha \beta \right\}\in A*$ where $A*=A*\cap A*\cap B*,$

$$A*=\{\{v\alpha \beta *\}\in Y*:(v\alpha \beta *),y\beta =,\mathrm{in}\Omega \},$$

$$A*=\left\{\{v\alpha \beta *\}\in Y*:{\displaystyle \left|\Omega \right|}\left(\int \Omega v\alpha \beta *dy\right),x\beta +P\alpha =,\mathrm{in}\Omega \right\},$$

$$B*=\left\{\{v\alpha \beta *\}\in Y*:\left\{v\alpha \beta *(x,y)\right\}\mathrm{is}\mathrm{positive}\mathrm{definite}\mathrm{in}\Omega \times \Omega \right\}.$$

and

$$C*=\left\{(M,Q)\in Y*\times Y*:M\alpha \beta ,\alpha \beta -\left(\int \Omega Q\alpha dy\right),x\alpha -P=,\mathrm{in}\Omega \right\}.$$

Also

$$\left\{v\alpha \beta *¯\right\}=\left\{v\alpha \beta *\right\}-,$$

and

$$\{H¯\alpha \beta \lambda \mu \}=\left\{H\alpha \beta \lambda \mu \right\}$$

in an appropriate tensor sense.

Here it is worth highlighting we have denoted,

$$\begin{array}{ccc}\hfill G*\left(M\right)& =& \underset{v\in Y}{sup}\{\langle \left(v\right)\alpha \beta ,M\alpha \beta \rangle L-G\left(v\right)\}\hfill \\ & =& {\displaystyle 12}\int \Omega h¯\alpha \beta \lambda \mu M\alpha \beta M\lambda \mu dx,\hfill \end{array}$$

(26)

where we recall that

$$\{h¯\alpha \beta \lambda \mu \}=\left\{h\alpha \beta \lambda \mu \right\}-$$

in an appropriate tensorial sense.

Summarizing, defining $J*:C*\times A*\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill J*((M,Q),v*)& =& -G*\left(M\right)-{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega \left(v\alpha \beta *¯\right)Q\alpha Q\beta dxdy\hfill \\ & & -{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega H¯\alpha \beta \lambda \mu v\alpha \beta *v\lambda \mu *dxdy,\hfill \end{array}$$

(27)

we have got

$$\underset{u\in U}{inf}J\left(u\right)\ge \underset{((M,Q),v*)\in C*\times A*}{sup}J*((M,Q),v*).$$

Remark 3.

This last dual functional is concave and such a concerning inequality corresponds a duality principle for the relaxed primal formulation.

We emphasize such results are extensions and in some sense complement the original duality principles in the works of Telega and Bielski, [1,2,3].

Moreover, if $((M,Q),v*)\in C*\times A*$ is such that

$$\delta J*((M,Q),v*)=\mathbf{0},$$

it is a well known result from the Legendre transform proprieties that the corresponding $(u,\varphi )\in V\times V$ such that

$$\left(w\right),\alpha \beta =h¯\alpha \beta \lambda \mu \left(M\right)\lambda \mu ,$$

and

$$(v*)\alpha \beta =H\alpha \beta \lambda \left({\displaystyle \left(u\right)\lambda ,\mu +\left(u\right)\mu ,\lambda }+{\displaystyle \left(\varphi \right)\lambda ,y\mu +\left(\varphi \right)\mu ,y\lambda }+{\displaystyle 12}\left(v20\right)\lambda \left(v20\right)\mu \right),$$

$$(v*)\alpha \beta ,y\beta =\mathbf{0},$$

is also such that

$$\delta J(u,\varphi )=\mathbf{0}$$

and

$$J(u,\varphi )=J*((M,Q),v*).$$

From this and

$$\underset{u\in V}{inf}J\left(u\right)=\underset{(u,\varphi )\in V\times V}{inf}J(u,\varphi )\ge \underset{((M,Q),v*)\in C*\times A*}{sup}J*((M,Q),v*),$$

we obtain

$$\begin{array}{ccc}\hfill J(u,\varphi )& =& \underset{(u,\varphi )\in V\times V}{inf}J(u,\varphi )\hfill \\ & =& \underset{((M,Q),v*)\in C*\times A*}{sup}J*((M,Q),v*)\hfill \\ & =& J*((M,Q),v*)\hfill \\ & =& \underset{u\in V}{inf}J\left(u\right).\hfill \end{array}$$

(28)

Also, from the modern calculus of variations theory, there exists a sequence $\left\{un\right\}\subset V$ such that

$$un\rightharpoonup u,\mathrm{weakly}\mathrm{in}V,$$

and

$$J\left(un\right)\to J(u,\varphi )=\underset{u\in V}{inf}J\left(u\right).$$

From this and the Ekeland variational principle, there exists $\left\{vn\right\}\subset V$ such that

$$\parallel un-vn\parallel V\le /n,$$

$$J\left(vn\right)\le \underset{u\in V}{inf}J\left(u\right)+/n,$$

and

$$\parallel \delta J\left(vn\right)\parallel V*\le /n,\forall n\in \mathbb{N},$$

so that

$$vn\rightharpoonup u,\mathrm{weakly}\mathrm{in}V,$$

and

$$J\left(vn\right)\to J(u,\varphi )=\underset{u\in V}{inf}J\left(u\right).$$

Assume now we are dealing with a finite dimensional version of such a model, in a finite elements of finite differences context, for example.

In such a case we have

$$vn\to u,\mathrm{strongly}\mathrm{in}\mathbb{R}N$$

for an appropriate $N\in \mathbb{N}.$

From continuity we obtain

$$\delta J\left(vn\right)\to \delta J\left(u\right)=\mathbf{0},$$

$$J\left(vn\right)\to J\left(u\right).$$

Summarizing, we have got

$$J\left(u\right)=\underset{u\in V}{inf}J\left(u\right),$$

$$\delta J\left(u\right)=\mathbf{0}.$$

Here we highlight such last results are valid just for this finite-dimensional model version.

6. A Duality Principle for a Related Relaxed Formulation Concerning the Vectorial Approach in the Calculus of Variations

In this section we develop a duality principle for a related vectorial model in the calculus of variations.

Let $\Omega \subset \mathbb{R}n$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega =\Gamma .$

For $<p<+\infty$, consider a functional $J:V\to \mathbb{R}$ where

$$J\left(u\right)=G(\nabla u)+F\left(u\right)-\langle u,f\rangle L,$$

where

$$V=\left\{u\in W,p(\Omega ;\mathbb{R}N):u=u\mathrm{on}\partial \Omega \right\},$$

$u\in C(\Omega ¯;\mathbb{R}N)$ and $f\in L(\Omega ;\mathbb{R}N).$

We assume $G:Y\to \mathbb{R}$ and $F:V\to \mathbb{R}$ are Fréchet differentiable and F is also convex.

Also

$$G(\nabla u)=\int \Omega g(\nabla u)dx,$$

where $g:\mathbb{R}N\times n\to \mathbb{R}$ it is supposed to be Fréchet differentiable. Here we have denoted $Y=Lp(\Omega ;\mathbb{R}N\times n)$.

We define also $J:V\times Y\to \mathbb{R}$ by

$$J(u,\varphi )=G(\nabla u+\nabla y\varphi )+F\left(u\right)-\langle u,f\rangle L,$$

where

$$Y=W,p(\Omega \times \Omega ;\mathbb{R}N)$$

and

$$G(\nabla u+\nabla y\varphi )={\displaystyle \left|\Omega \right|}\int \Omega \int \Omega g(\nabla u\left(x\right)+\nabla y\varphi (x,y))dxdy.$$

Moreover, we define the relaxed functional $J:V\to \mathbb{R}$ by

$$J\left(u\right)=\underset{\varphi \in V}{inf}J(u,\varphi ),$$

where

$$V=\{\varphi \in Y:\varphi (x,y)=,\mathrm{on}\Omega \times \partial \Omega \}.$$

Now observe that

$$\begin{array}{ccc}\hfill J(u,\varphi )& =& G(\nabla u+\nabla y\varphi )+F\left(u\right)-\langle u,f\rangle L\hfill \\ & =& -{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega v*(x,y)·(\nabla u+\nabla y\varphi (x,y))dydx+G(\nabla u+\nabla y\varphi )\hfill \\ & & +{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega v*(x,y)·(\nabla u+\nabla y\varphi (x,y))dydx+F\left(u\right)-\langle u,f\rangle L\hfill \\ & \ge & \underset{v\in Y}{inf}\left\{-{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega v*(x,y)·v(x,y)dydx+G\left(v\right)\right\}\hfill \\ & & +\underset{(v,\varphi )\in V\times V}{inf}\left\{{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega v*(x,y)·(\nabla u+\nabla y\varphi (x,y))dydx+F\left(u\right)-\langle u,f\rangle L\right\}\hfill \\ & =& -G*(v*)-F*\left(\mathrm{div}x\left({\displaystyle \left|\Omega \right|}\int \Omega v*(x,y)dy\right)+f\right)\hfill \\ & & +{\displaystyle \left|\Omega \right|}\int \partial \Omega \left(\int \Omega v*(x,y)dy\right)\otimes \mathbf{n}ud\Gamma ,\hfill \end{array}$$

(29)

$\forall (u,\varphi )\in V\times V,v*\in A*$, where

$$A*=\{v*\in Y*:\mathrm{div}yv*(x,y)=,\mathrm{in}\Omega \}.$$

Here we have denoted

$$G*(v*)=\underset{v\in Y}{sup}\left\{{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega v*(x,y)·v(x,y)dydx-G\left(v\right)\right\},$$

where $Y=Lp(\Omega \times \Omega ;\mathbb{R}N\times n),$$Y*=Lq(\Omega \times \Omega ;\mathbb{R}N\times n),$ and where

$${\displaystyle p}+{\displaystyle q}=.$$

Furthermore, for $v*\in A*$, we have

$$\begin{array}{ccc}& & F*\left(\mathrm{div}x\left({\displaystyle \left|\Omega \right|}\int \Omega v*(x,y)dy\right)+f\right)-{\displaystyle \left|\Omega \right|}\int \partial \Omega \left(\int \Omega v*(x,y)dy\right)\otimes \mathbf{n}ud\Gamma \hfill \\ & =& \underset{(v,\varphi )\in V\times V}{sup}\left\{-{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega v*(x,y)·(\nabla u+\nabla y\varphi (x,y))dydx-F\left(u\right)+\langle u,f\rangle L\right\},\hfill \end{array}$$

(30)

Therefore, denoting $J*:Y*\to \mathbb{R}$ by

$$J*(v*)=-G*(v*)-F*\left(\mathrm{div}x\left(\int \Omega v*(x,y)dy\right)+f\right)+{\displaystyle \left|\Omega \right|}\int \partial \Omega \left(\int \Omega v*(x,y)dy\right)\otimes \mathbf{n}ud\Gamma ,$$

we have got

$$\underset{u\in V}{inf}J\left(u\right)\ge \underset{v*\in A*}{sup}J*(v*).$$

Finally, we highlight such a dual functional $J$ is convex (in fact concave).

6.1. An Example in Finite Elasticity

In this section we develop an application of results obtained in the last section to a model in non-linear elasticity.

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Concerning a standard model in non-linear elasticity, consider a functional $J:V\to \mathbb{R}$ where

$$\begin{array}{ccc}& & J\left(u\right)\hfill \\ & =& {\displaystyle 12}\int \Omega Hijkl\left({\displaystyle ui,j+uj,i}+{\displaystyle 12}um,ium,j\right)\left({\displaystyle uk,l+uj,i}+{\displaystyle 12}um,kum,l\right)dx\hfill \\ & & -\langle ui,fi\rangle L\hfill \end{array}$$

(31)

where $f\in L(\Omega ;\mathbb{R})$ and $V=W,(\Omega ;\mathbb{R}).$

Here $\left\{Hijkl\right\}$ is a fourth-order and positive definite symmetric tensor (in an appropriate standard sense). Moreover, $u=(u,u,u)\in V$ is a field of displacements resulting from the f load field action on the volume comprised by $\Omega$.

At this point, we define the functional $J:V\times V\to \mathbb{R}$, where

$$\begin{array}{ccc}& & J(u,\varphi )\hfill \\ & =& {\displaystyle \left|\Omega \right|}\int \Omega \int \Omega Hijkl\left({\displaystyle ui,j+uj,i}+{\displaystyle \varphi i,yj+\varphi j,yi}{\displaystyle 12}(um,i+\varphi m,yi)(um,j+\varphi m,yj)\right)\hfill \\ & & \times \left({\displaystyle uk,l+ul,k}+{\displaystyle \varphi k,yl+\varphi l,yk}+{\displaystyle 12}(um,k+\varphi m,yk)(um,l+\varphi m,yl)\right)dxdy\hfill \\ & & -\langle ui,fi\rangle L,\hfill \end{array}$$

(32)

where

$$V=\{\varphi \in W,(\Omega \times \Omega ;\mathbb{R}):\varphi =\mathrm{on}\Omega \times \partial \Omega \}.$$

We define also the quasi-convex envelop of J, denoted by $QJ:V\to \mathbb{R}$, as

$$QJ\left(u\right)=\underset{\varphi \in V}{inf}J(u,\varphi ).$$

It is a well known result from the modern calculus of variations theory (please see [18] for details), that

$$\underset{u\in V}{inf}J\left(u\right)=\underset{u\in V}{inf}QJ\left(u\right).$$

Observe now that, denoting $Y=Y*=L(\Omega \times \Omega ;\mathbb{R})$, $Y=Y*=L(\Omega \times \Omega ;\mathbb{R}),$ and

$$\begin{array}{ccc}& & G\left({\displaystyle ui,j+uj,i}+{\displaystyle \varphi i,yj+\varphi j,yi}+{\displaystyle 12}(um,i+\varphi m,yi)(um,j+\varphi m,yj)\right)\hfill \\ & =& {\displaystyle \left|\Omega \right|}\int \Omega \int \Omega Hijkl\left({\displaystyle ui,j+uj,i}+{\displaystyle \varphi i,yj+\varphi j,yi}+{\displaystyle 12}(um,i+\varphi m,yi)(um,j+\varphi m,yj)\right)\hfill \\ & & \times \left({\displaystyle uk,l+ul,k}+{\displaystyle \varphi k,yl+\varphi l,yk}+{\displaystyle 12}(um,k+\varphi m,yk)(um,l+\varphi m,yl)\right)dxdy\hfill \end{array}$$

(33)

we have that

$$\begin{array}{ccc}& & J(u,\varphi )\hfill \\ & =& G\left({\displaystyle ui,j+uj,i}+{\displaystyle \varphi i,yj+\varphi j,yi}+{\displaystyle 12}(um,i+\varphi m,yi)(um,j+\varphi m,yj)\right)-\langle ui,fi\rangle L\hfill \\ & =& -{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega \left({\displaystyle ui,j+uj,i}+{\displaystyle \varphi i,yj+\varphi j,yi}+{\displaystyle 12}(um,i+\varphi m,yi)(um,j+\varphi m,yj)\right)\sigma ijdxdy\hfill \\ & & +G\left({\displaystyle ui,j+uj,i}+{\displaystyle \varphi i,yj+\varphi j,yi}+{\displaystyle 12}(um,i+\varphi m,yi)(um,j+\varphi m,yj)\right)\hfill \\ & & +{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega \left({\displaystyle ui,j+uj,i}+{\displaystyle \varphi i,yj+\varphi j,yi}+{\displaystyle 12}(um,i+\varphi m,yi)(um,j+\varphi m,yj)\right)\sigma ijdxdy-\langle ui,fi\rangle L\hfill \\ & \ge & \underset{v\in Y}{inf}\left\{-{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega vij\sigma ijdxdy-G\left(\left\{vij\right\}\right)\right\}\hfill \\ & & +\underset{v\in Y}{inf}\left\{-{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega \left(v\right)ijQijdxdy+{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega \left(\sigma ij{\displaystyle 12}\left(\left(v\right)mi\left(v\right)mj\right)\right)dxdy\right\}\hfill \\ & & +\underset{(u,\varphi )\in V\times V}{inf}\left\{{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega (\sigma ij+Qij)\left({\displaystyle ui,j+uj,i}+{\displaystyle \varphi i,yj+\varphi j,yi}\right)dxdy-\langle ui,fi\rangle L\right\}\hfill \\ & \ge & -{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega H¯ijkl\sigma ij\sigma kldxdy\hfill \\ & & -{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega \sigma ij¯QmiQmkdxdy,\hfill \end{array}$$

(34)

$\forall (u,\varphi )\in V\times V,(\sigma ,Q)\in A*$, where $A*=A*\cap A*\cap A*$,

$$A*=\{(\sigma ,Q)\in Y*\times Y*:\sigma ij,yj+Qij,yj=,\mathrm{in}\Omega \times \Omega \}.$$

$$A*=\left\{(\sigma ,Q)\in Y*\times Y*:{\displaystyle \left|\Omega \right|}\left(\int \Omega \left(\sigma ij\right)dy\right)xj+{\displaystyle \left|\Omega \right|}\left(\int \Omega \left(Qij\right)dy\right)xj+fi=,\mathrm{in}\Omega \right\},$$

$$A*=\{(\sigma ,Q)\in Y*\times Y*:\left\{\sigma ij\right\}\mathrm{is}\mathrm{positive}\mathrm{definite}\mathrm{in}\Omega \times \Omega \}.$$

Hence, denoting

$$J*(\sigma ,Q)=-{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega H¯ijkl\sigma ij\sigma kldxdy-{\displaystyle \left|\Omega \right|}\int \Omega \int \Omega \sigma ij¯QmiQmkdxdy,$$

we have obtained

$$\underset{u\in V}{inf}J\left(u\right)\ge \underset{(\sigma ,Q)\in A*}{sup}J*(\sigma ,Q).$$

Remark 4.

This last dual functional is concave and such a concerning inequality corresponds a duality principle for the relaxed primal formulation.

We emphasize again such results are also extensions and in some sense complement the original duality principles in the works of Telega and Bielski, [1,2,3].

Moreover, if $(\sigma ,Q)\in A*$ is such that

$$\delta J*(\sigma ,Q)=\mathbf{0},$$

it is a well known result from the Legendre transform proprieties that the corresponding $(u,\varphi )\in V\times V$ such that

$$\left(\sigma \right)ij=Hijkl\left({\displaystyle uk,l+ul,k}+{\displaystyle \varphi k,yl+\varphi l,yk}+{\displaystyle 12}(um,k+\varphi m,yk)(um,l+\varphi m,yl)\right)$$

and

$$\left(Q\right)ij=\left(\sigma \right)im\left(v20\right)mj,$$

is also such that

$$\delta J(u,\varphi )=\mathbf{0}$$

and

$$J(u,\varphi )=J*(\sigma ,Q).$$

From this and

$$\underset{u\in V}{inf}J\left(u\right)=\underset{(u,\varphi )\in V\times V}{inf}J(u,\varphi )\ge \underset{(\sigma ,Q)A*}{sup}J*(\sigma ,Q),$$

we obtain

$$\begin{array}{ccc}\hfill J(u,\varphi )& =& \underset{(u,\varphi )\in V\times V}{inf}J(u,\varphi )\hfill \\ & =& \underset{(\sigma ,Q)\in A*}{sup}J*(\sigma ,Q)\hfill \\ & =& J*(\sigma ,Q)\hfill \\ & =& \underset{u\in V}{inf}J\left(u\right).\hfill \end{array}$$

(35)

Also, from the modern calculus of variations theory, there exists a sequence $\left\{un\right\}\subset V$ such that

$$un\rightharpoonup u,\mathrm{weakly}\mathrm{in}V,$$

and

$$J\left(un\right)\to J(u,\varphi )=\underset{u\in V}{inf}J\left(u\right).$$

From this and the Ekeland variational principle, there exists $\left\{vn\right\}\subset V$ such that

$$\parallel un-vn\parallel V\le /n,$$

$$J\left(vn\right)\le \underset{u\in V}{inf}J\left(u\right)+/n,$$

and

$$\parallel \delta J\left(vn\right)\parallel V*\le /n,\forall n\in \mathbb{N},$$

so that

$$vn\rightharpoonup u,\mathrm{weakly}\mathrm{in}V,$$

and

$$J\left(vn\right)\to J(u,\varphi )=\underset{u\in V}{inf}J\left(u\right).$$

Assume now we are dealing with a finite dimensional version of such a model, in a finite elements of finite differences context, for example.

In such a case we have

$$vn\to u,\mathrm{strongly}\mathrm{in}\mathbb{R}N$$

for an appropriate $N\in \mathbb{N}.$

From continuity we obtain

$$\delta J\left(vn\right)\to \delta J\left(u\right)=\mathbf{0},$$

$$J\left(vn\right)\to J\left(u\right).$$

Summarizing, we have got

$$J\left(u\right)=\underset{u\in V}{inf}J\left(u\right),$$

$$\delta J\left(u\right)=\mathbf{0}.$$

Here we highlight such last results are valid just for this finite-dimensional model version.

7. An Exact Convex Dual Variational Formulation for a Non-Convex Primal One

In this section we develop a convex dual variational formulation suitable to compute a critical point for the corresponding primal one.

Let $\Omega \subset \mathbb{R}$ be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider a functional $J:V\to \mathbb{R}$ where

$$J\left(u\right)=F(ux,uy)-\langle u,f\rangle L,$$

$V=W,\left(\Omega \right)$ and $f\in L\left(\Omega \right).$

Here we denote $Y=Y*=L\left(\Omega \right)$ and $Y=Y*=L\left(\Omega \right)\times L\left(\Omega \right).$

Defining

$$V=\{u\in V:\parallel u\parallel ,\infty \le K\}$$

for some appropriate $K>$, suppose also F is twice Fréchet differentiable and

$$det\left\{{\displaystyle \partial F(ux,uy)\partial v\partial v}\right\}\ne ,$$

$\forall u\in V.$

Define now $F:V\to \mathbb{R}$ and $F:V\to \mathbb{R}$ by

$$F(ux,uy)=F(ux,uy)+{\displaystyle \epsilon }\int \Omega uxdx+{\displaystyle \epsilon }\int \Omega uydx,$$

and

$$F(ux,uy)={\displaystyle \epsilon }\int \Omega uxdx+{\displaystyle \epsilon }\int \Omega uydx,$$

where here we denote $dx=dxdx.$

Moreover, we define the respective Legendre transform functionals $F$ and $F$ as

$$F*(v*)=\langle v,v*\rangle L+\langle v,v*\rangle L-F(v,v),$$

where $v,v\in Y$ are such that

$$v*={\displaystyle \partial F(v,v)\partial v},$$

$$v*={\displaystyle \partial F(v,v)\partial v},$$

and

$$F*(v*)=\langle v,v*+f\rangle L+\langle v,v*\rangle L-F(v,v),$$

where $v,v\in Y$ are such that

$$v*+f={\displaystyle \partial F(v,v)\partial v},$$

$$v*={\displaystyle \partial F(v,v)\partial v}.$$

Here $f$ is any function such that

$$\left(f\right)x=f,\mathrm{in}\Omega .$$

Furthermore, we define

$$\begin{array}{ccc}\hfill J*(v*)& =& -F*(v*)+F*(v*)\hfill \\ & =& -F*(v*)+{\displaystyle \epsilon }\int \Omega (v*+f)dx+{\displaystyle \epsilon }\int \Omega (v*)dx.\hfill \end{array}$$

(36)

Observe that through the target conditions

$$v*+f=\epsilon ux,$$

$$v*=\epsilon uy,$$

we may obtain the compatibility condition

$$(v*+f)y-(v*)x=.$$

Define now

$$A*=\{v*=(v*,v*)\in Br(,)\subset Y*:(v*+f)y-(v*)x=,\mathrm{in}\Omega \},$$

for some appropriate $r>$ such that $J$ is convex in $Br(,).$

Consider the problem of minimizing $J$ subject to $v*\in A*.$

Assuming $r>$ is large enough so that the restriction in r is not active, at this point we define the associated Lagrangian

$$J*(v*,\phi )=J*(v*)+\langle \phi ,(v*+f)y-(v*)x\rangle L,$$

where $\phi$ is an appropriate Lagrange multiplier.

Therefore

$$\begin{array}{ccc}\hfill J*(v*)& =& -F*(v*)+{\displaystyle \epsilon }\int \Omega (v*+f)dx+{\displaystyle 12\epsilon }\int \Omega (v*)dx\hfill \\ & & +\langle \phi ,(v*+f)y-(v*)x\rangle L.\hfill \end{array}$$

(37)

The optimal point in question will be a solution of the corresponding Euler-Lagrange equations for $J*.$

From the variation of $J$ in $v$ we obtain

$$-{\displaystyle \partial F*(v*)\partial v*}+{\displaystyle v*+f\epsilon }-{\displaystyle \partial \phi \partial y}=.$$

(38)

From the variation of $J$ in $v$ we obtain

$$-{\displaystyle \partial F*(v*)\partial v*}+{\displaystyle v*\epsilon }+{\displaystyle \partial \phi \partial x}=.$$

(39)

From the variation of $J$ in $\phi$ we have

$$(v*+f)y-(v*)x=.$$

From this last equation, we may obtain $u\in V$ such that

$$v*+f=\epsilon ux,$$

and

$$v*=\epsilon uy.$$

From this and the previous extremal equations indicated we have

$$-{\displaystyle \partial F*(v*)\partial v*}+ux-{\displaystyle \partial \phi \partial y}=,$$

and

$$-{\displaystyle \partial F*(v*)\partial v*}+uy+{\displaystyle \partial \phi \partial x}=.$$

so that

$$v*+f={\displaystyle \partial F(ux-\phi y,uy+\phi x)\partial v},$$

and

$$v*={\displaystyle \partial F(ux-\phi y,uy+\phi x)\partial v}.$$

From this and equation (38) and (39) we have

$$\begin{array}{ccc}& & -\epsilon \left({\displaystyle \partial F*(v*)\partial v*}\right)x-\epsilon \left({\displaystyle \partial F*(v*)\partial v*}\right)y\hfill \\ & & +(v*+f)x+(v*)y\hfill \\ & =& -\epsilon uxx-\epsilon uyy+(v*)x+(v*)y+f=.\hfill \end{array}$$

(40)

Replacing the expressions of $v$ and $v$ into this last equation, we have

$$-\epsilon uxx-\epsilon uyy+\left({\displaystyle \partial F(ux-\phi y,uy+\phi x)\partial v}\right)x+\left({\displaystyle \partial F(ux-\phi y,uy+\phi x)\partial v}\right)y+f=,$$

so that

$$\left({\displaystyle \partial F(ux-\phi y,uy+\phi x)\partial v}\right)x+\left({\displaystyle \partial F(ux-\phi y,uy+\phi x)\partial v}\right)y+f=,\mathrm{in}\Omega .$$

(41)

Observe that if

$$\nabla \phi =$$

then there exists $u$ such that u and $\phi$ are also such that

$$ux-\phi y=u^x$$

and

$$uy+\phi x=u^y.$$

The boundary conditions for $\phi$ must be such that $u^\in W,.$

From this and equation (41) we obtain

$$\delta J(u^)=\mathbf{0}.$$

Summarizing, we may obtain a solution $u^\in W,$ of equation $\delta J(u^)=\mathbf{0}$ by minimizing $J$ on $A$.

Finally, observe that clearly $J$ is convex in an appropriate large ball $Br(,)$ for some appropriate $r>$

8. Another Primal Dual Formulation for a Related Model

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular boundary denoted by $\partial \Omega .$

Consider the functional $J:V\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J\left(u\right)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle \alpha }\int \Omega (u-\beta )dx\hfill \\ & & -\langle u,f\rangle L,\hfill \end{array}$$

(42)

$\alpha >,\beta >,\gamma >$, $V=W,\left(\Omega \right)$ and $f\in L\left(\Omega \right).$

Denoting $Y=Y*=L\left(\Omega \right)$, define now $J*:V\times Y*\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill J*(u,v*)& =& -{\displaystyle \gamma }\int \Omega \nabla u·\nabla udx-\langle u,v*\rangle L\hfill \\ & & +{\displaystyle K}\int \Omega (-\gamma \nabla u+v*u-f)dx+\langle u,f\rangle L\hfill \\ & & +{\displaystyle \alpha }\int \Omega (v*)dx+\beta \int \Omega v*dx,\hfill \end{array}$$

(43)

Define also

$$A+=\{u\in V:uf\ge ,\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega \},$$

$$V=\{u\in V:\parallel u\parallel \infty \le K\},$$

and

$$V=V\cap A+$$

for some appropriate $K>$ to be specified.

Moreover define

$$B*=\{v*\in Y*:\parallel v*\parallel \infty \le K\}$$

for some appropriate $K>$ to be specified.

Observe that, denoting

$$\phi =-\gamma \nabla u+v*u-f$$

we have

$${\displaystyle \partial J*(u,v*)\partial (v*)}={\displaystyle \alpha }+Ku$$

$${\displaystyle \partial J*(u,v*)\partial u}=\gamma \nabla -v*+K(-\gamma \nabla +v*)$$

and

$${\displaystyle \partial J*(u,v*)\partial u\partial v*}=K(\phi +(-\gamma \nabla u+v*u))-u$$

so that

$$\begin{array}{ccc}& & det\{\delta J*(u,v*)\}\hfill \\ & =& {\displaystyle \partial J*(u,v*)\partial (v*)}{\displaystyle \partial J*(u,v*)\partial u}-\left({\displaystyle \partial J*(u,v*)\partial u\partial v*}\right)\hfill \\ & =& {\displaystyle K(-\gamma \nabla +v*)\alpha }-{\displaystyle \gamma \nabla +v*+\alpha u\alpha }\hfill \\ & & -K12\phi -K\phi (-\gamma \nabla +v*)u+K\phi u\hfill \\ & & +K(-\gamma \nabla u+v*u)u.\hfill \end{array}$$

(44)

Observe now that a critical point $\phi =$ and $(-\gamma \nabla u+v*u)u=fu\ge$ in $\Omega$.

Therefore, for an appropriate large $K>$, also at a critical point, we have

$$\begin{array}{ccc}& & det\{\delta J*(u,v*)\}\hfill \\ & =& Kfu-{\displaystyle \delta J\left(u\right)\alpha }+K{\displaystyle (-\gamma \nabla +v*)\alpha }>\mathbf{0}.\hfill \end{array}$$

(45)

Remark 5.

From this last equation we may observe that $J$ has a large region of convexity about any critical point $(u,v^*)$, that is, there exists a large $r>$ such that $J$ is convex on $Br(u,v^*).$

With such results in mind, we may easily prove the following theorem.

Theorem 1.

Assume $K\gg max\{,K,K\}$ and suppose $(u,v^*)\in V\times B*$ is such that

$$\delta J*(u,v^*)=\mathbf{0}.$$

Under such hypotheses, there exists $r>$ such that $J$ is convex in $E*=Br(u,v^*)\cap (V\times B*)$,

$$\delta J\left(u\right)=\mathbf{0},$$

and

$$-J\left(u\right)=J(u,v^*)=\underset{(u,v*)\in E*}{inf}J*(u,v*).$$

9. A third primal dual formulation for a related model

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular boundary denoted by $\partial \Omega .$

Consider the functional $J:V\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J\left(u\right)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle \alpha }\int \Omega (u-\beta )dx\hfill \\ & & -\langle u,f\rangle L,\hfill \end{array}$$

(46)

$\alpha >,\beta >,\gamma >$, $V=W,\left(\Omega \right)$ and $f\in L\left(\Omega \right).$

Denoting $Y=Y*=L\left(\Omega \right)$, define now $J*:V\times Y*\times Y*\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill J*(u,v*,v*)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle 12}\int \Omega Kudx\hfill \\ & & -\langle u,v*\rangle L+{\displaystyle 12}\int \Omega {\displaystyle (v*)(-v*+K)}dx\hfill \\ & & +{\displaystyle (\alpha +\epsilon )}\int \Omega (v*-\alpha (u-\beta ))dx+\langle u,f\rangle L\hfill \\ & & -{\displaystyle \alpha }\int \Omega (v*)dx-\beta \int \Omega v*dx,\hfill \end{array}$$

(47)

where $\epsilon >$ is a small real constant.

Define also

$$A+=\{u\in V:uf\ge ,\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega \},$$

$$V=\{u\in V:\parallel u\parallel \infty \le K\},$$

and

$$V=V\cap A+$$

for some appropriate $K>$ to be specified.

Moreover define

$$B*=\{v*\in Y*:\parallel v*\parallel \infty \le K\}$$

and

$$D*=\{v*\in Y*:\parallel v*\parallel \le K\},$$

for some appropriate real constants $K,K>$ to be specified.

Remark 6.

Define now

$$H(u,v*)=-\gamma \nabla +v*+\alpha u$$

For an appropriate function (or, in a more general fashion, an appropriate bounded operator) $M$ define

$$B*=\{v*\in B*:v*+M\ge \epsilon \},$$

for some small parameter $\epsilon >.$

Moreover, define

$$E*=\{u\in V:\sqrt{\alpha }\left|u\right|\ge \sqrt{|M+\gamma \nabla |}.$$

Since for $(u,v*)\in V\times B*$ we have $uf\ge ,\mathrm{in}\Omega$, so that for $u,u\in V$ we have

$$\mathrm{sign}\left(u\right)=\mathrm{sign}\left(u\right)\mathrm{in}\Omega ,$$

we may infer that $E$ is a convex set.

Moreover if $(u,v*)\in E*\times B*$, then

$$\sqrt{\alpha }\left|u\right|\ge \sqrt{|M+\gamma \nabla |}$$

so that

$$\alpha u\ge M+\gamma \nabla$$

and

$$v*+M\ge \epsilon$$

so that

$$H(u,v*)=-\gamma \nabla +v*+\alpha u\ge \epsilon .$$

Such a result we will be used many times in the next sections.

Observe that, defining

$$\phi =v*-\alpha (u-\beta )$$

we may obtain

$${\displaystyle \partial J*(u,v*,v*)\partial u}=-\gamma \nabla +K+{\displaystyle \alpha \alpha +\epsilon }u-\phi {\displaystyle \alpha \alpha +\epsilon }$$

$${\displaystyle \partial J*(u,v*,v*)\partial (v*)}={\displaystyle -v*+K}$$

and

$${\displaystyle \partial J*(u,v*,v*)\partial u\partial v*}=-$$

so that

$$\begin{array}{ccc}& & det\left\{{\displaystyle \partial J*(u,v*,v*)\partial u\partial v*}\right\}\hfill \\ & =& {\displaystyle \partial J*(u,v*,v*)\partial (v*)}{\displaystyle \partial J*(u,v*,v*)\partial u}-\left({\displaystyle \partial J*(u,v*,v*)\partial u\partial v*}\right)\hfill \\ & =& {\displaystyle -\gamma \nabla +v*+\alpha \alpha +\epsilon u-\alpha \alpha +\epsilon \phi -v*+K}\hfill \\ & \equiv & H(u,v*).\hfill \end{array}$$

(48)

However, at a critical point, we have $\phi =\mathbf{0}$ so that, for a fixed $v*\in B*$ we define the non-active but convex restriction

$$\left(C\right)v**=\{u\in V:\left(\phi \right)\le \epsilon \},$$

for a small parameter $\epsilon >.$

From such results, assuming $K\gg max\{K,K,K\},$ and $<\epsilon \ll \epsilon \ll$, we have that

$$H(u,v*)>\mathbf{0},$$

for $v*\in B*$ and $u\in E*\cap \left(C\right)v**.$

With such results in mind, we may easily prove the following theorem.

Theorem 2.

Suppose $(u,v^*,v^*)\in (E*\cap \left(C\right)v^**)\times D*\times B*$ is such that

$$\delta J*(u,v^*,v^*)=\mathbf{0}.$$

Under such hypotheses, we have that

$$\delta J\left(u\right)=\mathbf{0}$$

and

$$\begin{array}{ccc}\hfill J\left(u\right)& =& \underset{u\in \left(C\right)v^**}{inf}J\left(u\right)\hfill \\ & =& J*(u,v^*,v^*)\hfill \\ & =& \underset{(u,v*)\in \left(C\right)v^**\times D*}{inf}\left\{\underset{v*\in B*}{sup}J*(u,v*,v*)\right\}\hfill \\ & =& \underset{v*\in B*}{sup}\left\{\underset{(u,v*)\in \left(C\right)v^**\times D*}{inf}J*(u,v*,v*)\right\}.\hfill \end{array}$$

(49)

Proof. 

The proof that

$$\delta J\left(u\right)=\mathbf{0}$$

and

$$J\left(u\right)=J*(u,v^*,v^*)$$

may be easily made similarly as in the previous sections.

Moreover, observe that for $K>$ sufficiently large, we have

$${\displaystyle \partial J*(u,v^*,v*)\partial (v*)}<\mathbf{0},\forall v*\in B*$$

so that this and the other hypotheses, we have also

$$J*(u,v^*,v^*)=\underset{(u,v*)\in \left(C\right)v^**\times D*}{inf}J*(u,v*,v^*)$$

and

$$J*(u,v^*,v^*)=\underset{v*\in B*}{sup}J*(u,v^*,v*).$$

From this, from a standard saddle point theorem and the remaining hypotheses, we may infer that

$$\begin{array}{ccc}\hfill J\left(u\right)& =& J*(u,v^*,v^*)\hfill \\ & =& \underset{(u,v*)\in \left(C\right)v^**\times D*}{inf}\left\{\underset{v*\in B*}{sup}J*(u,v*,v*)\right\}\hfill \\ & =& \underset{v*\in B*}{sup}\left\{\underset{(u,v*)\in \left(C\right)v^**\times D*}{inf}J*(u,v*,v*)\right\}.\hfill \end{array}$$

(50)

Moreover, observe that

$$\begin{array}{ccc}\hfill J*(u,v^*,v^*)& =& \underset{(u,v*)\in \left(C\right)v^**\times D*}{inf}J*(u,v*,v^*)\hfill \\ & \le & {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle K}\int \Omega udx\hfill \\ & & +\langle u,v^*\rangle L-{\displaystyle K}\int \Omega udx\hfill \\ & & -{\displaystyle \alpha }\int \Omega (v^*)dx-\beta \int \Omega v^*dx\hfill \\ & & +{\displaystyle (\alpha +\epsilon )}\int \Omega (v^*-\alpha (u-\beta ))dx-\langle u,f\rangle L\hfill \\ & \le & \underset{v*\in Y*}{sup}\left\{{\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+\langle u,v*\rangle \right.\hfill \\ & & -{\displaystyle \alpha }\int \Omega (v*)dx-\beta \int \Omega v*dx\hfill \\ & & \left.+{\displaystyle (\alpha +\epsilon )}\int \Omega (v*-\alpha (u-\beta ))dx-\langle u,f\rangle L\right\}\hfill \\ & =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle \alpha }\int \Omega (u-\beta )dx\hfill \\ & & -\langle u,f\rangle L,\forall u\in \left(C\right)v^**.\hfill \end{array}$$

(51)

Summarizing, we have got

$$J\left(u\right)=J*(u,v^*,v^*)\le \underset{u\in \left(C\right)v^**}{inf}J\left(u\right).$$

From such results, we may infer that

$$\begin{array}{ccc}\hfill J\left(u\right)& =& \underset{u\in \left(C\right)v^**}{inf}J\left(u\right)\hfill \\ & =& J*(u,v^*,v^*)\hfill \\ & =& \underset{(u,v*)\in \left(C\right)v^**\times D*}{inf}\left\{\underset{v*\in B*}{sup}J*(u,v*,v*)\right\}\hfill \\ & =& \underset{v*\in B*}{sup}\left\{\underset{(u,v*)\in \left(C\right)v^**\times D*}{inf}J*(u,v*,v*)\right\}.\hfill \end{array}$$

(52)

The proof is complete. □

10. An Algorithm for a Related Model in Shape Optimization

The next two subsections have been previously published by Fabio Silva Botelho and Alexandre Molter in [8], Chapter 21.

10.1. Introduction

Consider an elastic solid which the volume corresponds to an open, bounded, connected set, denoted by $\Omega \subset \mathbb{R}$ with a regular (Lipschitzian) boundary denoted by $\partial \Omega =\Gamma \cup \Gamma t$ where $\Gamma \cap \Gamma t=\varnothing .$ Consider also the problem of minimizing the functional $J^:U\times B\to \mathbb{R}$ where

$$J^(u,t)={\displaystyle 12}\langle ui,fi\rangle L\left(\Omega \right)+{\displaystyle 12}\langle ui,f^i\rangle L\left(\Gamma t\right),$$

subject to

$$\left\{\begin{array}{c}\left(Hijkl\left(t\right)ekl\left(u\right)\right),j+fi=\mathrm{in}\Omega ,\hfill \\ \\ Hijkl\left(t\right)ekl\left(u\right)nj-f^i=,\mathrm{on}\Gamma t,\forall i\in \{,,\}.\hfill \end{array}\right.$$

(53)

Here $\mathbf{n}=(n,n,n)$ denotes the outward normal to $\partial \Omega$ and

$$U=\{u=(u,u,u)\in W,(\Omega ;\mathbb{R}):u=(,,)=\mathbf{0}\mathrm{on}\Gamma \},$$

$$B=\left\{t:\Omega \to [,]\mathrm{measurable}:\int \Omega t\left(x\right)dx=t\left|\Omega \right|\right\},$$

where

$$<t<$$

and $\left|\Omega \right|$ denotes the Lebesgue measure of $\Omega .$

Moreover $u=(u,u,u)\in W,(\Omega ;\mathbb{R})$ is the field of displacements relating the cartesian system $(,x,x,x)$, resulting from the action of the external loads $f\in L(\Omega ;\mathbb{R})$ and $f^\in L(\Gamma t;\mathbb{R}).$

We also define the stress tensor $\left\{\sigma ij\right\}\in Y*=Y=L(\Omega ;\mathbb{R}\times ),$ by

$$\sigma ij\left(u\right)=Hijkl\left(t\right)ekl\left(u\right),$$

and the strain tensor $e:U\to L(\Omega ;\mathbb{R}\times )$ by

$$eij\left(u\right)={\displaystyle 12}(ui,j+uj,i),\forall i,j\in \{,,\}.$$

Finally,

$$\left\{Hijkl\left(t\right)\right\}=\{tHijkl+(-t)Hijkl\},$$

where $H$ corresponds to a strong material and $H$ to a very soft material, intending to simulate voids along the solid structure.

The variable t is the design one, which the optimal distribution values along the structure are intended to minimize its inner work with a volume restriction indicated through the set B.

The duality principle obtained is developed inspired by the works in [1,2]. Similar theoretical results have been developed in [7], however we believe the proof here presented, which is based on the min-max theorem is easier to follow (indeed we thank an anonymous referee for his suggestion about applying the min-max theorem to complete the proof). We highlight throughout this text we have used the standard Einstein sum convention of repeated indices.

Moreover, details on the Sobolev spaces addressed may be found in [6]. In addition, the primal variational development of the topology optimization problem has been described in [7].

The main contributions of this work are to present the detailed development, through duality theory, for such a kind of optimization problems. We emphasize that to avoid the check-board standard and obtain appropriate robust optimized structures without the use of filters, it is necessary to discretize more in the load direction, in which the displacements are much larger.

10.2. Mathematical formulation of the topology optimization problem

Our mathematical topology optimization problem is summarized by the following theorem.

Theorem 3.

Consider the statements and assumptions indicated in the last section, in particular those refereing to Ω and the functional $J^:U\times B\to \mathbb{R}.$

Define $J:U\times B\to \mathbb{R}$ by

$$J(u,t)=-G(e\left(u\right),t)+\langle ui,fi\rangle L\left(\Omega \right)+\langle ui,f^i\rangle L\left(\Gamma t\right),$$

where

$$G(e\left(u\right),t)={\displaystyle 12}\int \Omega Hijkl\left(t\right)eij\left(u\right)ekl\left(u\right)dx,$$

and where

$$dx=dxdxdx.$$

Define also $J*:U\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill J*\left(u\right)& =& \underset{t\in B}{inf}\left\{J(u,t)\right\}\hfill \\ & =& \underset{t\in B}{inf}\{-G(e\left(u\right),t)+\langle ui,fi\rangle L\left(\Omega \right)+\langle ui,f^i\rangle L\left(\Gamma t\right)\}.\hfill \end{array}$$

(54)

Assume there exists $c,c>$ such that

$$Hijklzijzkl>czijzij$$

and

$$Hijklzijzkl>czijzij,\forall z=\left\{zij\right\}\in \mathbb{R}\times ,\mathrm{such}\mathrm{that}z\ne \mathbf{0}.$$

Finally, define $J:U\times B\to \mathbb{R}\cup \{+\infty \}$ by

$$J(u,t)=J^(u,t)+Ind(u,t),$$

where

$$Ind(u,t)=\left\{\begin{array}{cc},\hfill & \mathrm{if}(u,t)\in A*,\hfill \\ +\infty ,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(55)

where $A*=A\cap A,$

$$A=\{(u,t)\in U\times B:\left(\sigma ij\left(u\right)\right),j+fi=,\mathrm{in}\Omega ,\forall i\in \{,,\}\}$$

and

$$A=\{(u,t)\in U\times B:\sigma ij\left(u\right)nj-f^i=,\mathrm{on}\Gamma t,\forall i\in \{,,\}\}.$$

Under such hypotheses, there exists $(u,t)\in U\times B$ such that

$$\begin{array}{ccc}\hfill J(u,t)& =& \underset{(u,t)\in U\times B}{inf}J(u,t)\hfill \\ & =& \underset{u^\in U}{sup}J*(u^)\hfill \\ & =& J*\left(u\right)\hfill \\ & =& J^(u,t)\hfill \\ & =& \underset{(t,\sigma )\in B\times C*}{inf}G*(\sigma ,t)\hfill \\ & =& G*(\sigma \left(u\right),t),\hfill \end{array}$$

(56)

where

$$\begin{array}{ccc}\hfill G*(\sigma ,t)& =& \underset{v\in Y}{sup}\{\langle vij,\sigma ij\rangle L\left(\Omega \right)-G(v,t)\}\hfill \\ & =& {\displaystyle 12}\int \Omega H¯ijkl\left(t\right)\sigma ij\sigma kldx,\hfill \end{array}$$

(57)

$$\{H¯ijkl\left(t\right)\}=\left\{Hijkl\left(t\right)\right\}-$$

and $C*=C\cap C,$ where

$$C=\{\sigma \in Y*:\sigma ij,j+fi=,\mathrm{in}\Omega ,\forall i\in \{,,\}\}$$

and

$$C=\{\sigma \in Y*:\sigma ijnj-f^i=,\mathrm{on}\Gamma t,\forall i\in \{,,\}\}.$$

Proof. 

Observe that

$$\begin{array}{ccc}\hfill \underset{(u,t)\in U\times B}{inf}J(u,t)& =& \underset{t\in B}{inf}\left\{\underset{u\in U}{inf}J(u,t)\right\}\hfill \\ & =& \underset{t\in B}{inf}\left\{\underset{u^\in U}{sup}\left\{\underset{u\in U}{inf}\left\{{\displaystyle }\int \Omega Hijkl\left(t\right)eij\left(u\right)ekl\left(u\right)dx\right.\right.\right.\hfill \\ & & +\langle u^i,\left(Hijkl\left(t\right)ekl\left(u\right)\right),j+fi\rangle L\left(\Omega \right)\hfill \\ & & \left.\left.\left.-\langle u^i,Hijkl\left(t\right)ekl\left(u\right)nj-f^i\rangle L\left(\Gamma t\right)\right\}\right\}\right\}\hfill \\ & =& \underset{t\in B}{inf}\left\{\underset{u^\in U}{sup}\left\{\underset{u\in U}{inf}\left\{{\displaystyle }\int \Omega Hijkl\left(t\right)eij\left(u\right)ekl\left(u\right)dx\right.\right.\right.\hfill \\ & & -\int \Omega Hijkl\left(t\right)eij(u^)ekl\left(u\right)dx\hfill \\ & & \left.\left.\left.+\langle u^i,fi\rangle L\left(\Omega \right)+\langle u^i,f^i\rangle L\left(\Gamma t\right)\right\}\right\}\right\}\hfill \\ & =& \underset{t\in B}{inf}\left\{\underset{u^\in U}{sup}\left\{-\int \Omega Hijkl\left(t\right)eij(u^)ekl(u^)dx\right.\right.\hfill \\ & & \left.\left.\langle u^i,fi\rangle L\left(\Omega \right)+\langle u^i,f^i\rangle L\left(\Gamma t\right)\right\}\right\}\hfill \\ & =& \underset{t\in B}{inf}\left\{\underset{\sigma \in C*}{inf}G*(\sigma ,t)\right\}.\hfill \end{array}$$

(58)

Also, from this and the min-max theorem, there exist $(u,t)\in U\times B$ such that

$$\begin{array}{ccc}\hfill \underset{(u,t)\in U\times B}{inf}J(u,t)& =& \underset{t\in B}{inf}\left\{\underset{u^\in U}{sup}J(u,t)\right\}\hfill \\ & =& \underset{u\in U}{sup}\left\{\underset{t\in B}{inf}J(u,t)\right\}\hfill \\ & =& J(u,t)\hfill \\ & =& \underset{t\in B}{inf}J(u,t)\hfill \\ & =& J*\left(u\right).\hfill \end{array}$$

(59)

Finally, from the extremal necessary condition

$${\displaystyle \partial J(u,t)\partial u}=\mathbf{0}$$

we obtain

$$\left(Hijkl\left(t\right)ekl\left(u\right)\right),j+fi=\mathrm{in}\Omega ,$$

and

$$Hijkl\left(t\right)ekl\left(u\right)nj-f^i=\mathrm{on}\Gamma t,\forall i\in \{,,\},$$

so that

$$G\left(e\left(u\right)\right)={\displaystyle 12}\langle \left(u\right)i,fi\rangle L\left(\Omega \right)+{\displaystyle 12}\langle \left(u\right)i,f^i\rangle L\left(\Gamma t\right).$$

Hence $(u,t)\in A*$ so that $Ind(u,t)=$ and $\sigma \left(u\right)\in C*.$

Moreover

$$\begin{array}{ccc}\hfill J*\left(u\right)& =& -G\left(e\left(u\right)\right)+\langle \left(u\right)i,fi\rangle L\left(\Omega \right)+\langle \left(u\right)i,f^i\rangle L\left(\Gamma t\right)\hfill \\ & =& G\left(e\left(u\right)\right)\hfill \\ & =& G\left(e\left(u\right)\right)+Ind(u,t)\hfill \\ & =& J(u,t)\hfill \\ & =& G*(\sigma \left(u\right),t).\hfill \end{array}$$

(60)

This completes the proof. □

10.3. About a concerning algorithm and related numerical method

For numerically solve this optimization problem in question, we present the following algorithm

Set $t=0.5\mathrm{in}\Omega$ and $n=$.

Calculate $un\in U$ such that

$$J(un,tn)=\underset{u\in U}{sup}J(u,tn).$$

Calculate $tn+\in B$ such that

$$J(un,tn+)=\underset{t\in B}{inf}J(un,t).$$

If $\parallel tn+-tn\parallel \infty <10-$ or $n>100$ then stop, else set $n:=n+$ and go to item 2.

We have developed a software in finite differences for solving such a problem.

Here the software.

**************************************

clear all

global P m8 d w u v Ea Eb Lo d1 z1 m9 du1 du2 dv1 dv2 c3

m8=27;

m9=24;

c3=0.95;

d=1.0/m8;

d1=0.5/m9;

Ea=$210*105$; (stronger material)

Eb=1000; (softer material simulating voids)

w=0.30;

P=-42000000;

z1=(m8-1)*(m9-1);

A3=zeros(z1,z1);

for i=1:z1

A3(1,i)=1.0;

end;

b=zeros(z1,1);

uo=0.000001*ones(z1,1);

u1=ones(z1,1);

b(1,1)=c3*z1;

for i=1:m9-1

for j=1:m8-1

Lo(i,j)=c3;

end; end;

for i=1:z1

x1(i)=c3*z1;

end;

for i=1:2*m8*m9

xo(i)=0.000;

end;

xw=xo;

xv=Lo;

for k2=1:24

c3=0.98*c3;

b(1,1)=c3*z1;

k2

b14=1.0;

k3=0;

while $(b14>10-3.5)$ and $(k<)$

k3=k3+1;

b12=1.0;

k=0;

while $(b12>10-4.0)$ and $(k<120)$

k=k+1;

k2

k3

k

X=fminunc(’funbeam’,xo);

xo=X;

b12=max(abs(xw-xo));

xw=X;

end;

for i=1:m9-1

for j=1:m8-1

$E=Lo(i,j)*(Ea-Eb);$

ex=du1(i,j);

ey=dv2(i,j);

exy=1/2*(dv1(i,j)+du2(i,j));

$Sx=E*(ex+w*ey)/(-w);$

$Sy=E*(w*ex+ey)/(-w);$

Sxy=E1/(2*(1+w))*exy;

dc3(i,j)=-(Sx*ex+Sy*ey+2*Sxy*exy);

end;

end;

for i=1:m9-1

for j=1:m8-1

f(j+(i-1)*(m8-1))=dc3(i,j);

end;

end;

for k1=1:1

k1

X1=linprog(f,$\left[\right]$,$\left[\right]$,A3,b,uo,u1,x1);

x1=X1;

end;

for i=1:m9-1

for j=1:m8-1

Lo(i,j)=X1(j+(m8-1)*(i-1));

end;

end;

b14=max(max(abs(Lo-xv)))

xv=Lo;

colormap(gray); imagesc(-Lo); axis equal; axis tight; axis off;pause(1e-6)

end;

end;

****************************************************

Here the auxiliary Function ’funbeam’

function S=funbeam(x)

global P m8 d w u v Ea Eb Lo d1 m9 du1 du2 dv1 dv2

for i=1:m9

for j=1:m8

u(i,j)=x(j+(m8)*(i-1));

v(i,j)=x(m8*m9+(i-1)*m8+j);

end;

end;

for i=1:m9

end;

u(m9-1,1)=0;

v(m9-1,1)=0;

u(m9-1,m8-1)=0;

v(m9-1,m8-1)=0;

for i=1:m9-1

for j=1:m8-1

du1(i,j)=(u(i,j+1)-u(i,j))/d;

du2(i,j)=(u(i+1,j)-u(i,j))/d1;

dv1(i,j)=(v(i,j+1)-v(i,j))/d;

dv2(i,j)=(v(i+1,j)-v(i,j))/d1;

end;

end;

S=0;

for i=1:m9-1

for j=1:m8-1

$E=Lo(i,j)*Ea+(-Lo(i,j))*Eb;$

ex=du1(i,j);

ey=dv2(i,j);

exy=1/2*(dv1(i,j)+du2(i,j));

$Sx=E*(ex+w*ey)/(-w);$

$Sy=E*(w*ex+ey)/(-w);$

Sxy=E1/(2*(1+w))*exy;

S=S+1/2*(Sx*ex+Sy*ey+2*Sxy*exy);

end;

end;

S=S*d*d1-P*v(2,(m8)/3)*d*d1;

***********************************************

For a two dimensional beam of dimensions $m\times 0.5m$ and $t=0.63$ we have obtained the following results:

Case A: For the optimal shape for a clamped beam at left (cantilever) and load $P=-106N\mathbf{j}$ at $(x,y)=(,0.25)$, please Figure 5.

Case B :For the optimal shape for a simply supported beam at $(,)$ and $(,)$ and load $P=-106N\mathbf{j}$ at $(x,y)=(/,0.5)$, please Figure 6.

In the first case the mesh was $28\times 24$. In the second one the mesh was $27\times 24$

11. A duality principle for a general vectorial case in the calculus of variations

In this section we develop a duality principle for a general vectorial case in variational optimization.

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega$. Let $J:V\to \mathbb{R}$ be a functional where

$$J\left(u\right)=G(\nabla u,\dots ,\nabla uN)-\langle u,f\rangle L,$$

where

$$V=W,(\Omega ;\mathbb{R}N)$$

and

$$f=(f,\dots ,fN)\in L(\Omega ;\mathbb{R}N).$$

Here we have denoted $u=(u,\dots ,uN)\in V$ and

$$\langle u,f\rangle L=\langle ui,fi\rangle L,$$

so that we may also denote

$$J\left(u\right)=G(\nabla u)-\langle u,f\rangle L.$$

Assume

$$G(\nabla u)=\int \Omega g(\nabla u)dx$$

where $g:\mathbb{R}N\to \mathbb{R}$ is a differentiable function such that

$$g\left(y\right)\to +\infty$$

as $\left|y\right|\to \infty$. Moreover, suppose there exists $\alpha \in \mathbb{R}$ such that

$$\alpha =\underset{u\in V}{inf}J\left(u\right).$$

It is well known that

$$\begin{array}{ccc}\hfill \alpha & =& \underset{u\in V}{inf}J\left(u\right)\hfill \\ & =& \underset{u\in V}{inf}J**\left(u\right)\hfill \\ & =& \underset{u\in V}{inf}\{(G\circ \nabla )**\left(u\right)-\langle u,f\rangle L\}.\hfill \end{array}$$

(61)

Under some mild hypotheses, from convexity, we have that

$$\begin{array}{ccc}& & \underset{u\in V}{inf}\{(G\circ \nabla )**\left(u\right)-\langle u,f\rangle L\}\hfill \\ & =& \underset{v*\in A*}{sup}\{-(G\circ \nabla )*(-divv*)\}=-(G\circ \nabla )*\left(f\right),\hfill \end{array}$$

(62)

where

$$A*=\{v*\in Y=Y*=L(\Omega ;\mathbb{R}N):divv*+f=\}.$$

Now observe that the restriction $v=\nabla u$ for some $u\in V$ is equivalent to the restriction

$$\mathrm{curl}vi=\mathbf{0},\mathrm{in}\Omega$$

where $v=\left\{vi\right\}=\left\{vij\right\}j=$, $\forall i\in \{,\dots ,N\},$ with appropriate boundary conditions, so that with an appropriate Lagrange multiplier $\varphi =\left\{\varphi i\right\}$, we obtain

$$\begin{array}{ccc}\hfill (G\circ \nabla )*(-divv*)& =& \underset{u\in V}{sup}\{\langle u,-divv*\rangle L-G(\nabla u)\}\hfill \\ & =& \underset{u\in V}{sup}\{\langle \nabla u,v*\rangle L-G(\nabla u)\}\hfill \\ & \le & \underset{\varphi \in Y*}{inf}\left\{\underset{v\in Y}{sup}\{\langle v,v*\rangle L-G\left(v\right)+\langle \varphi ,\mathrm{curl}v\rangle L\right\}\hfill \\ & =& \underset{\varphi \in Y*}{inf}G*(v*+\mathrm{curl}\varphi ).\hfill \end{array}$$

(63)

where we have denoted

$$\mathrm{curl}v=\left\{\mathrm{curl}vi\right\}$$

and

$$\mathrm{curl}\varphi =\left\{\mathrm{curl}\varphi i\right\}.$$

Joining the pieces, we have got

$$\begin{array}{ccc}\hfill \underset{u\in V}{inf}J\left(u\right)& =& \underset{u\in V}{inf}\{G(\nabla u)-\langle u,f\rangle L\}\hfill \\ & \ge & \underset{(v*,\varphi )\in A*\times Y*}{sup}\{-G*(v*+\mathrm{curl}\varphi )\},\hfill \end{array}$$

(64)

where we recall that $Y=Y*=L(\Omega ;\mathbb{R}N).$

We emphasize such a dual formulation in $(v*,\varphi )$ is convex (in fact concave).

12. A Note on the Galerkin Functional

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega$.

Consider the functional $J:V\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J\left(u\right)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle \alpha }\int \Omega udx\hfill \\ & & -{\displaystyle \beta }\int \Omega udx-\langle u,f\rangle L\hfill \end{array}$$

(65)

Here $V=W,\left(\Omega \right)$, $\gamma >,\alpha >,\beta >.$

We denote also

$$Y=Y*=L\left(\Omega \right).$$

At this point we define

$$A+=\{u\in V:uf\ge ,\mathrm{in}\Omega \},$$

$$V=\{u\in V:\parallel u\parallel \infty \le K\},$$

for some appropriate real constant $K>$ and

$$V=A+\cap V.$$

Observe that

$$J\prime \left(u\right)=-\gamma \nabla u+\alpha u-\beta -f,$$

so that we define the Galerkin functional $J:V\to \mathbb{R}$ by

$$J\left(u\right)={\displaystyle 12}\parallel J\prime \left(u\right)\parallel 22={\displaystyle 12}\int \Omega (-\gamma \nabla u+\alpha u-\beta u-f)dx.$$

From this, we get

$$\begin{array}{ccc}\hfill {\displaystyle \partial J\left(u\right)\partial u}& =& (-\gamma \nabla u+\alpha u-\beta u-f)\alpha u\hfill \\ & & +(-\gamma \nabla +\alpha u-\beta ).\hfill \end{array}$$

(66)

Define now

$$\phi =(-\gamma \nabla u+\alpha u-\beta u-f)\alpha u.$$

At this point, for an appropriate small real constant $\epsilon >$ and bounded constant operator $M>\epsilon$, we set the intended non-active restriction

$$\sqrt{\alpha }\left|u\right|\ge \sqrt{|M+\gamma \nabla +\beta |},$$

and define

$$B=\{u\in V:\sqrt{\alpha }\left|u\right|\ge \sqrt{|M+\gamma \nabla +\beta |}\}.$$

Observe that since for $u\in V$ we have $uf\ge$ in $\Omega$ so that if $u,u\in V$ then

$$\mathrm{sign}\left(u\right)=\mathrm{sign}\left(u\right),\mathrm{in}\Omega ,$$

we may infer that $B$ is a convex set.

Furthermore, if $u\in B$, then

$$\sqrt{\alpha }\left|u\right|\ge \sqrt{|M+\gamma \nabla +\beta |},$$

so that

$$\alpha u\ge M+\gamma \nabla +\beta ,$$

and hence

$$\delta J\left(u\right)=-\gamma \nabla +\alpha u-\beta \ge M>\epsilon >\mathbf{0}.$$

Observe now that

$${\displaystyle \partial \phi \partial u}=12\alpha (-\gamma \nabla +\alpha u-\beta ).$$

From such a result we may infer that

$${\displaystyle \partial \phi \partial u}\ge \mathbf{0},\mathrm{on}B,$$

so that $\phi$ is convex in $B.$

For a small parameter $\epsilon >$ we define the intended non-active restriction

$$|\phi |\le \epsilon ,\mathrm{in}\Omega ,$$

and define

$$B=\{u\in B:|\phi |\le \epsilon ,\mathrm{in}\Omega \}.$$

Assuming $<\epsilon \ll \epsilon \ll$,

Summarizing, if $u\in B$, then

$$\delta J\left(u\right)\ge \mathbf{0}.$$

With such results in mind, we define the following optimization problem for finding a critical point of J.

Minimize

$$J\left(u\right)={\displaystyle 12}\parallel J\prime \left(u\right)\parallel 22={\displaystyle 12}\int \Omega (-\gamma \nabla u+\alpha u-\beta u-f)dx,$$

subject to

$$u\in B.$$

Finally, we may also define the optimization problem of minimizing

$$\begin{array}{ccc}\hfill J\left(u\right)& =& KJ\left(u\right)+J\left(u\right)\hfill \\ & =& {\displaystyle K}\int \Omega (-\gamma \nabla u+\alpha u-\beta u-f)dx\hfill \\ & & +{\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle \alpha }\int \Omega udx\hfill \\ & & -{\displaystyle \beta }\int \Omega udx-\langle u,f\rangle L,\hfill \end{array}$$

(67)

subject to

$$u\in B.$$

Here $K>$ is a large real constant.

13. A Note on the Legendre-Galerkin Functional

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega$.

Consider the functional $J:V\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J\left(u\right)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle \alpha }\int \Omega udx\hfill \\ & & -{\displaystyle \beta }\int \Omega udx-\langle u,f\rangle L\hfill \end{array}$$

(68)

Here $V=W,\left(\Omega \right)$, $\gamma >,\alpha >,\beta >.$

We denote also

$$Y=Y*=L\left(\Omega \right)$$

and $F:V\to \mathbb{R}$, $F:V\to \mathbb{R}$ and $F:V\to \mathbb{R}$ by

$$F\left(u\right)={\displaystyle \gamma }\int \Omega \nabla u·\nabla udx,$$

$$F\left(u\right)={\displaystyle \alpha }\int \Omega udx,$$

$$F\left(u\right)={\displaystyle \beta }\int \Omega udx.$$

Moreover, we define $F*,F*,F*:Y*\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill F*(v*)& =& \underset{u\in V}{sup}\{\langle u,v*\rangle L-F\left(u\right)\}\hfill \\ & =& {\displaystyle 12}\int \Omega {\displaystyle (v*)-\gamma \nabla }dx,\hfill \end{array}$$

(69)

$$\begin{array}{ccc}\hfill F*(v*)& =& \underset{u\in V}{sup}\{\langle u,v*\rangle L-F\left(u\right)\}\hfill \\ & =& {\displaystyle 34}\int \Omega {\displaystyle (v*)/\alpha /}dx,\hfill \end{array}$$

(70)

$$\begin{array}{ccc}\hfill F*(v*)& =& \underset{u\in V}{sup}\{\langle u,v*\rangle L-F\left(u\right)\}\hfill \\ & =& {\displaystyle \beta }\int \Omega (v*)dx.\hfill \end{array}$$

(71)

Observe now that these three last suprema are attained through the equations,

$$v*={\displaystyle \partial F\left(u\right)\partial u}=-\gamma \nabla u,$$

$$v*={\displaystyle \partial F\left(u\right)\partial u}=\alpha u$$

$$v*={\displaystyle \partial F\left(u\right)\partial u}=\beta u.$$

From such results, at a critical point, we obtain the following compatibility conditions

$$u={\displaystyle v*-\gamma \nabla }=\left({\displaystyle v*\beta }\right)/={\displaystyle v*\beta }.$$

From such relations we have

$${\displaystyle v*-\gamma \nabla }={\displaystyle v*\beta },$$

and

$$v*=\alpha \left({\displaystyle v*\beta }\right),$$

so that

$$v*=-\gamma \nabla \left({\displaystyle v*\beta }\right),$$

and

$$v*=\alpha \left({\displaystyle v*\beta }\right).$$

Moreover, we define the functional $F*:Y*\to \mathbb{R},$ by

$$F*(v*)=\underset{u\in V}{sup}\{\langle u,v*+v*-v*\rangle L-\langle u,f\rangle L\}.$$

Therefore

$$F*(v*)=\left\{\begin{array}{cc},\hfill & \mathrm{if}v*+v*-v*-f=,\mathrm{in}\Omega ,\hfill \\ +\infty ,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(72)

Hence, a critical point of J corresponds to the solution of the following system of equations

$$v*=-\gamma \nabla \left({\displaystyle v*\beta }\right),$$

$$v*=\alpha \left({\displaystyle v*\beta }\right),$$

and

$$v*+v*-v*-f=,\mathrm{in}\Omega .$$

From this last equation we may obtain

$$v*=-v*+v*+f,$$

so that the final equations to be solved are

$$-v*+v*+f+\gamma \nabla \left({\displaystyle v*\beta }\right)=$$

and

$$v*-\alpha \left({\displaystyle v*\beta }\right)=,\mathrm{in}\Omega ,$$

with the boundary conditions

$$u={\displaystyle v*\beta }=,\mathrm{on}\partial \Omega .$$

With such results in mind, we define the Legendre-Galerkin functional $J*:[Y*]\to \mathbb{R}$, where

$$\begin{array}{ccc}\hfill J*(v*)& =& {\displaystyle 12}\int \Omega \left(-v*+v*+f+{\displaystyle \gamma \nabla v*\beta }\right)dx\hfill \\ & & +{\displaystyle 12}\int \Omega \left(v*-\alpha \left({\displaystyle v*\beta }\right)\right)dx.\hfill \end{array}$$

(73)

At this point, defining

$$\phi =v*-\alpha \left({\displaystyle v*\beta }\right),$$

we obtain

$${\displaystyle \partial J*(v*)\partial (v*)}=;$$

$${\displaystyle \partial J*(v*)\partial (v*)}=\left(--{\displaystyle \gamma \nabla \beta }\right)+{\displaystyle \alpha (v*)\beta }+\mathcal{O}\left(\phi \right),$$

$${\displaystyle \partial J*(v*)\partial v*\partial v*}={\displaystyle -\alpha (v*)\beta }+\left(--{\displaystyle \gamma \nabla \beta }\right).$$

From such results we may infer that

$$\begin{array}{ccc}\hfill det\left({\displaystyle \partial J*(v*)\partial v*\partial v*}\right)& =& {\displaystyle \partial J*(v*)\partial (v*)}{\displaystyle \partial J*(v*)\partial (v*)}-\left({\displaystyle \partial J*(v*)\partial v*\partial v*}\right)\hfill \\ & =& \left(--{\displaystyle \gamma \nabla \beta }+\alpha {\displaystyle (v*)\beta }\right)+\mathcal{O}\left(\phi \right)\hfill \end{array}$$

(74)

Observe that a critical point $\phi =$ so that $\delta J*(v*)>\mathbf{0}$ at a neighborhood of any critical point.

At this point we define

$$A+=\left\{v*=(v*,v*)\in [Y*]:{\displaystyle v*\beta }f\ge ,\mathrm{in}\Omega \right\},$$

$$D*=\{v*=(v*,v*)\in [Y*]:\parallel v*\parallel \infty \le K\},$$

for an appropriate real constant $K>$.

Define now $E*=A+\cap D*$,

$$C*=\{v*=(v*,v*)\in E*:\phi \le \epsilon ,\mathrm{in}\Omega \},$$

for a small real constant $\epsilon >,$

$$C*=\left\{v*=(v*,v*)\in E*:\left(--{\displaystyle \gamma \nabla \beta }+\alpha {\displaystyle (v*)\beta }\right)\ge \epsilon \right\},$$

and

$$C*=C*\cap C*.$$

Similarly as done in the previous section, we may prove that $C$ is a convex set.

Furthermore, for $<\epsilon \ll \epsilon \ll ,$ we have that $J$ is convex on $C$.

Summarizing, we may define the following convex optimization problem to obtain a critical point of the primal functional J,

$$\mathrm{Minimize}J*(v*,v*)\mathrm{subject}\mathrm{to}v*=(v*,v*)\in C*.$$

We call $J$ the Legendre-Galerkin functional associated to J.

13.1. Numerical examples

We have obtained numerical solutions for two one-dimensional examples.

For $\gamma =1.0,$$\alpha =3.0,$$\beta =30.0,$$f\equiv 10,\mathrm{in}\Omega =[,].$

For the respective solution please see Figure 7.

For $\gamma =0.01,$$\alpha =3.0,$$\beta =30.0,$$f\equiv 10,\mathrm{in}\Omega =[,].$

For the respective solution please see Figure 8.

14. A General Concave Dual Variational Formulation for Global Optimization

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider a functional $J:V\to \mathbb{R}$ where

$$J\left(u\right)=G\left(u\right)-\langle u,f\rangle L,\forall u\in V.$$

Here $V=W,\left(\Omega \right)$, $f\in L\left(\Omega \right)$ and we also denote $Y=Y*=L\left(\Omega \right).$

Assume there exists $\alpha \in \mathbb{R}$ such that

$$\alpha =\underset{u\in V}{inf}J\left(u\right).$$

Furthermore, suppose G is three times Fréchet differentiable and there exists $K>$ such that

$${\displaystyle \partial G\left(u\right)\partial u}+K>\mathbf{0},\forall u\in V.$$

Define now $J:V\times Y\to \mathbb{R}$ where,

$$J(u,v)=G(u,v)+F\left(u\right),$$

where

$$G(u,v)=G\left(v\right)-{\displaystyle \epsilon }\int \Omega vdx+{\displaystyle K}\int \Omega (v-u)dx,$$

and

$$F\left(u\right)={\displaystyle \epsilon }\int \Omega udx-\langle u,f\rangle L.$$

Moreover, we define the polar functionals $G*:Y*\times V\to \mathbb{R}$ and $F*:Y*\to \mathbb{R}$, where

$$\begin{array}{ccc}\hfill G*(v*,u)& =& \underset{v\in Y}{sup}\{\langle v,v*\rangle L-G(u,v)\}\hfill \\ & =& -GK\epsilon *(v*+Ku)+{\displaystyle K}\int \Omega udx,\hfill \end{array}$$

(75)

$$GK\epsilon *(v*+Ku)=\underset{v\in Y}{sup}\left\{\langle v,v*\rangle L-G\left(v\right)-{\displaystyle K}\int \Omega vdx+{\displaystyle \epsilon }\int \Omega vdx\right\},$$

and

$$\begin{array}{ccc}\hfill F*(-v*)& =& \underset{u\in V}{sup}\{-\langle u,v*\rangle L-F\left(u\right)\}\hfill \\ & =& {\displaystyle \epsilon }\int \Omega (v*-f)dx.\hfill \end{array}$$

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At this point we define the functional $J*:Y*\times V\to \mathbb{R}$ by

$$J*(v*,u)=-GK\epsilon *(v*+Ku)+{\displaystyle K}\int \Omega udx-F*(-v*).$$

With such results in mind we define

$$V=\{u\in V:\parallel u\parallel \infty \le K\},$$

and

$$D*=\{v*\in Y*:\parallel v*\parallel \infty \le K\},$$

for appropriated real constants $K>$ and $K>.$

Moreover, we define also the penalized functional $J*:Y*\times V\to \mathbb{R}$ where

$$J*(v*,u)=J*(v*,u)-{\displaystyle K12}\int \Omega \left(v*-{\displaystyle \partial G\left(u\right)\partial u}+\epsilon u\right)dx.$$

Finally, we remark that for $\epsilon >$ sufficiently small and $K>$ sufficiently large, $J$ is concave in $D*\times V$ around a concerning critical point. We recall that a critical point

$$v*-{\displaystyle \partial G\left(u\right)\partial u}+\epsilon u=,\mathrm{in}\Omega .$$

15. A Related Restricted Problem in Phase Transition

In this section we develop a convex (in fact concave) dual variational for a model similar to those found in phase transition problems.

Let $\Omega =[,]\subset \mathbb{R}.$ Consider the functional $J:V\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J\left(u\right)& =& {\displaystyle 12}\int \Omega min\{(u\prime +),(u\prime -)\}dx\hfill \\ & & +{\displaystyle 12}\int \Omega udx-\langle u,f\rangle L\hfill \\ & =& {\displaystyle 12}\int \Omega (u\prime )dx-\int \Omega |u\prime |dx+/\hfill \\ & & +{\displaystyle 12}\int \Omega udx-\langle u,f\rangle L.\hfill \end{array}$$

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Here

$$V=\{u\in W,\left(\Omega \right):u\left(\right)=\mathrm{and}u\left(\right)=/\}.$$

We also denote $V=W,\left(\Omega \right),$ and $Y=Y*=L\left(\Omega \right).$

Furthermore, we define the functionals G and $F:V\times V\to \mathbb{R}$ by

$$G(u\prime ,v\prime )={\displaystyle 12}\int \Omega (u\prime +v\prime )dx-\int \Omega |u\prime +v\prime |dx+/,$$

and

$$F(u,v)={\displaystyle 12}\int \Omega udx-\langle u,f\rangle L.$$

Moreover we define $J:V\times V\to \mathbb{R}$ by

$$J(u,v)=G(u\prime ,v\prime )+F(u,v),$$

and consider the problem of minimizing $J$ on the set

$$A=\{(u,v)\in V\times V:(v\prime )\le K,\mathrm{in}\Omega \}.$$

Already including the Lagrange multiplier $\varphi$ concerning such restrictions, we define

$$J(u,v,\varphi )=J(u,v)+{\displaystyle 12}\langle \varphi ,(v\prime )-K\rangle L.$$

Observe now that

$$\begin{array}{ccc}\hfill J(u,v,\varphi )& =& J(u,v)+{\displaystyle 12}\langle \varphi ,(v\prime )-K\rangle L\hfill \\ & =& G(u\prime ,v\prime )+{\displaystyle 12}\langle \varphi ,(v\prime )-K\rangle L\hfill \\ & & +F(u,v)\hfill \\ & =& -\langle u\prime ,v*\rangle L-\langle v\prime ,v*\rangle L+G(u\prime ,v\prime )\hfill \\ & & +{\displaystyle 12}\langle \varphi ,(v\prime )-K\rangle L\hfill \\ & & \langle u\prime ,v*\rangle L+\langle v\prime ,v*\rangle L+F(u,v)\hfill \\ & \ge & \underset{(v,v)\in Y\times Y}{inf}\left\{-\langle v,v*\rangle L-\langle v,v*\rangle L+G(v,v,\varphi )\right.\hfill \\ & & \left.+{\displaystyle 12}\langle \varphi ,\left(v\right)-K\rangle L\right\}\hfill \\ & & +\underset{(u,v)\in V\times V}{inf}\{\langle u\prime ,v*\rangle L+\langle v\prime ,v*\rangle L+F(u,v)\}\hfill \\ & =& -G*(v*,v*,\varphi )-F˜*(v*,v*),\forall (u,v)\in V\times V,(v*,v*,\varphi )\in [Y*],\hfill \end{array}$$

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where

$$G(u\prime ,v\prime ,\varphi )=G(u\prime ,v\prime )+{\displaystyle 12}\langle \varphi ,(v\prime )-K\rangle L.$$

Also,

$$\begin{array}{ccc}\hfill G*(v*,v*,\varphi )& =& \underset{(v,v)\in Y\times Y}{sup}\{\langle v,v*\rangle L+\langle v,v*\rangle L-G(v,v,\varphi )\}\hfill \\ & =& {\displaystyle 12}\int \Omega (v*)dx\hfill \\ & & +\int \Omega |v*|dx+{\displaystyle 12}\int \Omega {\displaystyle (v*-v*)\varphi }\hfill \\ & & +{\displaystyle K}\int \Omega \varphi dx,\hfill \end{array}$$

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where

$$F˜*(v*)=\left\{\begin{array}{cc}{\displaystyle 12}\int \Omega ((v*)\prime +f)dx-v*\left(\right)u\left(\right),\hfill & \mathrm{if}(v*)\prime =,\mathrm{in}\Omega ,\hfill \\ +\infty ,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

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From this we may infer that $v*=c,\mathrm{in}\Omega ,$ for some $c\in \mathbb{R}.$

Summarizing, denoting $v*=(v*,v*)=(v*,c)$, and

$$J*(v*,\varphi )=-G*(v*,\varphi )-F˜*(v*)$$

we have got

$$\underset{(u,v)\in A}{inf}J(u,v)\ge \underset{(v*,\varphi )\in Y*\times \mathbb{R}\times Y*}{sup}J*(v*\varphi ).$$

We have developed numerical results by maximizing the dual functional $J$ for two examples, namely.

Example A: In this case, we consider $f\left(x\right)=cos\left(\pi x\right)/$, $K=10-$.

For the optimal

$$u=(v*)\prime +f,$$

please see Figure 9.

Example B: In this case, we consider $f\left(x\right)=cos\left(\pi x\right)/$, $K=30$.

For the optimal

$$u=(v*)\prime +f,$$

please see Figure 10.

16. One More Dual Variational Formulation

In this section we develop one more dual variational formulation for a related model.

Let $\Omega =[,]\subset \mathbb{R}$ and consider the functional $J:V\to \mathbb{R}$ defined by

$$J\left(u\right)={\displaystyle 12}\int \Omega ((u\prime )-)dx+{\displaystyle 12}\int \Omega udx-\langle u,f\rangle L,$$

where

$$V=\{u\in W,\left(\Omega \right):u\left(\right)=\mathrm{and}u\left(\right)=/\}.$$

We define also the relaxed functional $J:V\times V\to \mathbb{R}$, already including a concerning restriction and corresponding non-negative Lagrange multiplier $\Lambda$, where

$$J(u,v,\Lambda )={\displaystyle 12}\int \Omega ((u\prime +v\prime )-)dx+{\displaystyle 12}\int \Omega udx-\langle u,f\rangle L+\langle \Lambda ,(v\prime )-K\rangle L.$$

where

$$V=\{v\in W,\left(\Omega \right):(v\prime )-K\le \mathrm{in}\Omega \}.$$

Observe that

$$\begin{array}{ccc}& & {\displaystyle 12}\int \Omega ((u\prime +v\prime )-)dx+{\displaystyle 12}\int \Omega udx-\langle u,f\rangle L+\langle \Lambda ,(v\prime )-K\rangle L\hfill \\ & =& -\langle v*,(u\prime +v\prime )-\rangle L+{\displaystyle 12}\int \Omega ((u\prime +v\prime )-)dx\hfill \\ & & +\langle v*,(u\prime +v\prime )-\rangle L+\langle \Lambda ,(v\prime )-K\rangle L-\langle u\prime ,v*\rangle L-\langle v\prime ,v*\rangle L\hfill \\ & & +\langle u\prime ,v*\rangle L+\langle v\prime ,v*\rangle L+{\displaystyle 12}\int \Omega udx-\langle u,f\rangle L\hfill \\ & \ge & \underset{w\in Y}{inf}\left\{-\langle v*,w\rangle L+{\displaystyle 12}\int \Omega \left(w\right)dx\right\}\hfill \\ & & \underset{(v,v)\in Y\times Y}{inf}\left\{\langle v*,(v+v)-\rangle L+\langle \Lambda ,\left(v\right)-K\rangle L-\langle v,v*\rangle L-\langle v,v*\rangle L\right\}\hfill \\ & & +\underset{(u,v)\in V\times V}{inf}\left\{\langle u\prime ,v*\rangle L+\langle v\prime ,v*\rangle L+{\displaystyle 12}\int \Omega udx-\langle u,f\rangle L\right\}\hfill \\ & =& -{\displaystyle 12}\int \Omega (v*)dx-\int \Omega v*dx\hfill \\ & & -{\displaystyle 14}\int \Omega {\displaystyle (v*)v*}dx-{\displaystyle 12}\int \Omega {\displaystyle (v*-v*)\Lambda }dx\hfill \\ & & -{\displaystyle 12}\int \Omega ((v*)\prime +f)dx-{\displaystyle 12}\int \Omega K\Lambda dx+v*\left(\right)u\left(\right).\hfill \end{array}$$

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Here, we highlight $v*=c\in \mathbb{R}\mathrm{in}\Omega$, for some real constant c.

Hence, denoting

$$\begin{array}{ccc}\hfill J*(v*,\Lambda )& =& -{\displaystyle 12}\int \Omega (v*)dx-\int \Omega v*dx\hfill \\ & & -{\displaystyle 14}\int \Omega {\displaystyle (v*)v*}dx-{\displaystyle 12}\int \Omega {\displaystyle (v*-v*)\Lambda }dx\hfill \\ & & -{\displaystyle 12}\int \Omega ((v*)\prime +f)dx-{\displaystyle 12}\int \Omega K\Lambda dx+v*\left(\right)u\left(\right)\hfill \end{array}$$

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and

$$J(u,v)={\displaystyle 12}\int \Omega ((u\prime +v\prime )-)dx+{\displaystyle 12}\int \Omega udx-\langle u,f\rangle L,$$

we have obtained

$$\underset{(u,v)\in V\times V}{inf}J(u,v)\}\ge \underset{(v*,\Lambda )\in A*\times [Y*]\times \mathbb{R}\times Y*}{sup}J*(v*,\Lambda ).$$

Finally, for

$$A*=\{v*\in Y*:v*\ge \epsilon \mathrm{in}\Omega \}$$

we emphasize $J$ is concave on $A*\times [Y*]\times \mathbb{R}\times Y*$.

Here $\epsilon >$ is a small regularizing real constant.

Remark 7.

The constraint $(v\prime )-K\le ,\mathrm{in}\Omega$ is included to restrict the action of v on the region where the primal functional is non-convex, through an appropriate constant $K>.$

17. A Model in Superconductivity Through an Eigenvalue Approach

In this section we intend to model superconductivity through a two phase eigenvalue approach.

Let $\Omega =[,]\subset \mathbb{R}$ be a straight wire corresponding to a one-dimensional super-conducting sample.

Consider the functional $J:V\times V\times \mathbb{R}\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J(u,v,E)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle \alpha }\int \Omega \left|u\right|dx\hfill \\ & & -{\displaystyle \omega }\int \Omega \left|u\right|dx\hfill \\ & & +{\displaystyle \gamma }\int \Omega \nabla v·\nabla vdx+{\displaystyle \alpha }\int \Omega \left|v\right|dx\hfill \\ & & -{\displaystyle \omega 12K32}\int \Omega \left|v\right|dx\hfill \\ & & -{\displaystyle E}\left(\int \Omega \left(\right|u|+\left|v\right|)dx-mT\right).\hfill \end{array}$$

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Here, in atomic units, $m$ is the total electronic charge, $V=W,\left(\Omega \right)$ and we set $\alpha =104$ corresponding to higher self-interacting energy which is related to a normal phase. We also set $\alpha =10-$ corresponding to a lower self-interacting energy which is related to a super-conducting phase and respective super-currents.

Moreover, we set $\gamma =\gamma =,$ and initially $\omega =1.8$ which is gradually decreased to $\omega =1.0$.

Furthermore, we define

$$|\varphi N|={\displaystyle \left|u\right|\left|u\right|+\left|v\right|}$$

and

$$|\varphi S|={\displaystyle \left|v\right|\left|u\right|+\left|v\right|}$$

where $\varphi$ corresponds to a normal phase and $\varphi$ to a super-conducting one.

At this point we observe that the temperature $T=T(x,t)$ is proportional the frequency $\omega /\left(\pi \right)$ of vibration for the normal phase.

We start the process with $\omega =1.8$ which in atomic units corresponds to a higher temperature and gradually decreases it to the value $\omega =1.0$

Between $\omega =1.2$ and $\omega =1.0$ the system changes from an almost total normal phase to an almost total super-conducting phase, as expected.

We highlight that the temperature is proportional to the vibrational kinetics energy

$$E\left(t\right)={\displaystyle 12}\int \Omega \left|u\right|{\displaystyle \partial \mathbf{r}N(x,t)\partial t}·{\displaystyle \partial \mathbf{r}N(x,t)\partial t}dx$$

so that for

$$\mathbf{r}N(x,t)=ei\omega t\mathbf{w}\left(x\right)$$

and for a suitable vectorial function $\mathbf{w}$, we have

$$T\propto E\propto \omega$$

so that we may model the decreasing of temperature T through the decreasing of $\omega$.

For $\omega =1.8$, for the corresponding normal phase $\varphi$ and super-conducting phase $\varphi$, please se Figure 11 and Figure 12, respectively.

For $\omega =1.0$, for the corresponding normal phase $\varphi$ and super-conducting phase $\varphi$, please se Figure 13 and Figure 14, respectively.

Finally, we have set $\omega /K\approx$ which for large $\omega$ corresponds to the super-currents.

18. A Simplified Qualitative Many Body Model for the Hydrogen Nuclear Fusion

In this section we develop a qualitative simple model for the hydrogen nuclear fusion.

Let $\Omega =[,L]\subset \mathbb{R}$ be a box in which is confined a gas comprised by an amount of ionized deuterium and tritium isotopes of hydrogen.

Though a suitable increasing in temperature, we intend to develop the following nuclear reaction

$$\mathrm{Deuterium}++\mathrm{Tritium}+\to \mathrm{Helium}+++\mathrm{Neutron}\left(\mathrm{energetic}\right).$$

We recall that the ionized Deuterium atom comprises a proton and a neutron and the ionized Tritium atom comprises a proton and two neutrons.

Under certain conditions and at a suitable high temperature the ionized Deuterium and Tritium atoms react chemically resulting in an ionized Helium atom, comprised by two protons and two neutrons and resulting also in one more single energetic neutron. We emphasize the higher kinetics neutron energy level has many potential practical applications, including its conversion in electric energy.

At this point we denote by $mD,mT,mHe\mathrm{and}mN$ the masses of the ionized Deuterium, Tritium and Helium atoms, and the single neutron, respectively.

Therefore, we have the following mass relation

$$mD+mT=mHe+mN.$$

To simplify our analysis, in such a chemical reaction, denoting the total masses of ionized Deuterium, Tritium, Helium and single Neutrons by $\left(mD\right)T,\left(mT\right)T,\left(mHe\right)T$ and $\left(mN\right)$ we assume there is a real constant $c>$ such that

$$\left(mD\right)T=cmD,\left(mT\right)T=cmT,\left(mHe\right)T=cmHe,\left(mN\right)T=cmN.$$

With such statements and definitions in mind, we define the following functional J, where

$$J(\varphi ,\mathbf{r})=J(\varphi D,\varphi T,\varphi He,\varphi N,\mathbf{r})=G(\nabla \varphi )+F\left(\varphi \right)+Ec(\varphi ,\mathbf{r}),$$

where, in a simplified many body context,

$$|\varphi D(x,y)|=|\varphi pD\left(y\right)|+|\varphi ND(x,y)|\left|\varphi pD\left(y\right)\right|{\displaystyle mp},$$

$$|\varphi T(x,y)|=|\varphi pT\left(y\right)|+\left(\right|\varphi NT(x,y)|+|\varphi NT(x,y)|\left)\right|\varphi pT\left(y\right)|{\displaystyle mp},$$

$$|\varphi He(x,y)|=|\varphi PHe\left(y\right)|+\left(\right|\varphi NHe(x,y)|+|\varphi NHe(x,y)|\left)\right|\varphi PHe\left(y\right)|{\displaystyle mp},$$

$$\varphi N=\varphi N\left(x\right).$$

Here $x,y\in \Omega \subset \mathbb{R}$ refers to the particle densities.

Furthermore, we assume $\gamma pD>,\gamma pT>,\gamma ND>,\gamma NT>,\gamma NT>,$$\gamma pHe>,\gamma NHe>,$ $\gamma NHe>,\gamma N>,$ and $\alpha D>,\alpha T>,\alpha He>,\alpha N>$, $\alpha DT>,\alpha HeN>,$ so that

$$\begin{array}{ccc}\hfill G(\nabla \varphi )& =& {\displaystyle \gamma pD}\int \Omega (\nabla \varphi pD)·(\nabla \varphi pD)dy\hfill \\ & & +{\displaystyle \gamma ND}\int \Omega (\nabla \varphi ND)·(\nabla \varphi ND)dxdy\hfill \\ & & {\displaystyle \gamma pT}\int \Omega (\nabla \varphi pT)·(\nabla \varphi pT)dy\hfill \\ & & +{\displaystyle \gamma NT}\int \Omega (\nabla \varphi NT)·(\nabla \varphi NT)dxdy\hfill \\ & & +{\displaystyle \gamma NT}\int \Omega (\nabla \varphi NT)·(\nabla \varphi NT)dxdy\hfill \\ & & +{\displaystyle \gamma pHe}\int \Omega (\nabla \varphi pHe)·(\nabla \varphi pHe)dy\hfill \\ & & +{\displaystyle \gamma NHe}\int \Omega (\nabla \varphi NHe)·(\nabla \varphi NHe)dxdy\hfill \\ & & +{\displaystyle \gamma NHe}\int \Omega (\nabla \varphi NHe)·(\nabla \varphi NHe)dxdy\hfill \\ & & +{\displaystyle \gamma N}\int \Omega (\nabla \varphi N)·(\nabla \varphi N)dx,\hfill \end{array}$$

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and,

$$\begin{array}{ccc}\hfill F\left(\varphi \right)& =& {\displaystyle \alpha D}\int \Omega {\displaystyle |\varphi D(x-\xi ,y-\xi )|\left|\varphi D(\xi ,\xi )\right||(x,y)-(\xi ,\xi )|}dxdyd\xi d\xi \hfill \\ & & +{\displaystyle \alpha T}\int \Omega {\displaystyle |\varphi T(x-\xi ,y-\xi )|\left|\varphi T(\xi ,\xi )\right||(x,y)-(\xi ,\xi )|}dxdyd\xi d\xi \hfill \\ & & +{\displaystyle \alpha DT}\int \Omega {\displaystyle |\varphi D(x-\xi ,y-\xi )|\left|\varphi T(\xi ,\xi )\right||(x,y)-(\xi ,\xi )|}dxdyd\xi d\xi \hfill \\ & & +{\displaystyle \alpha He}\int \Omega {\displaystyle |\varphi He(x-\xi ,y-\xi )|\left|\varphi He(\xi ,\xi )\right||(x,y)-(\xi ,\xi )|}dxdyd\xi d\xi \hfill \\ & & +{\displaystyle \alpha N}\int tf\int \Omega {\displaystyle |\varphi N(x-\xi )|\left|\varphi N\left(\xi \right)\right||x-\xi |}dxd\xi \hfill \\ & & +\sum j={\displaystyle \alpha HeN}\int \Omega {\displaystyle |\varphi He(x-\xi ,y-\xi )|\left|\varphi N\left(\xi j\right)\right||(x,y)-(\xi ,\xi )|}dxdyd\xi d\xi \hfill \end{array}$$

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and the kinetics energy is expressed by

$$\begin{array}{ccc}\hfill Ec(\varphi ,\mathbf{r})& =& {\displaystyle 12}\int \Omega \left|\varphi D\right|{\displaystyle \partial \mathbf{r}D\partial t}·{\displaystyle \partial \mathbf{r}D\partial t}dxdy\hfill \\ & & +{\displaystyle 12}\int \Omega \left|\varphi T\right|{\displaystyle \partial \mathbf{r}T\partial t}·{\displaystyle \partial \mathbf{r}T\partial t}dxdy\hfill \\ & & +{\displaystyle 12}\int \Omega \left|\varphi He\right|{\displaystyle \partial \mathbf{r}He\partial t}·{\displaystyle \partial \mathbf{r}He\partial t}dxdy\hfill \\ & & +{\displaystyle 12}\int \Omega \left|\varphi N\right|{\displaystyle \partial \mathbf{r}N\partial t}·{\displaystyle \partial \mathbf{r}N\partial t}dxdy,\hfill \end{array}$$

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where we also assume

$$\mathbf{r}D\approx ei\omega t\mathbf{w}(x,y),$$

$$\mathbf{r}T\approx ei\omega t\mathbf{w}(x,y),$$

so that considering such a vibrational motion, the temperature T is proportional to $\omega$, that is

$$T\propto \omega .$$

Therefore, an increasing in T corresponds to a proportional increasing in $\omega .$

Summarizing, we have supposed

$$Ec(\varphi ,\mathbf{r})\approx {\displaystyle 12}\omega \int \Omega |\varphi D|+|\varphi T|dxC+{\displaystyle 12}\omega 12\int \Omega \left|\varphi N\right|dxC,$$

so that we represent the increasing in T through an increasing in $\omega .$

Moreover, we denote by $m$ the mass of a single neutron and by $m$ the mass of a single proton.

Thus, denoting also by $\lambda ,\lambda$ the proportion of non-reacted and reacted masses respectively, we have the following constraints.

$$\int \Omega \left|\varphi ND(x,y)\right|dx=mN,$$

$$\int \Omega \left|\varphi NT(x,y)\right|dx=mN,$$

$$\int \Omega \left|\varphi NT(x,y)\right|dx=mN,$$

$$\int \Omega \left|\varphi NHe(x,y)\right|dx=mN,$$

$$\int \Omega \left|\varphi NHe(x,y)\right|dx=mN,$$

$$\int \Omega \left|\varphi pD\left(y\right)\right|dy=\lambda cmp,$$

$$\int \Omega \left|\varphi pT\left(y\right)\right|dy=\lambda cmp,$$

$$\int \Omega \left|\varphi PHe\left(y\right)\right|dy=\lambda \left(cmp\right),$$

Similar constraints are valid corresponding to the charge of a single proton.

We have also the following complementing constraints,

$$\int \Omega \left|\varphi D\right|dxdy=\lambda \left(mD\right)T,$$

$$\int \Omega \left|\varphi T\right|dxdy=\lambda \left(mT\right)T,$$

$$\int \Omega \left|\varphi He\right|dxdy=\lambda \left(mHe\right)T,$$

$$\int \Omega \left|\varphi N\right|dxdy=\lambda \left(mN\right)T,$$

$$\lambda +\lambda =.$$

With such results and statements in mind and simplifying the interacting terms, we re-define the functional J now denoting it by $J$, here already including the Lagrange multipliers concerning the constraints, where

$$\begin{array}{ccc}\hfill J(\varphi ,\omega ,E,\lambda )& =& {\displaystyle \gamma pD}\int \Omega (\nabla \varphi pD)·(\nabla \varphi pD)dy\hfill \\ & & +{\displaystyle \gamma ND}\int \Omega (\nabla \varphi ND)·(\nabla \varphi ND)dxdy\hfill \\ & & {\displaystyle \gamma pT}\int \Omega (\nabla \varphi pT)·(\nabla \varphi pT)dy\hfill \\ & & +{\displaystyle \gamma NT}\int \Omega (\nabla \varphi NT)·(\nabla \varphi NT)dxdy\hfill \\ & & +{\displaystyle \gamma NT}\int \Omega (\nabla \varphi NT)·(\nabla \varphi NT)dxdy\hfill \\ & & +{\displaystyle \gamma pHe}\int \Omega (\nabla \varphi pHe)·(\nabla \varphi pHe)dy\hfill \\ & & +{\displaystyle \gamma NHe}\int \Omega (\nabla \varphi NHe)·(\nabla \varphi NHe)dxdy\hfill \\ & & +{\displaystyle \gamma NHe}\int \Omega (\nabla \varphi NHe)·(\nabla \varphi NHe)dxdy\hfill \\ & & +{\displaystyle \gamma N}\int \Omega (\nabla \varphi N)·(\nabla \varphi N)dx\hfill \\ & & +{\displaystyle \alpha D}\int \Omega |\varphi D|dx+{\displaystyle \alpha T}\int \Omega \left|\varphi T\right|dx\hfill \\ & & +{\displaystyle \alpha He}\int \Omega |\varphi He|dx+{\displaystyle \alpha N}\int \Omega \left|\varphi N\right|dx\hfill \\ & & -\omega \int \Omega \left(\right|\varphi D|+|\varphi T\left|\right)dx\hfill \\ & & -\omega 12\int \Omega \left|\varphi N\right|dx+JAux,\hfill \end{array}$$

(87)

where the functional $J$ stands for

$$\begin{array}{ccc}\hfill JAux& =& -\int \Omega \left(END\right)\left(y\right)\left(\int \Omega \left|\varphi ND(x,y)\right|dx-mN\right)dy\hfill \\ & & -\int \Omega \left(ENT\right)\left(y\right)\left(\int \Omega \left|\varphi NT(x,y)\right|dx-mN\right)dy\hfill \\ & & -\int \Omega \left(ENT\right)\left(y\right)\left(\int \Omega \left|\varphi NT(x,y)\right|dx-mN\right)dy\hfill \\ & & -\int \Omega \left(ENHe\right)\left(y\right)\left(\int \Omega \left|\varphi NHe(x,y)\right|dx-mN\right)dy\hfill \\ & & -\int \Omega \left(ENHe\right)\left(y\right)\left(\int \Omega \left|\varphi NHe(x,y)\right|dx-mN\right)dy\hfill \\ & & -\left(ED\right)\left(\int \Omega \left|\varphi pD\left(y\right)\right|dy-\lambda cmp\right)\hfill \\ & & -\left(ET\right)\left(\int \Omega \left|\varphi pT\left(y\right)\right|dy-\lambda cmp\right)\hfill \\ & & -\left(EHe\right)\left(\int \Omega \left|\varphi PHe(x,y)\right|dy-\lambda cmp\right)\hfill \\ & & -E\left(\int \Omega \left|\varphi D\right|dxdy-\lambda \left(mD\right)T\right)\hfill \\ & & -E\left(\int \Omega \left|\varphi T\right|dxdy-\lambda \left(mT\right)T\right)\hfill \\ & & -E\left(\int \Omega \left|\varphi He\right|dxdy-\lambda \left(mHe\right)T\right)\hfill \\ & & -E\left(\int \Omega \left|\varphi N\right|dxdy-\lambda \left(mN\right)T\right)\hfill \\ & & -E(\lambda +\lambda -).\hfill \end{array}$$

(88)

Remark 8.

In order to obtain consistent results it is necessary to set

$$(\alpha N,\alpha He)\gg (\alpha D,\alpha T).$$

In such a case, a higher temperature corresponding to a large $\omega$, though such a nuclear reaction, will result in a small $\lambda$ and a higher kinetics energy for the neutron field, corresponding to a large $\omega$ and $\lambda$ closer to 1.

19. A More Detailed Mathematical Description of the Hydrogen Nuclear Fusion

In this section we develop in more details another model for the hydrogen nuclear fusion.

Remark 9.

Denoting by $i\in \mathbb{C}$ the imaginary unit, in this and in the subsequent sections, for the time-dependent case we generically define the gradient of a scalar function $u(x,t)$ with domain in $\mathbb{R}$, denoted by $\nabla u(x,t)$, as

$$\nabla u(x,t)=(iut(x,t),ux(x,t),ux(x,t),ux(x,t)),$$

so that

$$\nabla u·\nabla u=-ut+\sum j=uxj.$$

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Here such a set $\Omega$ stands for a control volume in which an ionized gas (plasma) flows. Such a gas comprises ionized Deuterium and Tritium atoms intended, through a suitable higher temperature, to chemically react resulting in atoms of Hellion and a field of single energetic Neutrons.

Symbolically such a reaction stands for

$$\mathrm{Deuterium}++\mathrm{Tritium}+\to \mathrm{Helium}+++\mathrm{Neutron}\left(\mathrm{energetic}\right).$$

We recall that the ionized Deuterium atom is comprised by a proton and a neutron and the ionized Tritium atom is comprised by a proton and two neutrons.

Moreover, the ionized Helium atom is comprised by two protons and two neutrons.

As previously mentioned, resulting from such a chemical reaction up surges also an energetic neutron which the higher kinetics energy has a great variety of applications, including its conversion in electric energy.

We highlight the model here presented includes electric and magnetic fields and the corresponding potential ones.

Denoting by t the time on the interval $[,tf],$ at this point we define the following density functions:

For the Deuterium field

$$|\varphi D(x,y,t)|=|\varphi pD(y,t)|+|\varphi ND(x,y,t)|\left|\varphi pD(y,t)\right|{\displaystyle mp},$$

For the Tritium field

$$|\varphi T(x,y,t)|=|\varphi pT(y,t)|+\left(\right|\varphi NT(x,y,t)|+|\varphi NT(x,y,t)|\left)\right|\varphi pT(y,t)|{\displaystyle mp},$$

For the Helium field

$$|\varphi He(x,y,t)|=|\varphi pHe(y,t)|+\left(\right|\varphi NHe(x,y,t)|+|\varphi NHe(x,y,t)|\left)\right|\varphi pHe(y,t)|{\displaystyle mp},$$

For the Neutron field

$$\varphi N=\varphi N(x,t),$$

For the electronic field resulting from the ionization

$$\varphi e=\varphi e(x,y,t).$$

Furthermore, we define also the related densities

$$\rho D(y,t)=\int \Omega \left|\varphi D(x,y,t)\right|dx,$$

$$\rho T(y,t)=\int \Omega \left|\varphi T(x,y,t)\right|dx,$$

$$\rho He(y,t)=\int \Omega \left|\varphi He(x,y,t)\right|dx,$$

$$\rho N(x,t)=\left|\varphi N(x,t)\right|,$$

$$\rho e(y,t)=\int \Omega \left|\varphi e(x,y,t)\right|dx.$$

For the chemical reaction in question we consider that one unit of mass of fractional proportion $\alpha$ of ionized Deuterium and $\alpha$ of ionized Tritium results in one unit of mass of fractional proportion $\alpha$ of ionized Helium and $\alpha$ of neutrons.

Symbolic, this stands for

$$=\alpha D+\alpha T=\alpha He+\alpha N.$$

Concerning the control volume $\Omega$ in question and related surface control $\partial \Omega ,$ we assume such a volume has an initial (fot $t=$) amount of ionized Deuterium of $\left(mD\right)$ and an initial amount of ionized Tritium of $\left(mT\right).$ The initial amount of ionized Helium and single neutrons are supposed to be zero.

On the other hand, about the surface control $\partial \Omega$, we assume there is a part $\Omega \subset \partial \Omega$ for which is allowed the entrance and exit of Deuterium and Tritium ionized atoms.

We assume also there is another part $\partial \Omega \subset \partial \Omega$ such that $\partial \Omega \cap \partial \Omega =\varnothing$ for which is allowed only the exit of ionized Helium atoms and neutrons, but not their entrance.

In $\partial \Omega$ is allowed the exit only (not the entrance) of ionized Deuterium and Tritium atoms.

Indeed, we assume the following relations for the masses:

$$\left(mHe,N\right)T\left(t\right)=mHe,N\left(t\right)+\int t\int \partial \Omega (\rho He(x,\tau )+\rho N(x,\tau ))\mathbf{u}·\mathbf{n}dSd\tau ,$$

$$mHe,N\left(t\right)=mHe\left(t\right)+mN\left(t\right),$$

$$mHe\left(t\right)=\int \Omega \rho He(x,t)dx,$$

$$mN\left(t\right)=\int \Omega \rho N(x,t)dx,$$

$$\left(mHe\right)T\left(t\right)=\int \Omega \rho He(x,t)dx+\int t\int \partial \Omega \rho He(x,\tau )\mathbf{u}·\mathbf{n}d\Gamma d\tau ,$$

$$\left(mN\right)T\left(t\right)=\int \Omega \rho N(x,t)dx+\int t\int \partial \Omega \rho N(x,\tau )\mathbf{u}·\mathbf{n}d\Gamma d\tau ,$$

$${\displaystyle \left(mN\right)T\left(t\right)\left(mHe\right)T\left(t\right)}={\displaystyle \alpha N\alpha He},$$

so that

$$\alpha NmHe)T\left(t\right)=\alpha He\left(mN\right)T\left(t\right),$$

$$\left(mD\right)\left(t\right)=\left(mD\right)-\int t\int \partial \Omega \cup \partial \Omega \left(\rho D(x,\tau )\right)\mathbf{u}·\mathbf{n}dSd\tau -\alpha D\left(mHe,N\right)T\left(t\right),$$

$$\left(mT\right)\left(t\right)=\left(mT\right)-\int t\int \partial \Omega \cup \partial \Omega \left(\rho T(x,\tau )\right)\mathbf{u}·\mathbf{n}dSd\tau -\alpha T\left(mHe,N\right)T\left(t\right),$$

$$\left(me\right)T\left(t\right)=me\left(t\right)+\int t\int \partial \Omega \left(\rho e(x,\tau )\right)\mathbf{u}·\mathbf{n}dSd\tau ,$$

$$me\left(t\right)=\int \Omega \rho e(x,t)dx.$$

$$me\left(t\right)=\int \Omega |\varphi pD(x,t)|dx{\displaystyle memp}+\int \Omega |\varphi pT(x,t)|dx{\displaystyle memp}+\int \Omega \left|\varphi pHe(x,t)\right|dx{\displaystyle memp}.$$

Here $\mathbf{n}$ denotes the outward normal vectorial fields to the concerning surfaces.

Having clarified such masses relations, we define the functional

$$J(\varphi ,\rho ,\mathbf{r},\mathbf{u},\mathbf{E},\mathbf{A},\mathbf{B})$$

where

$$J=G(\nabla u)+F\left(\varphi \right)+Ec(\varphi ,\mathbf{r})+F+F+F,$$

and where we assume $\gamma pD>,\gamma pT>,\gamma ND>,\gamma NT>,\gamma NT>,$$\gamma pHe>,\gamma NHe>,$ $\gamma NHe>,\gamma N>,\gamma e>$ and $\alpha D>,\alpha T>,\alpha He>,\alpha N>$, $\alpha DT>,\alpha HeN>,\alpha e,e>,\alpha He,e<$ so that

$$\begin{array}{ccc}\hfill G(\nabla \varphi )& =& {\displaystyle \gamma pD}\int tf\int \Omega (\nabla \varphi pD)·(\nabla \varphi pD)dydt\hfill \\ & & +{\displaystyle \gamma ND}\int tf\int \Omega (\nabla \varphi ND)·(\nabla \varphi ND)dxdydt\hfill \\ & & {\displaystyle \gamma pT}\int tf\int \Omega (\nabla \varphi pT)·(\nabla \varphi pT)dydt\hfill \\ & & +{\displaystyle \gamma NT}\int tf\int \Omega (\nabla \varphi NT)·(\nabla \varphi NT)dxdydt\hfill \\ & & +{\displaystyle \gamma NT}\int tf\int \Omega (\nabla \varphi NT)·(\nabla \varphi NT)dxdydt\hfill \\ & & +{\displaystyle \gamma pHe}\int tf\int \Omega (\nabla \varphi pHe)·(\nabla \varphi pHe)dydt\hfill \\ & & +{\displaystyle \gamma NHe}\int tf\int \Omega (\nabla \varphi NHe)·(\nabla \varphi NHe)dxdydt\hfill \\ & & +{\displaystyle \gamma NHe}\int tf\int \Omega (\nabla \varphi NHe)·(\nabla \varphi NHe)dxdydt\hfill \\ & & +{\displaystyle \gamma N}\int tf\int \Omega (\nabla \varphi N)·(\nabla \varphi N)dxdt\hfill \\ & & +{\displaystyle \gamma e}\int tf\int \Omega (\nabla \varphi e)·(\nabla \varphi e)dxdydt,\hfill \end{array}$$

(89)

and

$$\begin{array}{ccc}\hfill F\left(\varphi \right)& =& {\displaystyle \alpha D}\int tf\int \Omega {\displaystyle |\varphi D(x-\xi ,y-\xi ,t)|\left|\varphi D(\xi ,\xi ,t)\right||(x,y)-(\xi ,\xi )|}dxdyd\xi d\xi dt\hfill \\ & & +{\displaystyle \alpha T}\int tf\int \Omega {\displaystyle |\varphi T(x-\xi ,y-\xi ,t)|\left|\varphi T(\xi ,\xi ,t)\right||(x,y)-(\xi ,\xi )|}dxdyd\xi d\xi dt\hfill \\ & & +{\displaystyle \alpha DT}\int tf\int \Omega {\displaystyle |\varphi D(x-\xi ,y-\xi ,t)|\left|\varphi T(\xi ,\xi ,t)\right||(x,y)-(\xi ,\xi )|}dxdyd\xi d\xi dt\hfill \\ & & +{\displaystyle \alpha He}\int tf\int \Omega {\displaystyle |\varphi He(x-\xi ,y-\xi ,t)|\left|\varphi He(\xi ,\xi ,t)\right||(x,y)-(\xi ,\xi )|}dxdyd\xi d\xi dt\hfill \\ & & +{\displaystyle \alpha N}\int tf\int \Omega {\displaystyle |\varphi N(x-\xi ,t)|\left|\varphi N\left(\xi \right)\right||x-\xi ,t|}dxd\xi dt\hfill \\ & & +\sum j={\displaystyle \alpha HeN}\int tf\int \Omega {\displaystyle |\varphi He(x-\xi ,y-\xi ,t)|\left|\varphi N(\xi j,t)\right||(x,y)-(\xi ,\xi )|}dxdyd\xi d\xi dt\hfill \\ & & +{\displaystyle \alpha He,e}\int tf\int \Omega {\displaystyle |\varphi He(x-\xi ,y-\xi ,t)|\left|\varphi e(\xi ,\xi ,t)\right||(x,y)-(\xi ,\xi )|}dxdyd\xi d\xi dt\hfill \\ & & +{\displaystyle \alpha e,e}\int tf\int \Omega {\displaystyle |\varphi e(x-\xi ,y-\xi ,t)|\left|\varphi e(\xi ,\xi ,t)\right||(x,y)-(\xi ,\xi )|}dxdyd\xi d\xi dt\hfill \end{array}$$

(90)

and the internal kinetics energy is expressed by

$$\begin{array}{ccc}\hfill Ec(\varphi ,\mathbf{r})& =& {\displaystyle 12}\int tf\int \Omega \left|\varphi D\right|{\displaystyle \partial \mathbf{r}D\partial t}·{\displaystyle \partial \mathbf{r}D\partial t}dxdydt\hfill \\ & & +{\displaystyle 12}\int tf\int \Omega \left|\varphi T\right|{\displaystyle \partial \mathbf{r}T\partial t}·{\displaystyle \partial \mathbf{r}T\partial t}dxdydt\hfill \\ & & +{\displaystyle 12}\int tf\int \Omega \left|\varphi He\right|{\displaystyle \partial \mathbf{r}He\partial t}·{\displaystyle \partial \mathbf{r}He\partial t}dxdydt\hfill \\ & & +{\displaystyle 12}\int tf\int \Omega \left|\varphi N\right|{\displaystyle \partial \mathbf{r}N\partial t}·{\displaystyle \partial \mathbf{r}N\partial t}dxdydt\hfill \\ & & +{\displaystyle 12}\int tf\int \Omega \left|\varphi e\right|{\displaystyle \partial \mathbf{r}e\partial t}·{\displaystyle \partial \mathbf{r}e\partial t}dxdydt,\hfill \end{array}$$

(91)

Here it is worth highlighting we have approximated the initially discrete set of indices s of particles as a continuous positive real variable s.

Moreover,

$$F={\displaystyle \pi }\int tf\parallel \mathrm{curl}\mathbf{A}-\mathbf{B}\parallel dt,$$

$$\begin{array}{ccc}\hfill F& =& \int tf\int \Omega \mathbf{E}ind·Kp\left|\varphi pD\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}D\partial t}\right)dxdydt\hfill \\ & & +\int tf\int \Omega \mathbf{E}ind·Kp\left|\varphi pT\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}T\partial t}\right)dxdydt\hfill \\ & & +\int tf\int \Omega \mathbf{E}ind·Kp\left|\varphi pHe\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}He\partial t}\right)dxdydt\hfill \\ & & +\int tf\int \Omega \mathbf{E}ind·Ke\left|\varphi e\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}e\partial t}\right)dxdydt,\hfill \end{array}$$

(92)

where $K$ and $K$ are appropriate real constants related to the respective charges.

Here $\mathbf{u}=(u,u,u)$ is the fluid velocity field and

$$\mathbf{r}D,\mathbf{r}T,\mathbf{r}He,\mathbf{r}N,\mathbf{r}e$$

are fields of displacements for the corresponding atom fields.

Also $\mathbf{A}$ denotes the magnetic potential, $\mathbf{B}$ an external magnetic field and $\mathbf{B}$ is the total magnetic field.

Moreover, $\mathbf{E}$ is an induced electric field.

Finally,

$$\begin{array}{ccc}\hfill F& =& {\displaystyle CD}\int tf\int \Omega \nabla (x,y)\mathbf{r}D·\nabla (x,y)\mathbf{r}Ddxdydt+{\displaystyle CT}\int tf\int \Omega \nabla (x,y)\mathbf{r}T·\nabla (x,y)\mathbf{r}Tdxdydt\hfill \\ & & {\displaystyle CHe}\int tf\int \Omega \nabla (x,y)\mathbf{r}He·\nabla (x,y)\mathbf{r}Hedxdydt+{\displaystyle CN}\int tf\int \Omega \nabla (x,y)\mathbf{r}N·\nabla (x,y)\mathbf{r}Ndxdydt\hfill \\ & & {\displaystyle Ce}\int tf\int \Omega \nabla (x,y)\mathbf{r}e·\nabla (x,y)\mathbf{r}edxdydt,\hfill \end{array}$$

(93)

for appropriate real positive constants $CDCT,CHe,CN,Ce.$

Such a functional J is subject to the following constraints:

The momentum conservation equation for the fluid motion

$$\rho \left({\displaystyle \partial uk\partial t}+uj{\displaystyle \partial uk\partial xj}\right)=\rho fk-{\displaystyle \partial P\partial xk}+\tau kj,j+\left(FE\right)k+\left(FM\right)k,$$

$\forall k\in \{,,\}.$

Here $\rho =\rho D+\rho T+\rho He+\rho N+\rho e$ is the total density and P is the fluid pressure field.

Furthermore,

$$\tau ij=\mu \left({\displaystyle \partial ui\partial xj}+{\displaystyle \partial uj\partial xi}-{\displaystyle 23}\delta ij\sum k={\displaystyle \partial uk\partial xk}\right),$$

$\forall i,j\in \{,,\},$

$$\mathbf{F}E=\left\{\left(FE\right)k\right\}=\left(Kp\left(\right|\varphi pD|+|\varphi pT|+|\varphi pHe|)+Ke\int \Omega \left|\varphi e\right|dx\right)\mathbf{E},$$

and

$$\begin{array}{ccc}\hfill \mathbf{F}M& =& \left\{\left(FM\right)k\right\}\hfill \\ & =& \left(Kp\left(|\varphi pD|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}D\partial t}\right)\right.\right.\hfill \\ & & |\varphi pT|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}T\partial t}\right)\hfill \\ & & \left.+|\varphi pHe|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}He\partial t}\right)\right)\hfill \\ & & \left.+Ke\left|\varphi e\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}e\partial t}\right)\right)\times \mathbf{B}.\hfill \end{array}$$

(94)

Mass conservation equation:

$${\displaystyle \partial \rho \partial t}+\mathrm{div}\left(\rho \mathbf{u}\right)=.$$

Energy equation

$$\rho {\displaystyle DeDt}+\nabla x(E^)·\mathbf{u}+E^+P\left(\mathrm{div}\mathbf{u}\right)={\displaystyle \partial Q\partial t}-\mathrm{div}\mathbf{q}+\tau jk{\displaystyle \partial uj\partial xk},$$

where we assume the Fourier law

$$\mathbf{q}=-K\nabla T,$$

where $T=T(x,t)$ is the scalar field of temperature and Q is a standard heat function.

Also,

$$\begin{array}{ccc}\hfill e& \approx & {\displaystyle \rho }\mathbf{u}·\mathbf{u}+{\displaystyle \rho D}{\displaystyle \partial \mathbf{r}D\partial t}·{\displaystyle \partial \mathbf{r}D\partial t}\hfill \\ & & +{\displaystyle \rho T}{\displaystyle \partial \mathbf{r}T\partial t}·{\displaystyle \partial \mathbf{r}T\partial t}\hfill \\ & & +{\displaystyle \rho He}{\displaystyle \partial \mathbf{r}He\partial t}·{\displaystyle \partial \mathbf{r}He\partial t}\hfill \\ & & +{\displaystyle \rho N}{\displaystyle \partial \mathbf{r}N\partial t}·{\displaystyle \partial \mathbf{r}N\partial t}\hfill \\ & & +{\displaystyle \rho e}{\displaystyle \partial \mathbf{r}e\partial t}·{\displaystyle \partial \mathbf{r}e\partial t}\hfill \end{array}$$

(95)

where the densities $E$ and $E$ are defined through the expressions of $F\left(\varphi \right)$ and $F$ so that

$$F\left(\varphi \right)=\int tf\int \Omega E^dxdt$$

and

$$F=\int tf\int \Omega E^dxdt.$$

Here we recall that since $\mathbf{r}$ is highly oscillating in t we approximately have

$$\mathbf{u}·\mathbf{r}D\approx$$

in a weak or measure sense. The same remark is valid for the other internal velocity fields.

Moreover,

$${\displaystyle DeDt}={\displaystyle \partial e\partial t}+uj{\displaystyle \partial e\partial xj}.$$

Finally, for a calorically perfect gas we may assume

$$e=CvT$$

where

$$Cv={\displaystyle R\gamma -},$$

for appropriate constants $R>$, $\gamma >.$

$$P=F(\rho ,T),$$

for an appropriate scalar function $F$.

Mass relations

$$mD\left(t\right)=\int \Omega \rho D(x,t)dx,$$

$$mT\left(t\right)=\int \Omega \rho T(x,t)dx,$$

$$mHe\left(t\right)=\int \Omega \rho He(x,t)dx,$$

$$mN\left(t\right)=\int \Omega \rho N(x,t)dx,$$

$$me\left(t\right)=\int \Omega \rho e(x,t)dx,$$

$$\left(mHe\right)T\left(t\right)=\int \Omega \rho He(x,t)dx+\int t\int \partial \Omega \rho He(x,\tau )\mathbf{u}·\mathbf{n}d\Gamma d\tau ,$$

$$\left(mN\right)T\left(t\right)=\int \Omega \rho N(x,t)dx+\int t\int \partial \Omega \rho N(x,\tau )\mathbf{u}·\mathbf{n}d\Gamma d\tau ,$$

$${\displaystyle \left(mN\right)T\left(t\right)\left(mHe\right)T\left(t\right)}={\displaystyle \alpha N\alpha He},$$

so that

$$\alpha NmHe)T\left(t\right)=\alpha He\left(mN\right)T\left(t\right),$$

where,

$$\left(mHe,N\right)T\left(t\right)=mHe,N\left(t\right)+\int t\int \partial \Omega \left(\rho He(x,\tau )\right)\mathbf{u}·\mathbf{n}dSd\tau ,$$

$$mHe,N\left(t\right)=mHe\left(t\right)+mN\left(t\right),$$

$$\left(mD\right)\left(t\right)=\left(mD\right)-\int t\int \partial \Omega \cup \partial \Omega \left(\rho D(x,\tau )\right)\mathbf{u}·\mathbf{n}dSd\tau -\alpha D\left(mHe,N\right)T\left(t\right),$$

$$\left(mT\right)\left(t\right)=\left(mT\right)-\int t\int \partial \Omega \cup \partial \Omega \left(\rho T(x,\tau )\right)\mathbf{u}·\mathbf{n}dSd\tau -\alpha T\left(mHe,N\right)T\left(t\right),$$

$$\left(me\right)T\left(t\right)=me\left(t\right)+\int t\int \partial \Omega \left(\rho T(x,\tau )\right)\mathbf{u}·\mathbf{n}dSd\tau .$$

$$me\left(t\right)=\int \Omega |\varphi pD(x,t)|dx{\displaystyle memp}+\int \Omega |\varphi pT(x,t)|dx{\displaystyle memp}+\int \Omega \left|\varphi pHe(x,t)\right|dx{\displaystyle memp}.$$

Other mass constraints

$$\int \Omega \left|\varphi ND(x,y,t)\right|dx=mN,$$

$$\int \Omega \left|\varphi NT(x,y,t)\right|dx=mN,$$

$$\int \Omega \left|\varphi NT(x,y,t)\right|dx=mN,$$

$$\int \Omega \left|\varphi NHe(x,y,t)\right|dx=mN,$$

$$\int \Omega \left|\varphi NHe(x,y,t)\right|dx=mN.$$

For the induced electric field, we must have

$$\begin{array}{ccc}& & \mathrm{curl}\mathbf{E}ind+{\displaystyle c}\mathrm{curl}\left(K^p\left|\varphi pD\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}D\partial t}\right)\right.\hfill \\ & & +K^p\left|\varphi pT\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}T\partial t}\right)\hfill \\ & & +K^p\left|\varphi pHe\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}He\partial t}\right)\hfill \\ & & +\left.K^e\int \Omega \left|\varphi e(x,y,t)\right|\left(\mathbf{u}(y,t)+{\displaystyle \partial \mathbf{r}e(x,y,t)\partial t}dx\right)\right)\hfill \\ & & \times \left(\mathrm{curl}\mathbf{A}-\mathbf{B}\right)-{\displaystyle c}{\displaystyle \partial \partial t}\left(\mathrm{curl}\mathbf{A}-\mathbf{B}\right)=\mathbf{0},\hfill \end{array}$$

(96)

where $K$ and $K$ are appropriate real constants related to the respective charges.

A Maxwell equation:

$$\mathrm{div}\mathbf{B}=,$$

where

$$\mathbf{B}=\mathbf{B}-\mathrm{curl}\mathbf{A}.$$

Another Maxwell equation:

$$\mathrm{div}\mathbf{E}=\pi \left(Kp\left(\right|\varphi pD|+|\varphi pT|+|\varphi pHe|)+Ke\int \Omega \left|\varphi e(x,y,t)\right|dx\right),$$

where the total electric field $\mathbf{E}$ stands for

$$\mathbf{E}=\mathbf{E}ind+\mathbf{E}\rho ,$$

and where generically denoting

$$F\left(\varphi \right)=\int tf\int \Omega f(\varphi ,x,\xi ,t)dxd\xi dt,$$

we have also

$$\mathbf{E}\rho =\left\{\int \Omega {\displaystyle \partial f(\varphi ,x,\xi ,t)\partial xk}d\xi \right\}.$$

At this point we generically denote

$$\langle h,h\rangle L=\int tf\int \Omega hhdxdydt.$$

Thus, already including the Lagrange multipliers concerning the restrictions indicated, the extended functional $J$ stands for

$$\begin{array}{ccc}& & J=J(\varphi ,\mathbf{u},\mathbf{r},P,\mathbf{A},\mathbf{B},\mathbf{E},\Lambda ,E)\hfill \\ & =& G(\nabla \varphi )+F\left(\varphi \right)+Ec(\varphi ,\mathbf{r})+F+F+F\hfill \\ & & +〈\Lambda k,\rho \left({\displaystyle \partial uk\partial t}+uj{\displaystyle \partial uk\partial xj}\right)-\rho fk+{\displaystyle \partial P\partial xk}-\tau kj,j-\left(FE\right)k-\left(FM\right)k〉L\hfill \\ & & +〈\Lambda ,{\displaystyle \partial \rho \partial t}+\mathrm{div}\left(\rho \mathbf{u}\right)〉L+JAux+JAux+JAux+JAux,\hfill \end{array}$$

(97)

where,

$$\begin{array}{ccc}& & JAux\hfill \\ & =& 〈\Lambda ,\rho {\displaystyle DeDt}+\nabla x(E^)·\mathbf{u}+E^++P\left(\mathrm{div}\mathbf{u}\right)-{\displaystyle \partial Q\partial t}+\mathrm{div}\mathbf{q}-\tau jk{\displaystyle \partial uj\partial xk}〉L\hfill \\ & & +〈\Lambda ,P-F(\rho ,T)〉L,\hfill \end{array}$$

(98)

$$\begin{array}{ccc}\hfill JAux& =& 〈\Lambda ,mD\left(t\right)-\int \Omega \rho D(x,t)dx〉L\hfill \\ & & +〈\Lambda ,mT\left(t\right)-\int \Omega \rho T(x,t)dx〉L\hfill \\ & & 〈\Lambda ,mHe\left(t\right)-\int \Omega \rho He(x,t)dx〉L\hfill \\ & & 〈\Lambda 10,mN\left(t\right)-\int \Omega \rho N(x,t)dx〉L\hfill \\ & & 〈\Lambda 11,me\left(t\right)-\int \Omega \rho e(x,t)dx〉L\hfill \\ & & \int tfE\left(t\right)\left(\alpha NmHe\right)T\left(t\right)-\alpha He\left(mN\right)T\left(t\right))dt,\hfill \end{array}$$

(99)

$$\begin{array}{ccc}\hfill JAux& =& -\int tf\int \Omega \left(END\right)(y,t)\left(\int \Omega \left|\varphi ND(x,y,t)\right|dx-mN\right)dydt\hfill \\ & & -\int tf\int \Omega \left(ENT\right)(y,t)\left(\int \Omega \left|\varphi NT(x,y,t)\right|dx-mN\right)dydt\hfill \\ & & -\int tf\int \Omega \left(ENT\right)(y,t)\left(\int \Omega \left|\varphi NT(x,y,t)\right|dx-mN\right)dydt\hfill \\ & & -\int tf\int \Omega \left(ENHe\right)(y,t)\left(\int \Omega \left|\varphi NHe(x,y,t)\right|dx-mN\right)dydt\hfill \\ & & -\int tf\int \Omega \left(ENHe\right)(y,t)\left(\int \Omega \left|\varphi NHe(x,y,t)\right|dx-mN\right)dydt,\hfill \end{array}$$

(100)

$$\begin{array}{ccc}\hfill JAux& =& 〈\Lambda 12,\mathrm{curl}\mathbf{E}ind\hfill \\ & & +{\displaystyle c}\mathrm{curl}\left(K^p\left|\varphi pD\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}D\partial t}\right)\right.\hfill \\ & & +K^p\left|\varphi pT\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}T\partial t}\right)\hfill \\ & & +K^p\left|\varphi pHe\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}He\partial t}\right)\hfill \\ & & +\left.K^e\int \Omega \left|\varphi e(x,y,t)\right|\left(\mathbf{u}(y,t)+{\displaystyle \partial \mathbf{r}e(x,y,t)\partial t}dx\right)\right)\hfill \\ & & \times \left(\mathrm{curl}\mathbf{A}-\mathbf{B}\right)-{\displaystyle c}{\displaystyle \partial \partial t}\left(\mathrm{curl}\mathbf{A}-\mathbf{B}\right)〉L\hfill \\ & & +〈\Lambda 13,\mathrm{div}\mathbf{B}〉L\hfill \\ & & +〈\Lambda 14,\mathrm{div}\mathbf{E}-\pi \left(Kp\left(\right|\varphi pD|+|\varphi pT|+|\varphi pHe|)+Ke\int \Omega \left|\varphi e(x,y,t)\right|dx\right)〉L.\hfill \end{array}$$

(101)

Here we recall the following definitions and relations:

For the Deuterium field

$$|\varphi D(x,y,t)|=|\varphi pD(y,t)|+|\varphi ND(x,y,t)|\left|\varphi pD(y,t)\right|{\displaystyle mp},$$

For the Tritium field

$$|\varphi D(x,y,t)|=|\varphi pD(y,t)|+\left(\right|\varphi ND(x,y,t)|+|\varphi ND(x,y,t)|\left)\right|\varphi pD(y,t)|{\displaystyle mp},$$

For the Helium field

$$|\varphi He(x,y,t)|=|\varphi pHe(y,t)|+\left(\right|\varphi NHe(x,y,t)|+|\varphi NHe(x,y,t)|\left)\right|\varphi pHe(y,t)|{\displaystyle mp},$$

For the Neutron field

$$\varphi N=\varphi N(x,t),$$

For the electronic field resulting from the ionization

$$\varphi e=\varphi e(x,y,t).$$

$$\rho D(y,t)=\int \Omega \left|\varphi D(x,y,t)\right|dx,$$

$$\rho T(y,t)=\int \Omega \left|\varphi T(x,y,t)\right|dx,$$

$$\rho He(y,t)=\int \Omega \left|\varphi He(x,y,t)\right|dx,$$

$$\rho N(x,t)=\left|\varphi N(x,t)\right|,$$

$$\rho e(y,t)=\int \Omega \left|\varphi e(x,y,t)\right|dx.$$

Also,

$$\rho =\rho D+\rho T+\rho He+\rho N+\rho e,$$

$$\left(mHe,N\right)T\left(t\right)=mHe,N\left(t\right)+\int t\int \partial \Omega (\rho He(x,\tau )+\rho N(x,\tau ))\mathbf{u}·\mathbf{n}dSd\tau ,$$

$$mHe,N\left(t\right)=mHe\left(t\right)+mN\left(t\right),$$

$$mHe\left(t\right)=\int \Omega \rho He(x,t)dx,$$

$$mN\left(t\right)=\int \Omega \rho N(x,t)dx,$$

$$\left(mD\right)\left(t\right)=\left(mD\right)-\int t\int \partial \Omega \cup \partial \Omega \left(\rho D(x,\tau )\right)\mathbf{u}·\mathbf{n}dSd\tau -\alpha D\left(mHe,N\right)T\left(t\right),$$

$$\left(mT\right)\left(t\right)=\left(mT\right)-\int t\int \partial \Omega \cup \partial \Omega \left(\rho T(x,\tau )\right)\mathbf{u}·\mathbf{n}dSd\tau -\alpha T\left(mHe,N\right)T\left(t\right),$$

$$\left(mHe\right)T\left(t\right)=\int \Omega \rho He(x,t)dx+\int t\int \partial \Omega \rho He(x,\tau )\mathbf{u}·\mathbf{n}d\Gamma d\tau ,$$

$$\left(mN\right)T\left(t\right)=\int \Omega \rho N(x,t)dx+\int t\int \partial \Omega \rho N(x,\tau )\mathbf{u}·\mathbf{n}d\Gamma d\tau ,$$

$${\displaystyle \left(mN\right)T\left(t\right)\left(mHe\right)T\left(t\right)}={\displaystyle \alpha N\alpha He},$$

so that

$$\alpha NmHe)T\left(t\right)=\alpha He\left(mN\right)T\left(t\right),$$

$$\left(me\right)T\left(t\right)=me\left(t\right)-\int t\int \partial \Omega \left(\rho e(x,\tau )\right)\mathbf{u}·\mathbf{n}dSd\tau ,$$

$$me\left(t\right)=\int \Omega \rho e(x,t)dx.$$

$$me\left(t\right)=\int \Omega |\varphi pD(x,t)|dx{\displaystyle memp}+\int \Omega |\varphi pT(x,t)|dx{\displaystyle memp}+\int \Omega \left|\varphi pHe(x,t)\right|dx{\displaystyle memp}.$$

Finally,

$$\mathbf{E}=\mathbf{E}ind+\mathbf{E}\rho ,$$

and where generically denoting

$$F\left(\varphi \right)=\int \Omega f(\varphi ,x,\xi )dxd\xi ,$$

we have also

$$\mathbf{E}\rho =\left\{\int \Omega {\displaystyle \partial f(\varphi ,x,\xi )\partial xk}d\xi \right\}.$$

and,

$$\mathbf{B}=\mathbf{B}-\mathrm{curl}\mathbf{A}.$$

20. A Final Mathematical Description of the Hydrogen Nuclear Fusion

In this section we develop in even more details another model for the hydrogen nuclear fusion.

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Here such a set $\Omega$ stands for a control volume in which an ionized gas (plasma) flows. Such a gas comprises ionized Deuterium and Tritium atoms intended, through a suitable higher temperature, to chemically react resulting in atoms of Helium and a field of single energetic Neutrons.

Symbolically such a reaction stands for

$$\mathrm{Deuterium}++\mathrm{Tritium}+\to \mathrm{Helium}+++\mathrm{Neutron}\left(\mathrm{energetic}\right).$$

We recall that the ionized Deuterium atom is comprised by a proton and a neutron and the ionized Tritium atom is comprised by a proton and two neutrons.

Moreover, the ionized Helium atom is comprised by two protons and two neutrons.

As previously mentioned, resulting from such a chemical reaction up surges also an energetic neutron which the higher kinetics energy has a great variety of applications, including its conversion in electric energy.

We highlight the model here presented includes electric and magnetic fields and the corresponding potential ones.

Denoting by t the time on the interval $[,tf],$ at this point we define the following density functions:

For a single Deuterium atom indexed by s:

$$|\varphi D(x,y,t,s)|=|\varphi pD(y,t,s)|+|\varphi ND(x,y,t,s)|\left|\varphi pD(y,t,s)\right|{\displaystyle mp},$$

For a single Tritium atom indexed by s:

$$|\varphi T(x,y,t,s)|=|\varphi pT(y,t,s)|+\left(\right|\varphi NT(x,y,t,s)|+|\varphi NT(x,y,t,s)|\left)\right|\varphi pT(y,t,s)|{\displaystyle mp},$$

For a single Helium atom indexed by s:

$$|\varphi He(x,y,t,s)|=|\varphi pHe(y,t,s)|+\left(\right|\varphi NHe(x,y,t,s)|+|\varphi NHe(x,y,t,s)|\left)\right|\varphi pHe(y,t,s)|{\displaystyle mp},$$

For the Neutron field:

$$\varphi N=\varphi N(x,t,s),$$

For the electronic field resulting from the ionization

$$\varphi e=\varphi e(x,y,t,s).$$

Furthermore, we define also the related densities

$$\rho D(y,t)=\int ND\left(t\right)\int \Omega \left|\varphi D(x,y,t,s)\right|dxds,$$

$$\rho T(y,t)=\int NT\left(t\right)\int \Omega \left|\varphi T(x,y,t,s)\right|dxds,$$

$$\rho He(y,t)=\int NHe\left(t\right)\int \Omega \left|\varphi He(x,y,t,s)\right|dxds,$$

$$\rho N(x,t)=\int NN\left(t\right)\left|\varphi N(x,t,s)\right|ds,$$

$$\rho e(y,t)=\int Ne\left(t\right)\int \Omega \left|\varphi e(x,y,t,s)\right|dxds.$$

For the chemical reaction in question we consider that one unit of mass of fractional proportion $\alpha$ of ionized Deuterium and $\alpha$ of ionized Tritium results in one unit of mass of fractional proportion $\alpha$ of ionized Helium and $\alpha$ of neutrons.

Symbolically, this stands for

$$=\alpha D+\alpha T=\alpha He+\alpha N.$$

Concerning the control volume $\Omega$ in question and related surface control $\partial \Omega ,$ we assume such a volume has an initial (fot $t=$) amount of ionized Deuterium of $\left(mD\right)$ and an initial amount of ionized Tritium of $\left(mT\right).$ The initial amount of ionized Helium and single neutrons are supposed to be zero.

On the other hand, about the surface control $\partial \Omega$, we assume there is a part $\Omega \subset \partial \Omega$ for which is allowed the entrance and exit of Deuterium and Tritium ionized atoms.

We assume also there is another part $\partial \Omega \subset \partial \Omega$ such that $\partial \Omega \cap \partial \Omega =\varnothing$ for which is allowed only the exit of ionized Helium atoms and neutrons, but not their entrance.

In $\partial \Omega$ is allowed the exit only (not the entrance) of ionized Deuterium and Tritium atoms.

Indeed, we assume the following relations for the masses:

$$\left(mHe,N\right)T\left(t\right)=mHe,N\left(t\right)+\int t\int \partial \Omega (\rho He(x,\tau )+\rho N(x,\tau ))\mathbf{u}·\mathbf{n}dSd\tau ,$$

$$mHe,N\left(t\right)=mHe\left(t\right)+mN\left(t\right),$$

$$mHe\left(t\right)=\int \Omega \rho He(x,t)dx,$$

$$mN\left(t\right)=\int \Omega \rho N(x,t)dx,$$

$$\left(mD\right)\left(t\right)=\left(mD\right)-\int t\int \partial \Omega \cup \partial \Omega \left(\rho D(x,\tau )\right)\mathbf{u}·\mathbf{n}dSd\tau -\alpha D\left(mHe,N\right)T\left(t\right),$$

$$\left(mT\right)\left(t\right)=\left(mT\right)-\int t\int \partial \Omega \cup \partial \Omega \left(\rho T(x,\tau )\right)\mathbf{u}·\mathbf{n}dSd\tau -\alpha T\left(mHe,N\right)T\left(t\right),$$

$$\left(mHe\right)T\left(t\right)=\int \Omega \rho He(x,t)dx+\int t\int \partial \Omega \rho He(x,\tau )\mathbf{u}·\mathbf{n}d\Gamma d\tau ,$$

$$\left(mN\right)T\left(t\right)=\int \Omega \rho N(x,t)dx+\int t\int \partial \Omega \rho N(x,\tau )\mathbf{u}·\mathbf{n}d\Gamma d\tau ,$$

$${\displaystyle \left(mN\right)T\left(t\right)\left(mHe\right)T\left(t\right)}={\displaystyle \alpha N\alpha He},$$

so that

$$\alpha NmHe)T\left(t\right)=\alpha He\left(mN\right)T\left(t\right),$$

$$\left(me\right)T\left(t\right)=me\left(t\right)+\int t\int \partial \Omega \left(\rho e(x,\tau )\right)\mathbf{u}·\mathbf{n}dSd\tau ,$$

$$me\left(t\right)=\int \Omega \rho e(x,t)dx.$$

$$\begin{array}{ccc}\hfill me\left(t\right)& =& \int ND\left(t\right)\int \Omega |\varphi pD(y,t,s)|dyds{\displaystyle memp}+\int NT\left(t\right)\int \Omega \left|\varphi pT(y,t,s)\right|dyds{\displaystyle memp}\hfill \\ & & +\int Np\left(t\right)\int \Omega \left|\varphi pHe(y,t,s)\right|dyds{\displaystyle memp}.\hfill \end{array}$$

(102)

Here $\mathbf{n}$ denotes the outward normal vectorial fields to the concerning surfaces.

Having clarified such masses relations, denoting by $ND\left(t\right)NT\left(t\right),NHe\left(t\right),NN\left(t\right),Ne\left(t\right)$ the respective indexed number of particles at time t, we define the functional

$$J(\varphi ,\rho ,\mathbf{r},\mathbf{u},\mathbf{E},\mathbf{A},\mathbf{B},\{ND,NT,NHe,NN,Ne\})$$

where

$$J=G(\nabla u)+F\left(\varphi \right)+Ec(\varphi ,\mathbf{r})+F+F+F+F,$$

and where we assume $\gamma pD>,\gamma pT>,\gamma ND>,\gamma NT>,\gamma NT>,$$\gamma pHe>,\gamma NHe>,$ $\gamma NHe>,\gamma N>,\gamma e>$ and $\alpha D>,\alpha T>,\alpha He>,\alpha N>$, $\alpha DT>,\alpha HeN>,\alpha e,e>,\alpha He,e<$ so that

$$\begin{array}{ccc}\hfill G(\nabla \varphi )& =& {\displaystyle \gamma pD}\int tf\int ND\left(t\right)\int \Omega (\nabla \varphi pD)·(\nabla \varphi pD)dydsdt\hfill \\ & & +{\displaystyle \gamma ND}\int tf\int ND\left(t\right)\int \Omega (\nabla \varphi ND)·(\nabla \varphi ND)dxdydsdt\hfill \\ & & {\displaystyle \gamma pT}\int tf\int NT\left(t\right)\int \Omega (\nabla \varphi pT)·(\nabla \varphi pT)dydsdt\hfill \\ & & +{\displaystyle \gamma NT}\int tf\int NT\left(t\right)\int \Omega (\nabla \varphi NT)·(\nabla \varphi NT)dxdydsdt\hfill \\ & & +{\displaystyle \gamma NT}\int tf\int NT\left(t\right)\int \Omega (\nabla \varphi NT)·(\nabla \varphi NT)dxdydsdt\hfill \\ & & +{\displaystyle \gamma pHe}\int tf\int NHe\left(t\right)\int \Omega (\nabla \varphi pHe)·(\nabla \varphi pHe)dydsdt\hfill \\ & & +{\displaystyle \gamma NHe}\int tf\int NHe\left(t\right)\int \Omega (\nabla \varphi NHe)·(\nabla \varphi NHe)dxdydsdt\hfill \\ & & +{\displaystyle \gamma NHe}\int tf\int NHe\left(t\right)\int \Omega (\nabla \varphi NHe)·(\nabla \varphi NHe)dxdydsdt\hfill \\ & & +{\displaystyle \gamma N}\int tf\int NN\left(t\right)\int \Omega (\nabla \varphi N)·(\nabla \varphi N)dxdsdt\hfill \\ & & +{\displaystyle \gamma e}\int tf\int Ne\left(t\right)\int \Omega (\nabla \varphi e)·(\nabla \varphi e)dxdydsdt,\hfill \end{array}$$

(103)

and

$$\begin{array}{ccc}& & F\left(\varphi \right)=\hfill \\ & & {\displaystyle \alpha D}\int tf\int ND\left(t\right)\int ND\left(t\right)\int \Omega {\displaystyle |\varphi D(x-\xi ,y-\xi ,t,s-s)|\left|\varphi D(\xi ,\xi ,t,s)\right||(x,y)-(\xi ,\xi )|}dxdyd\xi d\xi dsdsdt\hfill \\ & & +{\displaystyle \alpha T}\int tf\int NT\left(t\right)\int NT\left(t\right)\int \Omega {\displaystyle |\varphi T(x-\xi ,y-\xi ,t,s-s)|\left|\varphi T(\xi ,\xi ,t,s)\right||(x,y)-(\xi ,\xi )|}dxdyd\xi d\xi dsdsdt\hfill \\ & & +{\displaystyle \alpha DT}\int tf\int ND\left(t\right)\int NT\left(t\right)\int \Omega {\displaystyle |\varphi D(x-\xi ,y-\xi ,t,s-s)|\left|\varphi T(\xi ,\xi ,t,s)\right||(x,y)-(\xi ,\xi )|}dxdyd\xi d\xi dt\hfill \\ & & +{\displaystyle \alpha He}\int tf\int NHe\left(t\right)\int NHe\left(t\right)\int \Omega {\displaystyle |\varphi He(x-\xi ,y-\xi ,t,s-s)|\left|\varphi He(\xi ,\xi ,t)\right||(x,y)-(\xi ,\xi ,s)|}dxdyd\xi d\xi dsdsdt\hfill \\ & & +{\displaystyle \alpha N}\int tf\int NN\left(t\right)\int NN\left(t\right)\int \Omega {\displaystyle |\varphi N(x-\xi ,t,s-s)|\left|\varphi N(\xi ,t,s)\right||x-\xi |}dxd\xi dsdsdt\hfill \\ & & +\sum j={\displaystyle \alpha HeN}\int tf\int NHe\left(t\right)\int ND\left(t\right)\int \Omega {\displaystyle |\varphi He(x-\xi ,y-\xi ,t)|\left|\varphi N(\xi j,t)\right||(x,y)-(\xi ,\xi )|}dxdyd\xi d\xi dsdsdt\hfill \\ & & +{\displaystyle \alpha He,e}\int tf\int NHe\left(t\right)\int Ne\left(t\right)\int \Omega {\displaystyle |\varphi He(x-\xi ,y-\xi ,t,s-s)|\left|\varphi e(\xi ,\xi ,t,s)\right||(x,y)-(\xi ,\xi )|}dxdyd\xi d\xi dsdsdt\hfill \\ & & +{\displaystyle \alpha e,e}\int tf\int Ne\left(t\right)\int Ne\left(t\right)\int \Omega {\displaystyle |\varphi e(x-\xi ,y-\xi ,t,s-s)|\left|\varphi e(\xi ,\xi ,t,s)\right||(x,y)-(\xi ,\xi )|}dxdyd\xi d\xi dsdsdt\hfill \end{array}$$

(104)

and the internal kinetics energy is expressed by

$$\begin{array}{ccc}\hfill Ec(\varphi ,\mathbf{r})& =& {\displaystyle 12}\int tf\int ND\left(t\right)\int \Omega \left|\varphi D\right|{\displaystyle \partial \mathbf{r}D\partial t}·{\displaystyle \partial \mathbf{r}D\partial t}dxdydsdt\hfill \\ & & +{\displaystyle 12}\int tf\int NT\left(t\right)\int \Omega \left|\varphi T\right|{\displaystyle \partial \mathbf{r}T\partial t}·{\displaystyle \partial \mathbf{r}T\partial t}dxdydsdt\hfill \\ & & +{\displaystyle 12}\int tf\int NHe\left(t\right)\int \Omega \left|\varphi He\right|{\displaystyle \partial \mathbf{r}He\partial t}·{\displaystyle \partial \mathbf{r}He\partial t}dxdydsdt\hfill \\ & & +{\displaystyle 12}\int tf\int NN\left(t\right)\int \Omega \left|\varphi N\right|{\displaystyle \partial \mathbf{r}N\partial t}·{\displaystyle \partial \mathbf{r}N\partial t}dxdydsdt\hfill \\ & & +{\displaystyle 12}\int tf\int Ne\left(t\right)\int \Omega \left|\varphi e\right|{\displaystyle \partial \mathbf{r}e\partial t}·{\displaystyle \partial \mathbf{r}e\partial t}dxdydsdt,\hfill \end{array}$$

(105)

Moreover,

$$F={\displaystyle \pi }\int tf\parallel \mathrm{curl}\mathbf{A}-\mathbf{B}\parallel dt,$$

$$\begin{array}{ccc}\hfill F& =& \int tf\int ND\left(t\right)\int \Omega \mathbf{E}ind·Kp\left|\varphi pD\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}D\partial t}\right)dxdydsdt\hfill \\ & & +\int tf\int NT\left(t\right)\int \Omega \mathbf{E}ind·Kp\left|\varphi pT\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}T\partial t}\right)dxdydsdt\hfill \\ & & +\int tf\int NHe\left(t\right)\int \Omega \mathbf{E}ind·Kp\left|\varphi pHe\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}He\partial t}\right)dxdydsdt\hfill \\ & & +\int tf\int Ne\left(t\right)\int \Omega \mathbf{E}ind·Ke\left|\varphi e\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}e\partial t}\right)dxdydsdt,\hfill \end{array}$$

(106)

where $K$ and $K$ are appropriate real constants related to the respective charges.

Here $\mathbf{u}=(u,u,u)$ is the fluid velocity field and

$$\mathbf{r}D,\mathbf{r}T,\mathbf{r}He,\mathbf{r}N,\mathbf{r}e$$

are fields of displacements for the corresponding particle fields.

Also $\mathbf{A}$ denotes the magnetic potential, $\mathbf{B}$ an external magnetic field and $\mathbf{B}$ is the total magnetic field.

Moreover, $\mathbf{E}$ is an induced electric field.

Also,

$$\begin{array}{ccc}\hfill F& =& {\displaystyle CD}\int tf\int ND\left(t\right)\int \Omega \nabla (x,y)\mathbf{r}D·\nabla (x,y)\mathbf{r}Ddxdydsdt\hfill \\ & & +{\displaystyle CT}\int tf\int NT\left(t\right)\int \Omega \nabla (x,y)\mathbf{r}T·\nabla (x,y)\mathbf{r}Tdxdydsdt\hfill \\ & & +{\displaystyle CHe}\int tf\int NHe\left(t\right)\int \Omega \nabla (x,y)\mathbf{r}He·\nabla (x,y)\mathbf{r}Hedxdydsdt\hfill \\ & & +{\displaystyle CN}\int tf\int NN\left(t\right)\int \Omega \nabla (x,y)\mathbf{r}N·\nabla (x,y)\mathbf{r}Ndxdydsdt\hfill \\ & & {\displaystyle Ce}\int tf\int Ne\left(t\right)\int \Omega \nabla (x,y)\mathbf{r}e·\nabla (x,y)\mathbf{r}edxdydsdt,\hfill \end{array}$$

(107)

for appropriate real positive constants $CDCT,CHe,CN,Ce.$

Finally,

$$\begin{array}{ccc}\hfill F& =& {\displaystyle \epsilon D}\int tf\left({\displaystyle \partial ND\left(t\right)\partial t}\right)dt+{\displaystyle \epsilon T}\int tf\left({\displaystyle \partial ND\left(t\right)\partial t}\right)dt\hfill \\ & & +{\displaystyle \epsilon N}\int tf\left({\displaystyle \partial NN\left(t\right)\partial t}\right)dt+{\displaystyle \epsilon He}\int tf\left({\displaystyle \partial NHe\left(t\right)\partial t}\right)dt\hfill \\ & & +{\displaystyle \epsilon e}\int tf\left({\displaystyle \partial Ne\left(t\right)\partial t}\right)dt,\hfill \end{array}$$

(108)

where $\epsilon D,\epsilon T,\epsilon N,\epsilon He,\epsilon e$ are small real positive constants.

Such a functional J is subject to the following constraints:

The momentum conservation equation for the fluid motion

$$\rho \left({\displaystyle \partial uk\partial t}+uj{\displaystyle \partial uk\partial xj}\right)=\rho fk-{\displaystyle \partial P\partial xk}+\tau kj,j+\left(FE\right)k+\left(FM\right)k,$$

$\forall k\in \{,,\}.$

Here $\rho =\rho D+\rho T+\rho He+\rho N+\rho e$ is the total density and P is the fluid pressure field.

Furthermore,

$$\tau ij=\mu \left({\displaystyle \partial ui\partial xj}+{\displaystyle \partial uj\partial xi}-{\displaystyle 23}\delta ij\sum k={\displaystyle \partial uk\partial xk}\right),$$

$\forall i,j\in \{,,\},$

$$\begin{array}{ccc}& & \mathbf{F}E=\left\{\left(FE\right)k\right\}=\hfill \\ & & \left(Kp\left(\int ND\left(t\right)|\varphi pD|ds+\int NT\left(t\right)|\varphi pT|ds+\int NHe\left(t\right)\left|\varphi pHe\right|ds\right)+Ke\int Ne\left(t\right)\left|\varphi e\right|ds\right)\mathbf{E},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \mathbf{F}M& =& \left\{\left(FM\right)k\right\}\hfill \\ & =& \left(Kp\left(\int ND\left(t\right)\left|\varphi pD\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}D\partial t}\right)ds\right.\right.\hfill \\ & & \int NT\left(t\right)\left|\varphi pT\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}T\partial t}\right)ds\hfill \\ & & \left.+\int NHe\left(t\right)\left|\varphi pHe\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}He\partial t}\right)ds\right)\hfill \\ & & \left.+Ke\int Ne\left(t\right)\left|\varphi e\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}e\partial t}\right)ds\right)\times \mathbf{B}.\hfill \end{array}$$

(109)

Mass conservation equation:

$${\displaystyle \partial \rho \partial t}+\mathrm{div}\left(\rho \mathbf{u}\right)=.$$

Energy equation

$$\rho {\displaystyle DeDt}+\nabla x(E^)·\mathbf{u}+E^+P\left(\mathrm{div}\mathbf{u}\right)={\displaystyle \partial Q\partial t}-\mathrm{div}\mathbf{q}+\tau jk{\displaystyle \partial uj\partial xk},$$

where we assume the Fourier law

$$\mathbf{q}=-K\nabla T,$$

where $T=T(x,t)$ is the scalar field of temperature and Q is a standard heat function.

Also,

$$\begin{array}{ccc}\hfill e& =& {\displaystyle \rho }\mathbf{u}·\mathbf{u}+{\displaystyle \rho D}{\displaystyle \partial \mathbf{r}D\partial t}·{\displaystyle \partial \mathbf{r}D\partial t}\hfill \\ & & +{\displaystyle \rho T}{\displaystyle \partial \mathbf{r}T\partial t}·{\displaystyle \partial \mathbf{r}T\partial t}\hfill \\ & & +{\displaystyle \rho He}{\displaystyle \partial \mathbf{r}He\partial t}·{\displaystyle \partial \mathbf{r}He\partial t}\hfill \\ & & +{\displaystyle \rho N}{\displaystyle \partial \mathbf{r}N\partial t}·{\displaystyle \partial \mathbf{r}N\partial t}\hfill \\ & & +{\displaystyle \rho e}{\displaystyle \partial \mathbf{r}e\partial t}·{\displaystyle \partial \mathbf{r}e\partial t},\hfill \end{array}$$

(110)

where the densities $E$ and $E$ are defined through the expressions of $F\left(\varphi \right)$ and $F$ so that

$$F\left(\varphi \right)=\int tf\int \Omega E^dxdt$$

and

$$F=\int tf\int \Omega E^dxdt.$$

Here we recall that since $\mathbf{r}$ is highly oscillating in t we approximately have

$$\mathbf{u}·\mathbf{r}D\approx$$

in a weak or measure sense. The same remark is valid for the other internal velocity fields.

Moreover,

$${\displaystyle DeDt}={\displaystyle \partial e\partial t}+uj{\displaystyle \partial e\partial xj}.$$

$$P=F(\rho ,T),$$

for an appropriate scalar function $F$.

Mass relations

$$mD\left(t\right)=\int \Omega \rho D(x,t)dx,$$

$$mT\left(t\right)=\int \Omega \rho T(x,t)dx,$$

$$mHe\left(t\right)=\int \Omega \rho He(x,t)dx,$$

$$mN\left(t\right)=\int \Omega \rho N(x,t)dx,$$

$$me\left(t\right)=\int \Omega \rho e(x,t)dx,$$

where,

$$\left(mHe,N\right)T\left(t\right)=mHe,N\left(t\right)+\int t\int \partial \Omega \left(\rho He(x,\tau )\right)\mathbf{u}·\mathbf{n}dSd\tau ,$$

$$mHe,N\left(t\right)=mHe\left(t\right)+mN\left(t\right),$$

$$\left(mD\right)\left(t\right)=\left(mD\right)-\int t\int \partial \Omega \cup \partial \Omega \left(\rho D(x,\tau )\right)\mathbf{u}·\mathbf{n}dSd\tau -\alpha D\left(mHe,N\right)T\left(t\right),$$

$$\left(mT\right)\left(t\right)=\left(mT\right)-\int t\int \partial \Omega \cup \partial \Omega \left(\rho T(x,\tau )\right)\mathbf{u}·\mathbf{n}dSd\tau -\alpha T\left(mHe,N\right)T\left(t\right),$$

$$\left(mHe\right)T\left(t\right)=\int \Omega \rho He(x,t)dx+\int t\int \partial \Omega \rho He(x,\tau )\mathbf{u}·\mathbf{n}d\Gamma d\tau ,$$

$$\left(mN\right)T\left(t\right)=\int \Omega \rho N(x,t)dx+\int t\int \partial \Omega \rho N(x,\tau )\mathbf{u}·\mathbf{n}d\Gamma d\tau ,$$

$${\displaystyle \left(mN\right)T\left(t\right)\left(mHe\right)T\left(t\right)}={\displaystyle \alpha N\alpha He},$$

so that

$$\alpha NmHe)T\left(t\right)=\alpha He\left(mN\right)T\left(t\right),$$

$$\left(me\right)T\left(t\right)=me\left(t\right)+\int t\int \partial \Omega \left(\rho T(x,\tau )\right)\mathbf{u}·\mathbf{n}dSd\tau .$$

$$\begin{array}{ccc}\hfill me\left(t\right)& =& \int ND\left(t\right)\int \Omega |\varphi pD(y,t,s)|dydyds{\displaystyle memp}+\int NT\left(t\right)\int \Omega \left|\varphi pT(y,t,s)\right|dyds{\displaystyle memp}\hfill \\ & & +\int Np\left(t\right)\int \Omega \left|\varphi pHe(y,t,s)\right|dyds{\displaystyle memp}.\hfill \end{array}$$

(111)

Other mass constraints

$$\int \Omega \left|\varphi ND(x,y,t,s)\right|dx=mN,$$

$$\int \Omega \left|\varphi NT(x,y,t,s)\right|dx=mN,$$

$$\int \Omega \left|\varphi NT(x,y,t,s)\right|dx=mN,$$

$$\int \Omega \left|\varphi NHe(x,y,t,s)\right|dx=mN,$$

$$\int \Omega \left|\varphi NHe(x,y,t,s)\right|dx=mN,$$

$$\int \Omega \left|\varphi pD(x,t,s)\right|dx=mp,$$

$$\int \Omega \left|\varphi pT(x,t,s)\right|dx=mp,$$

$$\int \Omega \left|\varphi pHe(x,t,s)\right|dx=mp,$$

$$mD\left(t\right)=mpND\left(t\right)+mNND\left(t\right)$$

$$mT\left(t\right)=mpNT\left(t\right)+mNNT\left(t\right),$$

$$mHe\left(t\right)=mpNHe\left(t\right)+mNNHe\left(t\right),$$

$$me\left(t\right)=meND\left(t\right)+meNT\left(t\right)+meNHe\left(t\right).$$

For the induced electric field, we must have

$$\begin{array}{ccc}& & \mathrm{curl}\mathbf{E}ind+{\displaystyle c}\mathrm{curl}\left(K^p\int ND\left(t\right)\left|\varphi pD\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}D\partial t}\right)ds\right.\hfill \\ & & +K^p\int NT\left(t\right)\left|\varphi pT\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}T\partial t}\right)ds\hfill \\ & & +K^p\int NHe\left(t\right)\left|\varphi pHe\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}He\partial t}\right)ds\hfill \\ & & +\left.K^e\int Ne\left(t\right)\int \Omega \left|\varphi e(x,y,t,s)\right|\left(\mathbf{u}(y,t)+{\displaystyle \partial \mathbf{r}e(x,y,t)\partial t}dx\right)ds\right)\hfill \\ & & \times \left(\mathrm{curl}\mathbf{A}-\mathbf{B}\right)-{\displaystyle c}{\displaystyle \partial \partial t}\left(\mathrm{curl}\mathbf{A}-\mathbf{B}\right)=\mathbf{0},\hfill \end{array}$$

(112)

where $K$ and $K$ are appropriate real constants related to the respective charges.

A Maxwell equation:

$$\mathrm{div}\mathbf{B}=,$$

where

$$\mathbf{B}=\mathbf{B}-\mathrm{curl}\mathbf{A}.$$

Another Maxwell equation:

$$\begin{array}{ccc}\hfill \mathrm{div}\mathbf{E}& =& \pi \left(Kp\left(\int ND\left(t\right)|\varphi pD|ds+\int NT\left(t\right)|\varphi pT|ds+\int NHe\left(t\right)\left|\varphi pHe\right|ds\right)\right.\hfill \\ & & \left.+Ke\int Ne\left(t\right)\int \Omega \left|\varphi e(x,y,t,s)\right|dxds\right),\hfill \end{array}$$

(113)

where the total electric field $\mathbf{E}$ stands for

$$\mathbf{E}=\mathbf{E}ind+\mathbf{E}\rho ,$$

and where generically denoting

$$F\left(\varphi \right)=\int tf\int \Omega f(\varphi ,x,t\xi ,s)dxd\xi dsdt,$$

we have also

$$\mathbf{E}\rho =\left\{\int \Omega {\displaystyle \partial f(\varphi ,x,t,\xi ,s)\partial xk}d\xi ds\right\}.$$

At this point we generically denote

$$\langle h,h\rangle L=\int tf\int \Omega hhdxdydt.$$

Thus, already including the Lagrange multipliers concerning the restrictions indicated, the extended functional $J$ stands for

$$\begin{array}{ccc}& & J=J(\varphi ,\mathbf{u},\mathbf{r},P,\mathbf{A},\mathbf{B},\mathbf{E},\Lambda ,E,\{ND,NT,NHe,NN,Ne\})\hfill \\ & =& G(\nabla \varphi )+F\left(\varphi \right)+Ec(\varphi ,\mathbf{r})+F+F+F+F\hfill \\ & & +〈\Lambda k,\rho \left({\displaystyle \partial uk\partial t}+uj{\displaystyle \partial uk\partial xj}\right)-\rho fk+{\displaystyle \partial P\partial xk}-\tau kj,j-\left(FE\right)k-\left(FM\right)k〉L\hfill \\ & & +〈\Lambda ,{\displaystyle \partial \rho \partial t}+\mathrm{div}\left(\rho \mathbf{u}\right)〉L+JAux+JAux+JAux+JAux+JAux,\hfill \end{array}$$

(114)

where,

$$\begin{array}{ccc}\hfill JAux& =& 〈\Lambda ,\rho {\displaystyle DeDt}+\nabla x(E^)·\mathbf{u}+P\left(\mathrm{div}\mathbf{u}\right)-{\displaystyle \partial Q\partial t}+\mathrm{div}\mathbf{q}-\tau jk{\displaystyle \partial uj\partial xk}〉L\hfill \\ & & +〈\Lambda ,P-F(\rho ,T)〉L,\hfill \end{array}$$

(115)

$$\begin{array}{ccc}\hfill JAux& =& 〈\Lambda ,mD\left(t\right)-\int \Omega \rho D(x,t)dx〉L\hfill \\ & & +〈\Lambda ,mT\left(t\right)-\int \Omega \rho T(x,t)dx〉L\hfill \\ & & 〈\Lambda ,mHe\left(t\right)-\int \Omega \rho He(x,t)dx〉L\hfill \\ & & 〈\Lambda 10,mN\left(t\right)-\int \Omega \rho N(x,t)dx〉L\hfill \\ & & 〈\Lambda 11,me\left(t\right)-\int \Omega \rho e(x,t)dx〉L\hfill \\ & & \int tfE12\left(t\right)\left(\alpha NmHe\right)T\left(t\right)-\alpha He\left(mN\right)T\left(t\right))dt,\hfill \end{array}$$

(116)

$$\begin{array}{ccc}\hfill JAux& =& -\int tf\int \Omega \left(END\right)(y,t,s)\left(\int \Omega \left|\varphi ND(x,y,t,s)\right|dx-mN\right)dydt\hfill \\ & & -\int tf\int \Omega \left(ENT\right)(y,t,s)\left(\int \Omega \left|\varphi NT(x,y,t,s)\right|dx-mN\right)dydt\hfill \\ & & -\int tf\int \Omega \left(ENT\right)(y,t,s)\left(\int \Omega \left|\varphi NT(x,y,t,s)\right|dx-mN\right)dydt\hfill \\ & & -\int tf\int \Omega \left(ENHe\right)(y,t,s)\left(\int \Omega \left|\varphi NHe(x,y,t,s)\right|dx-mN\right)dydt\hfill \\ & & -\int tf\int \Omega \left(ENHe\right)(y,t,s)\left(\int \Omega \left|\varphi NHe(x,y,t,s)\right|dx-mN\right)dydt,\hfill \\ & & -\int tf\int \Omega \left(EpD\right)(t,s)\left(\int \Omega \left|\varphi pD(y,t,s)\right|dy-mp\right)dsdt,\hfill \\ & & -\int tf\int \Omega \left(EpT\right)(t,s)\left(\int \Omega \left|\varphi pT(y,t,s)\right|dy-mp\right)dsdt,\hfill \\ & & -\int tf\int \Omega \left(EpHe\right)(t,s)\left(\int \Omega \left|\varphi pHe(y,t,s)\right|dy-mp\right)dsdt,\hfill \end{array}$$

(117)

$$\begin{array}{ccc}\hfill JAux& =& 〈\Lambda 12,\mathrm{curl}\mathbf{E}ind\hfill \\ & & +{\displaystyle c}\mathrm{curl}\left(K^p\int ND\left(t\right)\left|\varphi pD\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}D\partial t}\right)ds\right.\hfill \\ & & +K^p\int NT\left(t\right)\left|\varphi pT\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}T\partial t}\right)ds\hfill \\ & & +K^p\int NHe\left(t\right)\left|\varphi pHe\right|\left(\mathbf{u}+{\displaystyle \partial \mathbf{r}He\partial t}\right)ds\hfill \\ & & +\left.K^e\int Ne\left(t\right)\int \Omega \left|\varphi e(x,y,t,s)\right|\left(\mathbf{u}(y,t)+{\displaystyle \partial \mathbf{r}e(x,y,t,s)\partial t}dx\right)ds\right)\hfill \\ & & \times \left(\mathrm{curl}\mathbf{A}-\mathbf{B}\right)-{\displaystyle c}{\displaystyle \partial \partial t}\left(\mathrm{curl}\mathbf{A}-\mathbf{B}\right)〉L\hfill \\ & & +〈\Lambda 13,\mathrm{div}\mathbf{B}〉L\hfill \\ & & +〈\Lambda 14,\mathrm{div}\mathbf{E}-\pi \left(Kp\left(\int ND\left(t\right)|\varphi pD|ds+\int NT\left(t\right)|\varphi pT|ds+\int NHe\left(t\right)\left|\varphi pHe\right|ds\right)\right.\hfill \\ & & \left.+Ke\int \Omega \left|\varphi e\right|dxds\right)〉L.\hfill \end{array}$$

(118)

$$\begin{array}{ccc}\hfill JAux& =& \langle \Lambda 15,mD\left(t\right)-(mpND\left(t\right)+mNND\left(t\right))\rangle L\hfill \\ & & +\langle \Lambda 16,mT\left(t\right)-(mpNT\left(t\right)+mNNT\left(t\right))\rangle L\hfill \\ & & +\langle \Lambda 17,mHe\left(t\right)-(mpNHe\left(t\right)+mNNHe\left(t\right))\rangle L\hfill \\ & & +\langle \Lambda 18,me\left(t\right)-(meND\left(t\right)+meNT\left(t\right)+meNHe\left(t\right))\rangle L.\hfill \end{array}$$

(119)

Here we recall the following definitions and relations:

For the Deuterium field

$$|\varphi D(x,y,t,s)|=|\varphi pD(y,t,s)|+|\varphi ND(x,y,t,s)|\left|\varphi pD(y,t,s)\right|{\displaystyle mp},$$

For the Tritium field

$$|\varphi T(x,y,t,s)|=|\varphi pT(y,t,s)|+\left(\right|\varphi NT(x,y,t,s)|+|\varphi NT(x,y,t,s)|\left)\right|\varphi pD(y,t,s)|{\displaystyle mp},$$

For the Helium field

$$|\varphi He(x,y,t,s)|=|\varphi pHe(y,t,s)|+\left(\right|\varphi NHe(x,y,t,s)|+|\varphi NHe(x,y,t,s)|\left)\right|\varphi pHe(y,t,s)|{\displaystyle mp},$$

For the Neutron field

$$\varphi N=\varphi N(x,t,s),$$

For the electronic field resulting from the ionization

$$\varphi e=\varphi e(x,y,t,s).$$

$$\rho D(y,t)=\int ND\left(t\right)\int \Omega \left|\varphi D(x,y,t,s)\right|dxds,$$

$$\rho T(y,t)=\int NT\left(t\right)\int \Omega \left|\varphi T(x,y,t,s)\right|dxds,$$

$$\rho He(y,t)=\int NHe\left(t\right)\int \Omega \left|\varphi He(x,y,t,s)\right|dxds,$$

$$\rho N(x,t)=\int NN\left(t\right)\left|\varphi N(x,t,s)\right|ds,$$

$$\rho e(y,t)=\int Ne\left(t\right)\int \Omega \left|\varphi e(x,y,t,s)\right|dxds.$$

Also,

$$\rho =\rho D+\rho T+\rho He+\rho N+\rho e,$$

$$\left(mHe,N\right)T\left(t\right)=mHe,N\left(t\right)+\int t\int \partial \Omega (\rho He(x,\tau )+\rho N(x,\tau ))\mathbf{u}·\mathbf{n}dSd\tau ,$$

$$mHe,N\left(t\right)=mHe\left(t\right)+mN\left(t\right),$$

$$mHe\left(t\right)=\int \Omega \rho He(x,t)dx,$$

$$mN\left(t\right)=\int \Omega \rho N(x,t)dx,$$

$$\left(mD\right)\left(t\right)=\left(mD\right)-\int t\int \partial \Omega \cup \partial \Omega \left(\rho D(x,\tau )\right)\mathbf{u}·\mathbf{n}dSd\tau -\alpha D\left(mHe,N\right)T\left(t\right),$$

$$\left(mT\right)\left(t\right)=\left(mT\right)-\int t\int \partial \Omega \cup \partial \Omega \left(\rho T(x,\tau )\right)\mathbf{u}·\mathbf{n}dSd\tau -\alpha T\left(mHe,N\right)T\left(t\right),$$

$$\left(mHe\right)T\left(t\right)=\int \Omega \rho He(x,t)dx+\int t\int \partial \Omega \rho He(x,\tau )\mathbf{u}·\mathbf{n}d\Gamma d\tau ,$$

$$\left(mN\right)T\left(t\right)=\int \Omega \rho N(x,t)dx+\int t\int \partial \Omega \rho N(x,\tau )\mathbf{u}·\mathbf{n}d\Gamma d\tau ,$$

$${\displaystyle \left(mN\right)T\left(t\right)\left(mHe\right)T\left(t\right)}={\displaystyle \alpha N\alpha He},$$

so that

$$\alpha N\left(mHe\right)T\left(t\right)=\alpha He\left(mN\right)T\left(t\right),$$

$$\left(me\right)T\left(t\right)=me\left(t\right)-\int t\int \partial \Omega \left(\rho e(x,\tau )\right)\mathbf{u}·\mathbf{n}d\Gamma d\tau ,$$

$$me\left(t\right)=\int \Omega \rho e(x,t)dx.$$

$$\begin{array}{ccc}\hfill me\left(t\right)& =& \int ND\left(t\right)\int \Omega |\varphi pD(y,t,s)|dydyds{\displaystyle memp}+\int NT\left(t\right)\int \Omega \left|\varphi pT(y,t,s)\right|dyds{\displaystyle memp}\hfill \\ & & +\int Np\left(t\right)\int \Omega \left|\varphi pHe(y,t,s)\right|dyds{\displaystyle memp}.\hfill \end{array}$$

(120)

Finally,

$$\mathbf{E}=\mathbf{E}ind+\mathbf{E}\rho ,$$

and where generically denoting

$$F\left(\varphi \right)=\int tf\int \Omega f(\varphi ,x,t,\xi ,s)dxd\xi ds,$$

we have also

$$\mathbf{E}\rho =\left\{\int \Omega {\displaystyle \partial f(\varphi ,x,t,\xi ,s)\partial xk}d\xi ds\right\}.$$

and,

$$\mathbf{B}=\mathbf{B}-\mathrm{curl}\mathbf{A}.$$

21. A Qualitative Modeling for a General Phase Transition Process

In this section we develop a general qualitative modeling for a phase transition process.

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Such a set $\Omega$ is supposed to a be a fixed volume in which an amount of mass of a substance A with a density function u will develop phase a transition for another phase with corresponding density function $v.$ The total mass $m$ is suppose to be kept constant throughout such a process.

We model such transition in phase through a functional $J:V\times V\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J(u,v)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle \alpha }\int \Omega udx\hfill \\ & & {\displaystyle \gamma }\int \Omega \nabla v·\nabla vdx+{\displaystyle \alpha }\int \Omega vdx\hfill \\ & & -{\displaystyle 12}\int \Omega \omega (u+v)dx-{\displaystyle E}\left(\int \Omega (u+v)dx-mT\right).\hfill \end{array}$$

(121)

Here $\gamma >,\gamma >,\alpha >,\alpha >$ and $V=W,\left(\Omega \right).$

The phases corresponding to u and v are connected through a Lagrange multiplier E, which represents the chemical potential of the chemical process in question.

We assume the temperature is directly proportional to the internal kinetics $E$ energy where

$$EC={\displaystyle 12}\int \Omega u{\displaystyle \partial \mathbf{r}u\partial t}·{\displaystyle \partial \mathbf{r}u\partial t}dx.$$

For a internal vibrational motion, we assume approximately

$$\mathbf{r}u\approx ei\omega t\mathbf{w}\left(x\right),$$

for an appropriate frequency $\omega$ and vectorial function $\mathbf{w}.$

Thus, the temperature $T=T(x,t)$ is indeed proportional to $\omega$, that is, symbolically, we may write

$$T\propto E\propto \omega .$$

Therefore, we start with the system with a phase corresponding to $u\approx$ and $v\approx$ at $\omega =$. Gradually increasing the temperature to a corresponding $\omega =15$, we obtain a transition to a phase corresponding to $u\approx$ and $v\approx$.

At this point, we also define the index normalized corresponding densities

$$\varphi u={\displaystyle uu+v}$$

and

$$\varphi v={\displaystyle vu+v}.$$

Finally, we have obtained some numerical results for the following parameters:

$\Omega =[,]\subset \mathbb{R},$$\gamma =\gamma =$, $\alpha =0.1$, $\alpha =103.$

We start with $\omega =$ corresponding to $\varphi u\approx$ and $\varphi v\approx$ in $\Omega$.

For the corresponding solutions $\varphi$ and $\varphi$, please see Figure 15 and Figure 16, respectively.

We end the process with $\omega =15$ corresponding to $\varphi u\approx$ and $\varphi v\approx$ in $\Omega$.

For the corresponding solutions $\varphi$ and $\varphi$, please see Figure 17 and Figure 18, respectively.

22. A Mathematical Description of a Hydrogen Molecule in a Quantum Mechanics Context

In this section we develop a mathematical description for a hydrogen molecule.

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega$.

Observe that a single hydrogen molecule comprises two hydrogen atoms physically linked through their electrons.

We recall that each hydrogen atom comprises one proton, one neutron and one electron.

Since the electric charge interaction effects are much higher than those related to the respective masses, in a first analysis we neglect the single neutron densities.

Denoting $(x,y,z)\in \Omega \times \Omega \times \Omega$ and time $t\in [,tf]$, generically, for a particle $p$ at the atom $A$ in the molecule $M$, we define the following general density:

$$|\varphi \left(pjkl\right)T(x,y,z,t)|={\displaystyle |\varphi pjkl(x,y,z,t)||\varphi Akl(y,z,t)|\left|\varphi Ml(z,t)\right|mAjkmMl}.$$

Here we have the particle density $|\varphi pjkl(x,y,z,t)|$ in the atom $A$ with density $|\varphi Akl(y,z,t)|$, at the molecule $M$ with a global density $|\varphi Ml(z,t)|$.

Here we have also denoted, $m$ the particle mass, $m$ the mass of atom $A$ and $m$ the mass of molecule $M$, so that we set the following constraints:

$$\int \Omega \left|\varphi pjkl(x,y,z,t)\right|dx=mpjkl,$$

$$\int \Omega \left|\varphi Akl(y,z,t)\right|dy=mAkl,$$

$$\int \Omega \left|\varphi Ml(z,t)\right|dz=mMl.$$

At this point we denote for the atoms $A$ e $A$ of a hydrogen molecule:

$mej=me$: mass of electron $e$ in the atom $A$, where $j\in \{,\}.$

$mpj=mp$ : mass of proton $p$ in the atom $A$, where $j\in \{,\}.$

Therefore, considering the respective indexed densities for the particles in question, we define the total hydrogen molecule density, denoted by $|\varphi H(x,y,z,t)|$ as

$$\begin{array}{ccc}\hfill |\varphi H(x,y,z,t)|& =& {\displaystyle |\varphi p(x,y,z,t)||\varphi A(y,z,t)|\left|\varphi M(z,t)\right|mAmM}\hfill \\ & & +{\displaystyle |\varphi e(x,y,z,t)||\varphi A(y,z,t)|\left|\varphi M(z,t)\right|mAmM}\hfill \\ & & +{\displaystyle |\varphi p(x,y,z,t)||\varphi A(y,z,t)|\left|\varphi M(z,t)\right|mAmM}\hfill \\ & & +{\displaystyle |\varphi e(x,y,z,t)||\varphi A(y,z,t)|\left|\varphi M(z,t)\right|mAmM}.\hfill \end{array}$$

(122)

Such system is subject to the following constraints:

From the proton $p$ in the atom $A$:

$$\int \Omega \left|\varphi p(x,y,z,t)\right|dx=mp,$$

For the proton $p$ in the atom $A$:

$$\int \Omega \left|\varphi p(x,y,z,t)\right|dx=mp,$$

For the atom $A$:

$$\int \Omega \left|\varphi A(y,z,t)\right|dy=mA,$$

For the atom $A$:

$$\int \Omega \left|\varphi A(y,z,t)\right|dy=mA,$$

For the electrons $e$ and $e$, concerning the physical electronic link between the atoms:

$$\int \Omega |\varphi e(x,y,z,t)|dx+\int \Omega \left|\varphi e(x,y,z,t)\right|dx=me.$$

For the total molecular density:

$$\int \Omega \left|\varphi M(z,t)\right|dz=mM.$$

Therefore, already including the Lagrange multipliers, the corresponding variational formulation for such a system stands for $J:V\to \mathbb{R}$, where

$$J(\varphi ,E)=G(\nabla \varphi )+F\left(\varphi \right)+JAux(\varphi ,E).$$

Here we denote

$$|\left(\varphi pj\right)T|={\displaystyle |\varphi pj(x,y,z,t)||\varphi Aj(y,z,t)|\left|\varphi M(z,t)\right|mAjmM},$$

$$|\left(\varphi ej\right)T|={\displaystyle |\varphi ej(x,y,z,t)||\varphi Aj(y,z,t)|\left|\varphi M(z,t)\right|mAjmM},\forall j\in \{,\}$$

we assume $\gamma \left(pj\right)>,\gamma ej>,\gamma Aj>,\gamma M>,\alpha \left(pj\right)T>,\alpha \left(ej\right)T>\alpha \left(pjek\right)T<,\forall j,k\in \{,\},$

$$\begin{array}{ccc}\hfill G(\nabla \varphi )& =& {\displaystyle \gamma pj}\int tf\int \Omega (\nabla \varphi pj)·(\nabla \varphi pj)dxdydzdt\hfill \\ & & +{\displaystyle \gamma ej}\int tf\int \Omega (\nabla \varphi ej)·(\nabla \varphi ej)dxdydzdt\hfill \\ & & {\displaystyle \gamma Aj}\int \Omega (\nabla \varphi Aj)·(\nabla \varphi Aj)dydzdt\hfill \\ & & +{\displaystyle \gamma M}\int tf\int \Omega (\nabla \varphi M)·(\nabla \varphi M)dzdt\hfill \end{array}$$

(123)

and

$$\begin{array}{ccc}& & F\left(\varphi \right)=\hfill \\ & & {\displaystyle \alpha \left(pj\right)T}\int tf\int \Omega {\displaystyle |\varphi \left(pj\right)T(x-\xi ,y-\xi ,z-\xi ,t)|\left|\varphi \left(pj\right)T(\xi ,\xi ,\xi ,t)\right||(x,y,z)-(\xi ,\xi ,\xi )|}dxdydz;d\xi d\xi d\xi dt\hfill \\ & & +{\displaystyle \alpha \left(ej\right)T}\int tf\int \Omega {\displaystyle |\varphi \left(ej\right)T(x-\xi ,y-\xi ,z-\xi ,t)|\left|\varphi \left(ej\right)T(\xi ,\xi ,\xi ,t)\right||(x,y,z)-(\xi ,\xi ,\xi )|}dxdydzd\xi d\xi d\xi dt\hfill \\ & & +{\displaystyle \alpha \left(pjek\right)T}\int tf\int \Omega {\displaystyle |\varphi \left(pj\right)T(x-\xi ,y-\xi ,z-\xi ,t)|\left|\varphi \left(ek\right)T(\xi ,\xi ,\xi ,t)\right||(x,y,z)-(\xi ,\xi ,\xi )|}dxdydzd\xi d\xi d\xi dt\hfill \end{array}$$

Finally,

$$\begin{array}{ccc}\hfill JAux(\varphi ,E)& =& \int tf\int \Omega \left(Ep\right)j(y,z,t)\left(\int \Omega \left|\varphi pj(x,y,z,t)\right|dx-mp\right)dydzdt\hfill \\ & & \int tf\int \Omega \left(Ee\right)(y,z,t)\left(\int \Omega \left(\right|\varphi e(x,y,z,t)|+|\varphi e(x,y,z,t)\left|\right)dx-me\right)dydzdt\hfill \\ & & \int tf\int \Omega \left(EA\right)j(z,t)\left(\int \Omega \left|\varphi Aj(y,z,t)\right|dy-mAj\right)dzdt\hfill \\ & & \int tf\left(EM\right)\left(t\right)\left(\int \Omega \left|\varphi M(z,t)\right|dz-mM\right)dt.\hfill \end{array}$$

(124)

Remark 10.

We highlight the two electrons which link the atoms are at same level of energy $E$. Morever, each atom has its energy level $E$ and the molecule as a whole has also its energy level $EM.$

23. A Mathematical Model for the Water Hydrolysis

In this section we develop a modeling for a chemical reaction known as the water hydrolysis.

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

In such a volume $\Omega$ containing a total mass $m$ of water initially at the temperature 25 C with pressure 1 atm, we intend to model the following reaction

$$HO\rightleftharpoons OH-+H+$$

which as previously mentioned is the well known water hydrolysis.

We highlight $HO$ stand for a water molecule which subject to an appropriate electric potential is decomposed into a ionized $OH-$ molecule and ionized $H$ atom.

It is also well known that the water symbol $HO$ corresponds to a molecule with two hydrogen (H) atoms and one oxygen (O) atom.

Moreover, the oxygen atom O has 8 protons, 8 neutrons and 8 electrons whereas the hydrogen atom H has one proton, one neutron and one electron.

Remark 11.

Here we have assumed that a unit mass of $HO$ reacts into a fractional mass $\alpha$ of $OH-$ and a fractional mass $\alpha$ of $H+.$

Symbolically, we have:

$$=\alpha B+\alpha C.$$

To clarify the notation we set the conventions:

$HO$ molecule generically corresponds to wave function $\varphi$.

$OH-$ molecule corresponds to wave function $\varphi .$

$H$ hydrogen atom corresponds to wave function $\varphi .$

At this point we define the following densities:

For the $HO$ water density (for charges), denoted by $|\varphi |$, we have

$$\begin{array}{ccc}\hfill |\varphi (x,y,z,t)|& =& Kp\sum j=\left|\left(\varphi H\right)pj(x,y,z,t)\right|{\displaystyle |\left(\varphi H\right)Aj(y,z,t)|\left|\left(\varphi \right)M(z,t)\right|\left(m\right)AjH\left(m\right)M}\hfill \\ & & +Ke\sum j=\left|\left(\varphi H\right)ej(x,y,z,t)\right|{\displaystyle |\left(\varphi H\right)Aj(y,z,t)|\left|\left(\varphi \right)M(z,t)\right|\left(m\right)AjH\left(m\right)M}\hfill \\ & & +Kp\sum j=\left|\left(\varphi O\right)pj(x,y,z,t)\right|{\displaystyle |\left(\varphi O\right)A(y,z,t)|\left|\left(\varphi \right)M(z,t)\right|\left(m\right)AO\left(m\right)M}\hfill \\ & & +Ke\sum j=\left|\left(\varphi O\right)ej(x,y,z,t)\right|{\displaystyle |\left(\varphi O\right)A(y,z,t)|\left|\left(\varphi \right)M(z,t)\right|\left(m\right)AO\left(m\right)M}\hfill \end{array}$$

(125)

where $\left(m\right)$ is the mass of a single water molecule and generically $|\left(\varphi H\right)pj(x,y,z,t)|$ refers to the hydrogen proton $p$ at the hydrogen atom $A$ concerning the $HO$ molecular density and so on.

For the $OH-$ density, denoted by $|\varphi |$, we have

$$\begin{array}{ccc}\hfill |\varphi (x,y,z,t)|& =& Kp\left|\left(\varphi H\right)p(x,y,z,t)\right|{\displaystyle |\left(\varphi H\right)A(y,z,t)|\left|\left(\varphi \right)M(z,t)\right|\left(m\right)AH\left(m\right)M}\hfill \\ & & +Ke\left|\left(\varphi H\right)e(x,y,z,t)\right|{\displaystyle |\left(\varphi H\right)A(y,z,t)|\left|\left(\varphi \right)M(z,t)\right|\left(m\right)AH\left(m\right)M}\hfill \\ & & +Ke\left|\left(\varphi OH-\right)e(x,z,t)\right|{\displaystyle |\left(\varphi \right)M(z,t)|\left(m\right)M}\hfill \\ & & +Kp\sum j=\left|\left(\varphi O\right)pj(x,y,z,t)\right|{\displaystyle |\left(\varphi O\right)A(y,z,t)|\left|\left(\varphi \right)M(z,t)\right|\left(m\right)AO\left(m\right)M}\hfill \\ & & +Ke\sum j=\left|\left(\varphi O\right)ej(x,y,z,t)\right|{\displaystyle |\left(\varphi O\right)A(y,z,t)|\left|\left(\varphi \right)M(z,t)\right|\left(m\right)AO\left(m\right)M},\hfill \end{array}$$

(126)

where $\left(m\right)$ is the mass of a single molecule of $OH-$.

For the ionized hydrogen atom have

$$|\varphi (x,y,t)|=Kp\left|\left(\varphi H\right)p(x,y,t)\right|{\displaystyle |\left(\varphi H\right)A(y,t)|\left(m\right)A}.$$

where we have denoted $\left(m\right)$ is the mass of a single atom of $H$.

Here $Kp>$ and $Ke<$ are appropriate real constants concerning a proton and an electron charge, respectively.

The system is subject to the following constraints:

$$\int \Omega \left|\left(\varphi H\right)pj(x,y,z,t)\right|dx=mp,\forall j\in \{,\},$$

$$\int \Omega \left|\left(\varphi H\right)ej(x,y,z,t)\right|dx=me,\forall j\in \{,\},$$

$$\int \Omega \left|\left(\varphi O\right)pj(x,y,z,t)\right|dx=mp,\forall j\in \{,\},$$

$$\int \Omega \left|\left(\varphi O\right)ej(x,y,z,t)\right|dx=me,\forall j\in \{,\},$$

$$\int \Omega \left|\left(\varphi H\right)p(x,y,z,t)\right|dx=mp,$$

$$\int \Omega \left|\left(\varphi H\right)e(x,y,z,t)\right|dx=me,$$

$$\int \Omega \left|\left(\varphi H\right)e(x,y,z,t)\right|dx=me,$$

$$\int \Omega \left|\left(\varphi O\right)pj(x,y,z,t)\right|dx=mp,\forall j\in \{,\},$$

$$\int \Omega \left|\left(\varphi O\right)ej(x,y,z,t)\right|dx=me,\forall j\in \{,\},$$

$$\int \Omega \left|\left(\varphi H\right)p(x,z,t)\right|dx=mp,$$

$$\int \Omega \left|\left(\varphi H\right)Aj(y,z,t)\right|dy=mAH,\forall j\in \{,\},$$

$$\int \Omega \left|\left(\varphi O\right)A(y,z,t)\right|dy=mAO,$$

$$\int \Omega \left|\left(\varphi H\right)A(y,z,t)\right|dy=mAH,$$

$$\int \Omega \left|\left(\varphi O\right)A(y,z,t)\right|dy=mAO,$$

$$\int \Omega \left|\left(\varphi H\right)A(y,z,t)\right|dy=mAH,$$

$$\int \Omega \left(\right|\left(\varphi \right)M(z,t)|+|\left(\varphi \right)M(z,t)|+|\left(\varphi \right)M(z,t)\left|\right)dz=mT,$$

$$\int \Omega (\alpha C|\left(\varphi \right)M(z,t)|-\alpha B|\left(\varphi \right)M(z,t)\left|\right)dz=.$$

Already including the Lagrange multipliers for the constraints, the variational formulation for such system. denoted by the functional $J(\varphi ,E)$ stands for

$$J(\varphi ,E)=G(\nabla \varphi )+F\left(\varphi \right)+F\left(\varphi \right)-JAux(\varphi ,E),$$

where

$$\begin{array}{ccc}\hfill G(\nabla \varphi )& =& {\displaystyle \gamma p}\sum j=\int tf\int \Omega \nabla \left(\varphi H\right)pj·\nabla \left(\varphi H\right)pjdxdydzdt\hfill \\ & & +{\displaystyle \gamma e}\sum j=\int tf\int \Omega \nabla \left(\varphi H\right)ej·\nabla \left(\varphi H\right)ejdxdydzdt\hfill \\ & & +{\displaystyle \gamma p}\sum j=\int tf\int \Omega \nabla \left(\varphi O\right)pj·\nabla \left(\varphi O\right)pjdxdydzdt\hfill \\ & & +{\displaystyle \gamma e}\sum j=\int tf\int \Omega \nabla \left(\varphi O\right)ej·\nabla \left(\varphi O\right)ejdxdydzdt\hfill \\ & & +{\displaystyle \gamma p}\int tf\int \Omega \nabla \left(\varphi H\right)p·\nabla \left(\varphi H\right)pdxdydzdt\hfill \\ & & +{\displaystyle \gamma e}\int tf\int \Omega \nabla \left(\varphi H\right)e·\nabla \left(\varphi H\right)edxdydzdt\hfill \\ & & +{\displaystyle \gamma e}\sum j=\int tf\int \Omega \nabla \left(\varphi OH-\right)e·\nabla \left(\varphi OH-\right)edxdzdt\hfill \\ & & +{\displaystyle \gamma p}\sum j=\int tf\int \Omega \nabla \left(\varphi O\right)pj·\nabla \left(\varphi O\right)pjdxdydzdt\hfill \\ & & +{\displaystyle \gamma e}\sum j=\int tf\int \Omega \nabla \left(\varphi O\right)ej·\nabla \left(\varphi O\right)ejdxdydzdt\hfill \\ & & +{\displaystyle \gamma p}\sum j=\int tf\int \Omega \nabla \left(\varphi H\right)p·\nabla \left(\varphi O\right)pdxdydt\hfill \\ & & +{\displaystyle \gamma AH}\sum j=\int tf\int \Omega \nabla \left(\varphi H\right)Aj·\nabla \left(\varphi H\right)Ajdydzdt\hfill \\ & & +{\displaystyle \gamma AO}\int tf\int \Omega \nabla \left(\varphi O\right)A·\nabla \left(\varphi O\right)Adydzdt\hfill \\ & & +{\displaystyle \gamma AH}\int tf\int \Omega \nabla \left(\varphi H\right)A·\nabla \left(\varphi H\right)Adydzdt\hfill \\ & & +{\displaystyle \gamma AO}\int tf\int \Omega \nabla \left(\varphi O\right)A·\nabla \left(\varphi O\right)Adydzdt\hfill \\ & & +{\displaystyle \gamma M}\int tf\int \Omega \nabla \left(\varphi \right)M·\nabla \left(\varphi \right)Mdzdt\hfill \\ & & +{\displaystyle \gamma M}\int tf\int \Omega \nabla \left(\varphi \right)M·\nabla \left(\varphi \right)Mdzdt\hfill \\ & & {\displaystyle \gamma A}\int tf\int \Omega \nabla \left(\varphi \right)A·\nabla \left(\varphi \right)Adydt.\hfill \end{array}$$

Here $\gamma p>$, $\gamma e>$, $\gamma AH>,$, $\gamma AO>,$$\gamma M>$, $\gamma M>$, $\gamma A>.$

Moreover,

$$\begin{array}{ccc}& & F\left(\varphi \right)\hfill \\ & =& {\displaystyle \alpha }\int tf\int \Omega {\displaystyle |\varphi (x-\xi ,y-\xi ,z-\xi ,t)|\left|\varphi (\xi ,\xi ,\xi ,t)\right||(x,y,z)-(\xi ,\xi ,\xi )|}dxdydzdxdxdxdt\hfill \\ & & +{\displaystyle \alpha }\int tf\int \Omega {\displaystyle |\varphi (x-\xi ,y-\xi ,z-\xi ,t)|\left|\varphi (\xi ,\xi ,\xi ,t)\right||(x,y,z)-(\xi ,\xi ,\xi )|}dxdydzdxdxdxdt\hfill \\ & & +{\displaystyle \alpha }\int tf\int \Omega {\displaystyle |\varphi (x-\xi ,z-\xi ,t)|\left|\varphi (\xi ,\xi ,t)\right||(x,y,z)-(\xi ,\xi ,\xi )|}dxdydzdxdxdt\hfill \\ & & +{\displaystyle \alpha 23}\int tf\int \Omega {\displaystyle |\varphi (x-\xi ,y-\xi ,z-\xi ,t)|\left|\varphi (\xi ,\xi ,t)\right||(x,y,z)-(\xi ,\xi ,\xi )|}dxdydzdxdxdxdt\hfill \end{array}$$

where $\alpha >$, $\alpha >$, $\alpha >$ and $\alpha 23>.$

Furthermore,

$$\begin{array}{ccc}\hfill F\left(\varphi \right)& =& \int tf\int \Omega V(x,y,z,t)\left(\right|\varphi |+|\varphi |+|\varphi \left|\right)dxdydzdt,\hfill \end{array}$$

(127)

where $V=V(x,y,z,t)$ is an electric potential originated from an external electric field $\mathbf{E}$ applied on $\Omega .$

Finally,

$$\begin{array}{ccc}& & JAux(\varphi ,E)\hfill \\ & =& \sum j=\int tf\int \Omega \left(E\right)pjH(y,z,t)\left(\int \Omega \left|\left(\varphi H\right)pj(x,y,z,t)\right|dx-mp\right)dydzdt\hfill \\ & & +\sum j=\int tf\int \Omega \left(E\right)ejH(y,z,t)\left(\int \Omega \left|\left(\varphi H\right)ej(x,y,z,t)\right|dx-me\right)dydzdt\hfill \\ & & +\sum j=\int tf\int \Omega \left(E\right)pjO(y,z,t)\left(\int \Omega \left|\left(\varphi O\right)pj(x,y,z,t)\right|dx-mp\right)dydzdt\hfill \\ & & +\sum j=\int tf\int \Omega \left(E\right)ejO(y,z,t)\left(\int \Omega \left|\left(\varphi O\right)ej(x,y,z,t)\right|dx-me\right)dydzdt\hfill \\ & & +\int tf\int \Omega \left(E\right)pH(y,z,t)\left(\int \Omega \left|\left(\varphi H\right)p(x,y,z,t)\right|dx-mp\right)dydzdt\hfill \\ & & +\sum j=\int tf\int \Omega \left(E\right)pjO(y,z,t)\left(\int \Omega \left|\left(\varphi O\right)pj(x,y,z,t)\right|dx-mp\right)dydzdt\hfill \\ & & +\sum j=\int tf\int \Omega \left(E\right)ejO(y,z,t)\left(\int \Omega \left|\left(\varphi O\right)ej(x,y,z,t)\right|dx-me\right)dydzdt\hfill \\ & & +\int tf\int \Omega \left(E\right)pH(y,t)\left(\int \Omega \left|\left(\varphi H\right)p(x,y,t)\right|dx-mp\right)dydt\hfill \\ & & +\sum j=\int tf\int \Omega \left(E\right)AjH(z,t)\left(\int \Omega \left(\right|\left(\varphi \right)AjH(y,z,t)|dy-mAjH\right)dzdt\hfill \\ & & +\int tf\int \Omega \int \Omega \left(E\right)AO(z,t)\left(\int \Omega \left(\right|\left(\varphi \right)AO(y,z,t)|dy-mAO\right)dzdt\hfill \\ & & +\int tf\int \Omega \left(E\right)AH(z,t)\left(\int \Omega \left(\right|\left(\varphi \right)AH(y,z,t)|dy-mAH\right)dzdt\hfill \\ & & +\int tf\int \Omega \left(E\right)AO(z,t)\left(\int \Omega \left(\right|\left(\varphi \right)AO(y,z,t)|dy-mAO\right)dzdt\hfill \\ & & \int tf\left(E\right)AH\left(t\right)\left(\int \Omega \left(\right|\left(\varphi \right)AH(y,t)|dy-mAH\right)dt\hfill \\ & & +\int tf\left(E\right)\left(t\right)\left(\int \Omega \left(\right|\left(\varphi \right)M(z,t)|+|\left(\varphi \right)M(z,t)|+|\left(\varphi \right)M(z,t)\left|\right)dz-mT\right)dt\hfill \\ & & +\int tf\left(E\right)\left(t\right)\left(\int \Omega (\alpha C|\left(\varphi \right)M(z,t)|-\alpha B|\left(\varphi \right)M(z,t)\left|\right)dz\right)dt.\hfill \end{array}$$

(128)

24. A Mathematical Model for the Austenite and Martensite Phase Transition

In this section we consider a phase transition of a solid solution of $\gamma -Fe$ ($\gamma -iron$) and carbon with a $.75/100$ proportion of carbon, known as austenite, initially at a temperature above and close to $723C$ and rapidly cooled to a temperature of about $25C$, developing a phase transition which generates a solid solution of $\alpha -Fe$ ($\alpha -iron$) and carbon known as martensite.

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular boundary denoted by $\partial \Omega$ which contains an amount of austenite at $723C$ and which, as previously mentioned, is rapidly cooled to a temperature $25C$ on a time interval $[,tf],$ resulting a phase known as martensite.

We recall the $\gamma -Fe$ of austenite phase presents a multi-faced cubic crystalline structure in a micro-structure with carbon atoms.

On the other hand, $\alpha -Fe$ structure of the martensite phase has a $CCC$ cubic centralized crystalline structure in a micro-structure with carbon atoms.

At this point, we also recall that the $F$ (iron) atom has 26 protons, 26 electrons and 30 neutrons.

On the other hand a $Carbon12$ atom has 6 protons and this same number of electrons and neutrons.

Here we define the density function $\varphi$, representing the Austenite phase, where:

$$\begin{array}{ccc}\hfill |\varphi (x,y,z,t)|& =& \sum j=26|\varphi pj\gamma -Fe(x,y,z,t)||\varphi A\gamma -Fe(y,z,t)|\left|\varphi \gamma (z,t)\right|{\displaystyle \left(mA\gamma \right)}\hfill \\ & & +\sum j=26|\varphi ej\gamma -Fe(x,y,z,t)||\varphi A\gamma -Fe(y,z,t)|\left|\varphi \gamma (z,t)\right|{\displaystyle \left(mA\gamma \right)}\hfill \\ & & +\sum j=30|\varphi Nj\gamma -Fe(x,y,z,t)||\varphi A\gamma -Fe(y,z,t)|\left|\varphi \gamma (z,t)\right|{\displaystyle \left(mA\gamma \right)}\hfill \\ & & +\sum j=|\left(\varphi C\right)pj(x,y,z,t)||\left(\varphi C\right)A(y,z,t)|\left|\varphi C(z,t)\right|{\displaystyle \left(mAC\right)}\hfill \\ & & +\sum j=|\left(\varphi C\right)ej(x,y,z,t)||\left(\varphi C\right)A(y,z,t)|\left|\varphi C(z,t)\right|{\displaystyle \left(mAC\right)}\hfill \\ & & +\sum j=|\left(\varphi C\right)Nj(x,y,z,t)||\left(\varphi C\right)A(y,z,t)|\left|\varphi C(z,t)\right|{\displaystyle \left(mAC\right)}.\hfill \end{array}$$

(129)

Similarly, we define the density function for the Martensite phase, which is denoted by $\varphi$, where:

$$\begin{array}{ccc}\hfill |\varphi (x,y,z,t)|& =& \sum j=26|\varphi pj\alpha -Fe(x,y,z,t)||\varphi A\gamma -Fe(y,z,t)|\left|\varphi \alpha (z,t)\right|{\displaystyle \left(mA\alpha \right)}\hfill \\ & & +\sum j=26|\varphi ej\alpha -Fe(x,y,z,t)||\varphi A\alpha -Fe(y,z,t)|\left|\varphi \gamma (z,t)\right|{\displaystyle \left(mA\alpha \right)}\hfill \\ & & +\sum j=30|\varphi Nj\alpha -Fe(x,y,z,t)||\varphi A\alpha -Fe(y,z,t)|\left|\varphi \alpha (z,t)\right|{\displaystyle \left(mA\alpha \right)}\hfill \\ & & +\sum j=|\left(\varphi C\right)pj(x,y,z,t)||\left(\varphi C\right)A(y,z,t)|\left|\varphi C(z,t)\right|{\displaystyle \left(mAC\right)}\hfill \\ & & +\sum j=|\left(\varphi C\right)ej(x,y,z,t)||\left(\varphi C\right)A(y,z,t)|\left|\varphi C(z,t)\right|{\displaystyle \left(mAC\right)}\hfill \\ & & +\sum j=|\left(\varphi C\right)Nj(x,y,z,t)||\left(\varphi C\right)A(y,z,t)|\left|\varphi C(z,t)\right|{\displaystyle \left(mAC\right)}.\hfill \end{array}$$

(130)

For the $CFC$$\gamma -Fe$$(\gamma -iron)$ corresponding to the Austenite phase, such density functions are subject to the following constraints:

Defining

$$C\gamma =\{(\epsilon ,,),(,\epsilon ,),(,,\epsilon ),:\epsilon j\in \{+,-\},\forall j\in \{,,\}\},$$

$$\left(C\gamma \right)=\{(\epsilon ,\epsilon ,\epsilon ),:\epsilon j\in \{+,-\},\forall j\in \{,,\}\},$$

and

$$\left(C\gamma \right)=\{(\epsilon ,\epsilon ,),(\epsilon ,,\epsilon ),(,\epsilon ,\epsilon ),:\epsilon j\in \{+,-\},\forall j\in \{,,\}\},$$

we must have

$$\varphi A\gamma -Fe(y,z+\epsilon \delta z,z+\epsilon \delta z,z+\epsilon \delta z,t)=\varphi A\gamma -Fe(y,z+\epsilon ˜\delta z,z+\epsilon ˜\delta z,z+\epsilon ˜\delta z,t),$$

$\forall \epsilon ,\epsilon ˜\in C\gamma ,$ where $\delta z\in \mathbb{R}+$ is a small real parameter related to $\gamma -Fe$ crystalline structure dimensions.

We must have also,

$$\varphi A\gamma -Fe(y,z+\epsilon \delta z,z+\epsilon \delta z,z+\epsilon \delta z,t)=\varphi A\gamma -Fe(y,z+\epsilon ˜\delta z,z+\epsilon ˜\delta z,z+\epsilon ˜\delta z,t),$$

$\forall \epsilon ,\epsilon ˜\in \left(C\gamma \right)$ and,

$$\left(\varphi C\right)A(y,z+\epsilon \delta z,z+\epsilon \delta z,z+\epsilon \delta z,t)=\left(\varphi C\right)A(y,z+\epsilon ˜\delta z,z+\epsilon ˜\delta z,z+\epsilon ˜\delta z,t),$$

$\forall \epsilon ,\epsilon ˜\in \left(C\gamma \right).$

For the $CCC$$\alpha -Fe$$(\alpha -iron)$ corresponding to the Austenite phase, such density functions are subject to the following constraints:

Defining

$$C\alpha =\{(\epsilon ,\epsilon ,\epsilon ),:\epsilon j\in \{+,-\},\forall j\in \{,,\}\},$$

$$\left(C\alpha \right)=\{(\epsilon ,\epsilon ,\epsilon ),:\epsilon ,\epsilon \in \{+,-\}\mathrm{and}\epsilon =\},$$

$$\left(C\alpha \right)=\{(\epsilon ,\epsilon ,\epsilon ),:\epsilon =\epsilon =\mathrm{and}\epsilon \in \{+,-\}\},$$

we must have

$$\varphi A\alpha -Fe(y,z+\epsilon \delta ^z,z+\epsilon \delta ^z,z+\epsilon \delta ^z,t)=\varphi A\alpha -Fe(y,z+\epsilon ˜\delta ^z,z+\epsilon ˜\delta ^z,z+\epsilon ˜\delta ^z,t),$$

$\forall \epsilon ,\epsilon ˜\in C\alpha ,$ where $\delta ^z\in \mathbb{R}+$ is a small real parameter related to $\alpha -Fe$ crystalline structure dimensions.

We must have also,

$$\left(\varphi C\right)A(y,z+\epsilon \delta ^z,z+\epsilon \delta ^z,z+\epsilon \delta ^z,t)=\left(\varphi C\right)A(y,z+\epsilon ˜\delta ^z,z+\epsilon ˜\delta ^z,z+\epsilon ˜\delta ^z,t),$$

$\forall \epsilon ,\epsilon ˜\in \left(C\alpha \right)\cup \left(C\alpha \right).$

The other constraints for the densities are given by:

For the Austenite phase:

$$\int \Omega \left|\varphi pj\gamma -Fe(x,y,z,t)\right|dx=mp,\forall j\in \{,26\},$$

$$\int \Omega \left|\varphi ej\gamma -Fe(x,y,z,t)\right|dx=me,\forall j\in \{,26\},$$

$$\int \Omega \left|\varphi Nj\gamma -Fe(x,y,z,t)\right|dx=mN,\forall j\in \{,30\},$$

$$\int \Omega \left|\varphi A\gamma -Fe(x,y,z,t)\right|dx=mA\gamma ,$$

$$\int \Omega \left|\left(\varphi C\right)pj(x,y,z,t)\right|dx=mp,\forall j\in \{,\},$$

$$\int \Omega \left|\left(\varphi C\right)ej(x,y,z,t)\right|dx=me,\forall j\in \{,\},$$

$$\int \Omega \left|\left(\varphi C\right)Nj(x,y,z,t)\right|dx=mN,\forall j\in \{,\},$$

$$\int \Omega \left|\left(\varphi C\right)A(x,y,z,t)\right|dx=mAC,$$

For the Martensite phase:

$$\int \Omega \left|\varphi pj\alpha -Fe(x,y,z,t)\right|dx=mp,\forall j\in \{,26\},$$

$$\int \Omega \left|\varphi ej\alpha -Fe(x,y,z,t)\right|dx=me,\forall j\in \{,26\},$$

$$\int \Omega \left|\varphi Nj\alpha -Fe(x,y,z,t)\right|dx=mN,\forall j\in \{,30\},$$

$$\int \Omega \left|\varphi A\alpha -Fe(x,y,z,t)\right|dx=mA\alpha ,$$

$$\int \Omega \left|\left(\varphi C\right)pj(x,y,z,t)\right|dx=mp,\forall j\in \{,\},$$

$$\int \Omega \left|\left(\varphi C\right)ej(x,y,z,t)\right|dx=me,\forall j\in \{,\},$$

$$\int \Omega \left|\left(\varphi C\right)Nj(x,y,z,t)\right|dx=mN,\forall j\in \{,\},$$

$$\int \Omega \left|\left(\varphi C\right)A(x,y,z,t)\right|dx=mAC.$$

For the total $F$ (iron) mass,

$$\int \Omega |\varphi \gamma (z,t)|dz+\int \Omega \left|\varphi \gamma (z,t)\right|dz=\left(mFe\right)T,$$

For the total Carbon mass

$$\int \Omega |\varphi C(z,t)|dz+\int \Omega \left|\varphi C(z,t)\right|dz=\left(mC\right)T.$$

At this point we define the functional J which models such a pahse transition in question, where

$$J(\varphi ,E)=G(\nabla \varphi )+F\left(\varphi \right)+F\left(\varphi \right)+JAux(\varphi ,E)$$

where

$$\begin{array}{ccc}\hfill G(\nabla \varphi )& =& \sum j=26{\displaystyle \gamma ^p\gamma -Fe}\int tf\int \Omega \nabla \varphi pj\gamma -Fe·\nabla \varphi pj\gamma -Fedxdydzdt\hfill \\ & & +\sum j=26{\displaystyle \gamma ^e\gamma -Fe}\int tf\int \Omega \nabla \varphi ej\gamma -Fe·\nabla \varphi ej\gamma -Fedxdydzdt\hfill \\ & & +\sum j=30{\displaystyle \gamma ^N\gamma -Fe}\int tf\int \Omega \nabla \varphi Nj\gamma -Fe·\nabla \varphi Nj\gamma -Fedxdydzdt\hfill \\ & & +\sum j=26{\displaystyle \gamma ^p\alpha -Fe}\int tf\int \Omega \nabla \varphi pj\alpha -Fe·\nabla \varphi pj\alpha -Fedxdydzdt\hfill \\ & & +\sum j=26{\displaystyle \gamma ^e\alpha -Fe}\int tf\int \Omega \nabla \varphi ej\alpha -Fe·\nabla \varphi ej\gamma -Fedxdydzdt\hfill \\ & & +\sum j=30{\displaystyle \gamma ^N\alpha -Fe}\int tf\int \Omega \nabla \varphi Nj\alpha -Fe·\nabla \varphi Nj\alpha -Fedxdydzdt\hfill \\ & & +{\displaystyle \gamma ^A\gamma }\int tf\int \Omega (\nabla \varphi A\gamma -Fe(y,z,t)·\nabla \varphi A\gamma -Fe(y,z,t))dydzdt\hfill \\ & & +{\displaystyle \gamma ^A\alpha }\int tf\int \Omega (\nabla \varphi A\alpha -Fe(y,z,t)·\nabla \varphi A\alpha -Fe(y,z,t))dydzdt\hfill \\ & & +\sum j={\displaystyle \gamma ^pC}\int tf\int \Omega \nabla \left(\varphi C\right)pj·\nabla \left(\varphi C\right)pjdxdydzdt\hfill \\ & & +\sum j={\displaystyle \gamma ^eC}\int tf\int \Omega \nabla \left(\varphi C\right)ej·\nabla \left(\varphi C\right)ejdxdydzdt\hfill \\ & & +\sum j={\displaystyle \gamma ^NC}\int tf\int \Omega \nabla \left(\varphi C\right)Nj·\nabla \left(\varphi C\right)Njdxdydzdt\hfill \\ & & +\sum j={\displaystyle \gamma ^pC}\int tf\int \Omega \nabla \left(\varphi C\right)pj·\nabla \left(\varphi C\right)pjdxdydzdt\hfill \\ & & +\sum j={\displaystyle \gamma ^eC}\int tf\int \Omega \nabla \left(\varphi C\right)ej·\nabla \left(\varphi C\right)ejdxdydzdt\hfill \\ & & +\sum j={\displaystyle \gamma ^NC}\int tf\int \Omega \nabla \left(\varphi C\right)Nj·\nabla \left(\varphi C\right)Njdxdydzdt\hfill \\ & & +{\displaystyle \gamma ^AC}\int tf\int \Omega (\nabla \left(\varphi C\right)A·\nabla \left(\varphi C\right)A)dydzdt+{\displaystyle \gamma ^AC}\int tf\int \Omega (\nabla \left(\varphi C\right)A·\nabla \left(\varphi C\right)A)dydzdt\hfill \\ & & +{\displaystyle \gamma ^T\gamma }\int tf\int \Omega (\nabla \left(\varphi \gamma \right)·\nabla \left(\varphi \gamma \right))dzdt+{\displaystyle \gamma ^T\gamma }\int tf\int \Omega (\nabla \left(\varphi \alpha \right)·\nabla \left(\varphi \alpha \right))dzdt\hfill \\ & & +{\displaystyle \gamma ^TC}\int tf\int \Omega (\nabla \left(\varphi C\right)·\nabla \left(\varphi C\right))dzdt+{\displaystyle \gamma ^T\gamma }\int tf\int \Omega (\nabla \left(\varphi C\right)·\nabla \left(\varphi C\right))dzdt\hfill \end{array}$$

(131)

Also,

$$\begin{array}{ccc}& & F\left(\varphi \right)\hfill \\ & =& {\displaystyle \alpha ^}\int tf\int \Omega {\displaystyle |\varphi (x-\xi ,y-\xi ,z-\xi ,t)|\left|\right|\varphi (\xi ,\xi ,\xi ,t)||(x,y,z)-(\xi ,\xi ,\xi )|}dxdydzd\xi d\xi d\xi dt\hfill \\ & & +{\displaystyle \alpha ^}\int tf\int \Omega {\displaystyle |\varphi (x-\xi ,y-\xi ,z-\xi ,t)|\left|\right|\varphi (\xi ,\xi ,\xi ,t)||(x,y,z)-(\xi ,\xi ,\xi )|}dxdydzd\xi d\xi d\xi dt,\hfill \end{array}$$

$$F\left(\varphi \right)=-\int tf\int \Omega w(z,t)\left(\right|\varphi (z,t)|+\varphi (z,t)\left|\right)dzdt,$$

Finally, $JAux=JAux+JAux+JAux+JAux+JAux$, where

$$\begin{array}{ccc}\hfill JAux& =& \sum j=126\int tf\int \Omega Epj\gamma -Fe(y,z,t)\left(\int \Omega \left|\varphi pj\gamma -Fe(x,y,z,t)\right|dx-mp\right)dydzdt\hfill \\ & & +\sum j=126\int tf\int \Omega Eej\gamma -Fe(y,z,t)\left(\int \Omega \left|\varphi ej\gamma -Fe(x,y,z,t)\right|dx-me\right)dydzdt\hfill \\ & & +\sum j=130\int tf\int \Omega ENj\gamma -Fe(y,z,t)\left(\int \Omega \left|\varphi Nj\gamma -Fe(x,y,z,t)\right|dx-mN\right)dydzdt\hfill \\ & & \sum j=26\int tf\int \Omega Epj\alpha -Fe(y,z,t)\left(\int \Omega \left|\varphi pj\alpha -Fe(x,y,z,t)\right|dx-mp\right)dydzdt\hfill \\ & & +\sum j=26\int tf\int \Omega Eej\alpha -Fe(y,z,t)\left(\int \Omega \left|\varphi ej\alpha -Fe(x,y,z,t)\right|dx-me\right)dydzdt\hfill \\ & & +\sum j=30\int tf\int \Omega ENj\alpha -Fe(y,z,t)\left(\int \Omega \left|\varphi Nj\alpha -Fe(x,y,z,t)\right|dx-mN\right)dydzdt\hfill \\ & & +\int tf\int \Omega EA\gamma -Fe(y,t)\left(\int \Omega \left|\varphi A\gamma -Fe(y,z,t)\right|dy-mA\gamma \right)dzdt\hfill \\ & & +\int tf\int \Omega EA\alpha -Fe(y,t)\left(\int \Omega \left|\varphi A\alpha -Fe(y,z,t)\right|dy-mA\alpha \right)dzdt\hfill \end{array}$$

(132)

$$\begin{array}{ccc}\hfill JAux& =& \sum j=26\int tf\int \Omega \left(EC\right)pj(y,z,t)\left(\int \Omega \left|\left(\varphi C\right)pj(x,y,z,t)\right|dx-mp\right)dydzdt\hfill \\ & & \sum j=26\int tf\int \Omega \left(EC\right)ej(y,z,t)\left(\int \Omega \left|\left(\varphi C\right)ej(x,y,z,t)\right|dx-me\right)dydzdt\hfill \\ & & \sum j=26\int tf\int \Omega \left(EC\right)Nj(y,z,t)\left(\int \Omega \left|\left(\varphi C\right)Nj(x,y,z,t)\right|dx-mN\right)dydzdt\hfill \\ & & \sum j=26\int tf\int \Omega \left(EC\right)pj(y,z,t)\left(\int \Omega \left|\left(\varphi C\right)pj(x,y,z,t)\right|dx-mp\right)dydzdt\hfill \\ & & \sum j=26\int tf\int \Omega \left(EC\right)ej(y,z,t)\left(\int \Omega \left|\left(\varphi C\right)ej(x,y,z,t)\right|dx-me\right)dydzdt\hfill \\ & & \sum j=26\int tf\int \Omega \left(EC\right)Nj(y,z,t)\left(\int \Omega \left|\left(\varphi C\right)Nj(x,y,z,t)\right|dx-mN\right)dydzdt\hfill \\ & & \int tf\int \Omega \left(EC\right)A(y,t)\left(\int \Omega \left|\left(\varphi C\right)A(y,z,t)\right|dy-mAC\right)dzdt\hfill \\ & & +\int tf\int \Omega \left(EC\right)A(y,t)\left(\int \Omega \left|\left(\varphi C\right)A(y,z,t)\right|dy-mAC\right)dzdt\hfill \end{array}$$

(133)

and,

$$\begin{array}{ccc}\hfill JAux& =& \int tfE\gamma ,\alpha \left(t\right)\left(\int \Omega \left(\right|\varphi \gamma (z,t)|+|\varphi \alpha (z,t)\left|\right)dz-\left(mFe\right)T\right)dt\hfill \\ & & +\int tfEC\left(t\right)\left(\int \Omega \left(\right|\varphi C(z,t)|+|\varphi C(z,t)\left|\right)dz-\left(mC\right)T\right)dt.\hfill \end{array}$$

(134)

$$\begin{array}{ccc}& & JAux\hfill \\ & =& +\sum _{\epsilon ,\epsilon ˜\in C\gamma }\int tf\int \Omega E\epsilon ,\epsilon ˜(y,z,t)(\varphi A\gamma -Fe(y,z+\epsilon \delta z,z+\epsilon \delta z,z+\epsilon \delta z,t)\hfill \\ & & -\varphi A\gamma -Fe(y,z+\epsilon ˜\delta z,z+\epsilon ˜\delta z,z+\epsilon ˜\delta z,t))dydzdt\hfill \\ & & \sum _{\epsilon ,\epsilon ˜\in \left(C\gamma \right)}\int tf\int \Omega E\epsilon ,\epsilon ˜(y,z,t)\varphi A\gamma -Fe(y,z+\epsilon \delta z,z+\epsilon \delta z,z+\epsilon \delta z,t)\hfill \\ & & -\varphi A\gamma -Fe(y,z+\epsilon ˜\delta z,z+\epsilon ˜\delta z,z+\epsilon ˜\delta z,t))dydzdt\hfill \\ & & +\sum _{\epsilon ,\epsilon ˜\in \left(C\gamma \right)}\int tf\int \Omega E\epsilon ,\epsilon ˜(y,z,t)\left(\varphi C\right)A(y,z+\epsilon \delta z,z+\epsilon \delta z,z+\epsilon \delta z,t)\hfill \\ & & -\left(\varphi C\right)A(y,z+\epsilon ˜\delta z,z+\epsilon ˜\delta z,z+\epsilon ˜\delta z,t))dydzdt\hfill \\ & & +\sum _{\epsilon ,\epsilon ˜\in \left(C\alpha \right)}\int tf\int \Omega E\epsilon ,\epsilon ˜(y,z,t)(\varphi A\alpha -Fe(y,z+\epsilon \delta ^z,z+\epsilon \delta ^z,z+\epsilon \delta ^z,t)\hfill \\ & & -\varphi A\alpha -Fe(y,z+\epsilon ˜\delta ^z,z+\epsilon ˜\delta ^z,z+\epsilon ˜\delta ^z,t))dydzdt\hfill \\ & & +\sum _{\epsilon ,\epsilon ˜\in \left(C\alpha \right)\cup \left(C\alpha \right)}\int tf\int \Omega E\epsilon ,\epsilon ˜(y,z,t)(\left(\varphi C\right)A(y,z+\epsilon \delta ^z,z+\epsilon \delta ^z,z+\epsilon \delta ^z,t)\hfill \\ & & -\left(\varphi C\right)A(y,z+\epsilon ˜\delta ^z,z+\epsilon ˜\delta ^z,z+\epsilon ˜\delta ^z,t))dydzdt.\hfill \end{array}$$

(135)

Finally, for a field of displacements $u=(u,u,u)$ resulting from the action of a external load field $f=(f,f,f)$ and temperature variations, we define

$$\begin{array}{ccc}& & JAux\hfill \\ & =& {\displaystyle 12}\int tf\int \Omega \left(\Lambda (x,t)Hijkl\left((eij\left(u\right)-eij\left(w\right))(ekl\left(u\right)-ekl\left(w\right))\right)\right.\hfill \\ & & +\left.\Lambda (z,t)Hijkl\left((eij\left(u\right)-eij\left(w\right))(ekl\left(u\right)-ekl\left(w\right))\right)\right)dxdt\hfill \\ & & -{\displaystyle 12}\int tf\int \Omega \rho (x,t)ut(x,t)·ut(x,t)dxdt\hfill \\ & & -\langle ui,fi\rangle L,\hfill \end{array}$$

(136)

where

$$eij\left(u\right)={\displaystyle 12}\left({\displaystyle \partial ui\partial xj}+{\displaystyle \partial uj\partial xi}\right),$$

$$\rho (z,t)=\int \Omega \left|\varphi (x,y,z,t)\right|dxdy,$$

$$\rho (z,t)=\int \Omega \left|\varphi (x,y,z,t)\right|dxdy,$$

$$\rho (z,t)=\rho (z,t)+\rho (z,t),$$

and

$$\Lambda (z,t)={\displaystyle \rho (z,t)\rho (z,t)+\rho (z,t)},$$

$$\Lambda (z,t)={\displaystyle \rho (z,t)\rho (z,t)+\rho (z,t)}.$$

Remark 12.

The system temperature is supposed to be directly proportional to $w(z,t)$, which in this model is a known function obtained experimentally. Finally, the strain tensors $\left\{eij\left(w\right)\right\}$ and $\left\{eij\left(w\right)\right\}$ refer to austenite and martensite phases, respectively. Such tensors also depend on the temperature and must be also obtained experimentally.

25. A Note on Classical Free Fields Through a Variational Perspective

This section is strongly based on the first chapter of the book [20], by N.N. Bogoliubov and D.V. Shirkov.

Therefore, the credit for this section is of these mentioned authors. This section is a kind of review of such a book chapter indicated. In fact, what we have done is simply to open more and clarify some calculations, specially about the first variation of the functional L, in order to improve their understanding.

Let $\Omega =\Omega ^\times [,T]\subset \mathbb{R}$ where $\Omega ^\subset \mathbb{R}$ is a bounded, open and connected set with a regular boundary denoted by $\partial \Omega ^.$

Consider the Lagrangian density $L:\mathbb{R}N\times \mathbb{R}N\times n\to \mathbb{R}$ and an action $A:V\to \mathbb{R}$ where

$$A\left(u\right)=\int \Omega L(u,\nabla u)dx,$$

$V=W,(\Omega ;\mathbb{R}N)$.

We denote

$$\nabla u=\left\{{\displaystyle \partial ui\partial xj}\right\}$$

and

$${\displaystyle \partial ui\partial xj}=\left(ui\right)xj.$$

Assume $u\in V$ is such that

$$\delta L(u,\nabla u)=\mathbf{0},$$

so that

$${\displaystyle \partial L(u,\nabla u)\partial ui}-\sum k=n{\displaystyle ddxk}\left({\displaystyle \partial L(u,\nabla u)\partial \left(ui\right)xk}\right)=,\mathrm{in}\Omega ,\forall i\in \{,\dots ,N\}.$$

We define a change of variables

$$(x\prime )k=xk+\delta xk,$$

where $xk=(x,x,x,x)$ and $x=t$ (here t denotes time).

Also

$$gjk=,\mathrm{if}j\ne k,g00=-\mathrm{and}g11=g22=g33=,\left\{gjk\right\}=\left\{gjk\right\}-,$$

$$\delta xk=\sum j=NXjk\epsilon wj,$$

where $\left|\epsilon \right|\ll$ denotes a small real parameter.

We define also

$$ui\prime (x\prime )=ui\left(x\right)+\delta ui\left(x\right),$$

where

$$\delta ui\left(x\right)=\sum j=N\psi ij\epsilon wj,$$

and

$$\delta ui¯=ui\prime \left(x\right)-ui\left(x\right).$$

Observe that

$$\begin{array}{ccc}\hfill \delta ui\left(x\right)& =& ui\prime \left(x\right)-ui\left(x\right)\hfill \\ & =& ui\prime (x\prime )-ui\prime \left(x\right)+ui\prime \left(x\right)-ui\left(x\right),\hfill \end{array}$$

(137)

so that

$$\begin{array}{ccc}\hfill \delta ui\left(x\right)¯& =& ui\prime \left(x\right)-ui\left(x\right)\hfill \\ & =& \delta ui\left(x\right)-(ui\prime (x\prime )-ui\prime \left(x\right))\hfill \\ & & \sum j=N\psi ij\epsilon wj-\sum k=n{\displaystyle \partial ui\prime (x˜i)dxk}\delta xk\hfill \\ & =& \sum j=N\psi ij\epsilon wj-\sum k=n{\displaystyle \partial ui\left(x\right)dxk}\delta xk+\mathcal{O}\left(\epsilon \right).\hfill \end{array}$$

(138)

Summarizing, we have got

$$\delta ui\left(x\right)¯=\epsilon \left(\sum j=N\left(\psi ijwj-\sum k=n{\displaystyle \partial ui\left(x\right)dxk}XjKwj\right)\right)+\mathcal{O}\left(\epsilon \right).$$

Define now

$$A˜(u,\phi ,\phi ,\epsilon )=\int \Omega L[u(x+\epsilon \phi \left(x\right))+\epsilon \phi \left(x\right)]detJ\left(x\right)dx.$$

where we have generically denoted

$$L\left[u\right]\equiv L(u,\nabla u),$$

$$L[u(x+\epsilon \phi \left(x\right))+\epsilon \phi \left(x\right)]\equiv L(u(x+\epsilon \phi \left(x\right))+\epsilon \phi \left(x\right),\nabla u(x+\epsilon \phi \left(x\right))+\epsilon \nabla \phi \left(x\right)),$$

and

$$\begin{array}{ccc}\hfill J\left(x\right)& =& \left\{{\displaystyle \partial xj\prime \partial xk}\right\}\hfill \\ & =& \left\{{\displaystyle \partial (xj+\epsilon \left(\phi \right)j\left(x\right))\partial xk}\right\}\hfill \\ & =& \left\{\delta jk+\epsilon {\displaystyle \partial \left(\phi \right)j\left(x\right)\partial xk}\right\}.\hfill \end{array}$$

(139)

From such a last definition we have

$$detJ\left(x\right)=+\epsilon \sum k=n{\displaystyle \partial \left(\phi \right)k\left(x\right)\partial xk}+\mathcal{O}\left(\epsilon \right).$$

so that

$${\displaystyle \partial detJ\left(x\right)\partial \epsilon }|\epsilon ==\sum k=n{\displaystyle \partial \left(\phi \right)k\left(x\right)\partial xk},$$

At this point we define

$$\delta A(u,\phi ,\phi )={\displaystyle dd\epsilon }\left(A˜(u,\phi ,\phi ,\epsilon )\right)|\epsilon =,$$

so that

$$\begin{array}{ccc}\hfill \delta A(u,\phi ,\phi )& =& \int \Omega \left(\sum i=N\left({\displaystyle \partial L(u,\nabla u)\partial ui}\left(\phi \right)i\right.\right.\hfill \\ & & +\sum k=n\left({\displaystyle \partial L(u,\nabla u)\partial \left(ui\right)xk}\left(\left(\phi \right)i\right)xk\right)\hfill \\ & & \left.\left.+\sum k=n{\displaystyle \delta L\left[u\right]\delta ui}{\displaystyle \partial ui\partial xk}\left(\phi \right)k\right)+\sum k=nL\left[u\right]{\displaystyle \partial \left(\phi \right)k\partial xk}\right)dx.\hfill \end{array}$$

(140)

From this and

$${\displaystyle \partial L(u,\nabla u)\partial ui}-{\displaystyle ddxk}\left({\displaystyle \partial L(u,\nabla u)\partial uxk}\right)=,\mathrm{in}\Omega ,\forall i\in \{,\dots ,N\},$$

we obtain

$$\begin{array}{ccc}\hfill \delta A(u,\phi ,\phi )& =& \sum i=N\sum k=n\left(\int \Omega {\displaystyle ddxk}\left({\displaystyle \partial L\left[u\right]\partial \left(ui\right)xk}\left(\phi \right)k\right)\right)dx\hfill \\ & & +\sum k=n\int \Omega {\displaystyle d\left(L\left[u\right]\left(\phi \right)k\right)dxk}dx.\hfill \end{array}$$

(141)

In particular, for

$$\left(\phi \right)k=\sum j=NXjkwj$$

and

$$\left(\phi \right)i=\sum j=N\left(\psi ijwj-\sum k=n{\displaystyle \partial ui\partial xk}Xjkwj\right),$$

we obtain

$$\begin{array}{ccc}& & \delta A(u,\phi ,\phi )\hfill \\ & =& \sum i=N\sum k=n\int \Omega \left({\displaystyle ddxk}\left({\displaystyle \partial L\left[u\right]\partial \left(ui\right)k}\left(\sum j=N\left(\psi ij-\sum l=n{\displaystyle \partial ui\left(x\right)\partial xl}Xjlwj\right)\right)\right)\right)dx\hfill \\ & & +\sum k=n\int \Omega {\displaystyle \partial L\left[u\right]Xjkwjdxk}dx\hfill \\ & =& \sum j=N\left(\sum k=n\left(\int \Omega {\displaystyle ddxk}\left(\sum i=N{\displaystyle \partial L\left[u\right]\partial \left(ui\right)k}\left(\sum j=N\left(\psi ij-\sum l=n{\displaystyle \partial ui\left(x\right)\partial xl}Xjlwj\right)\right)\right.\right.\right.\hfill \\ & & \left.\left.\left.+L\left[u\right]Xjkwj\right)dx\right)\right).\hfill \end{array}$$

(142)

Moreover, we define

$$\theta kj=\sum i=N\left({\displaystyle \partial L\left[u\right]\partial \left(ui\right)xk}\left(-\psi ij+\sum l=n{\displaystyle \partial ui\partial xl}Xjl\right)\right)-L\left(u\right)Xjk$$

so that

$$\delta A(u,\phi ,\phi )=-\int \Omega \sum j=N\sum k=n{\displaystyle d\left(\theta jkwj\right)dxk}dx,$$

$\forall \left\{wj\right\}\in Cc\infty (\Omega ;\mathbb{R}N).$

In particular, for

$$\psi ij=$$

and

$$Xjk=\delta jk$$

we obtain the Energy-Momentum tensor $T$, where

$$Tkj\equiv \theta kj=\sum i=N\sum l=n\left({\displaystyle \partial L\left[u\right]\partial \left(ui\right)xk}{\displaystyle \partial ui\partial xl}\delta jl\right)-L\left[u\right]\delta jk.$$

25.1. The Angular-Momentum tensor

In this subsection we define the following change of variables

$$xk\prime =xk+\sum _{m\ne k}gmmxm\epsilon wkm,$$

where

$$wkm=-wmk.$$

With such relations in mind, we set

$$\begin{array}{ccc}\hfill \delta xk& =& xk\prime -xk\hfill \\ & =& \epsilon \sum l=n\sum _{m<l}wml(gllxlgmk-gmmxmglk).\hfill \end{array}$$

(143)

We define also,

$$ui\prime (x\prime )=ui\left(x\right)+\delta ui\left(x\right)$$

where

$$\delta ui\left(x\right)=\sum l=n\sum _{j,p<l}Ai\left(pl\right)juj\left(x\right)\epsilon wpl.$$

Moreover, we define

$$\psi i\left(mn\right)=\sum j=nAi\left(mn\right)j,$$

where

$$Ai\left(pl\right)j=gip\delta lj-gil\delta pj.$$

Hence,

$$\psi i\left(mn\right)=\sum j=nAi\left(mn\right)juj\left(x\right)=ginum\left(x\right)-gjmun\left(x\right).$$

For the general variation, we define again

$$A˜(u,\phi ,\phi ,\epsilon )=\int \Omega L[u(x+\epsilon \phi \left(x\right))+\epsilon \phi \left(x\right)]detJ\left(x\right)dx.$$

where we have generically denoted

$$L\left[u\right]\equiv L(u,\nabla u),$$

$$L[u(x+\epsilon \phi \left(x\right))+\epsilon \phi \left(x\right)]\equiv L(u(x+\epsilon \phi \left(x\right))+\epsilon \phi \left(x\right),\nabla u(x+\epsilon \phi \left(x\right))+\epsilon \nabla \phi \left(x\right)),$$

$$\begin{array}{ccc}\hfill J\left(x\right)& =& \left\{{\displaystyle \partial xj\prime \partial xk}\right\}\hfill \\ & =& \left\{{\displaystyle \partial (xj+\epsilon \left(\phi \right)j\left(x\right))\partial xk}\right\}\hfill \\ & =& \left\{\delta jk+\epsilon {\displaystyle \partial \left(\phi \right)j\left(x\right)\partial xk}\right\}.\hfill \end{array}$$

(144)

and

$$\delta A(u,\phi ,\phi )={\displaystyle dd\epsilon }\left(A˜(u,\phi ,\phi ,\epsilon )\right)|\epsilon =,$$

Moreover, we set

$$\left(\phi \right)kml=wml(gllxl\delta mk-gmmxm\delta lk),$$

and

$$\delta ui¯=ui\prime \left(x\right)-ui\left(x\right).$$

Thus,

$$\begin{array}{ccc}\hfill \delta ui\left(x\right)& =& ui\prime \left(x\right)-ui\left(x\right)\hfill \\ & =& ui\prime (x\prime )-ui\prime \left(x\right)+ui\prime \left(x\right)-ui\left(x\right),\hfill \end{array}$$

(145)

so that

$$\begin{array}{ccc}& & \delta ui\left(x\right)¯=ui\prime \left(x\right)-ui\left(x\right)\hfill \\ & =& \delta ui\left(x\right)-(ui\prime (x\prime )-ui\prime \left(x\right))\hfill \\ & & \delta ui\left(x\right)-\sum k=n{\displaystyle \partial ui\left(x\right)dxk}\delta xk+\mathcal{O}\left(\epsilon \right)\hfill \\ & =& \delta ui\left(x\right)-\sum l=n\sum _{m<l}\sum k=n{\displaystyle \partial ui\left(x\right)\partial xk}\epsilon wml(gllxl\delta mk-gmmxm\delta lk)+\mathcal{O}\left(\epsilon \right)\hfill \\ & =& \epsilon \left(\sum l=n\sum _{j,k<l}Ai\left(kl\right)juj\left(x\right)wkl-\sum l=n\sum _{m<l}\sum k=n{\displaystyle \partial ui\left(x\right)\partial xk}wml(gllxl\delta mk-gmmxm\delta lk)\right)+\mathcal{O}\left(\epsilon \right),\hfill \end{array}$$

With such results in mind, we define

$$\begin{array}{ccc}\hfill \left(\phi \right)iml& =& \sum _{j,k<l}Ai\left(kl\right)juj\left(x\right)wml\hfill \\ & & -\sum k=n\left({\displaystyle \partial ui\left(x\right)\partial xk}wml(gllxl\delta mk-gmmxm\delta lk)\right).\hfill \end{array}$$

(146)

Similarly as in the previous section, we may obtain

$$\begin{array}{ccc}& & \delta A(u,\phi ,\phi )\hfill \\ & =& {\displaystyle dA˜(u,\phi ,\phi ,\epsilon )d\epsilon }|\epsilon =\hfill \\ & =& \sum l=n\sum _{j,m<l}\sum k=n\sum i=N\int \Omega {\displaystyle ddxk}\left({\displaystyle \partial L\left[u\right]\partial \left(ui\right)xk}(Ai(l,m)juj\left(x\right)+{\displaystyle \partial ui\partial xp}gmmxm\delta lp-{\displaystyle \partial ui\partial xp}gllxl\delta mp)wml\right)dx\hfill \\ & & +\sum k=n\sum l=n\sum _{j,m<l}\sum i=N\int \Omega {\displaystyle ddxk}\left(L\left[u\right](gllxl\delta mk-gmmxm\delta lk)wml\right)dx\hfill \end{array}$$

(147)

Thus,

$$\delta A(u,\phi ,\phi )=-\sum k=n\sum _{m<l}\int \Omega {\displaystyle ddxk}\left(Mmlkwml\right)dx,$$

where

$$\begin{array}{ccc}\hfill Mmlk& =& \sum i=N\sum _{j<l}{\displaystyle \partial L\left[u\right]\partial \left(ui\right)xk}\left(Ailmjuj-{\displaystyle \partial ui\partial xl}gmmxm+{\displaystyle \partial ui\partial xm}gllxl\right)\hfill \\ & & +L\left[u\right](gllxl\delta mk+gmmxm\delta lk),\hfill \end{array}$$

(148)

so that

$$\begin{array}{ccc}\hfill Mlmk& =& (gmmxmTlk-gllxlTmk)\hfill \\ & & -\sum i=n\sum _{j<l}{\displaystyle \partial L\left[u\right]\partial \left(ui\right)xk}Ai\left(lm\right)juj\left(x\right)\hfill \\ & =& Lmlk+Smlk,\hfill \end{array}$$

(149)

where

$$Lmlk=(gmmxmTlk-gllxlTmk)$$

and

$$Smlk=-\sum i=N\sum _{j<l}{\displaystyle \partial L\left[u\right]\partial \left(ui\right)xk}Ai\left(lm\right)juj\left(x\right).$$

The tensor $\left\{Lmlk\right\}$ is said to be the Orbital angular momentum tensor and $\left\{Smlk\right\}$ is said to be Spin one.

25.2. A note on the solution of the Klein-Gordon equation

For $\Omega =\mathbb{R}$, $\Omega =\mathbb{R}$ and denoting as usual by $i\in \mathbb{C}$ the imaginary unit, consider the Klein-Gordon equation in distributional sense

$$-{\displaystyle \partial u\partial t}+\sum j={\displaystyle \partial u\partial xj}-mu=,\mathrm{in}\Omega ,$$

where $u\in V=W,\left(\Omega \right).$

Defining the Fourier transform of u, by

$$\varphi \left(p\right)={\displaystyle \left(\pi \right)/}\int \Omega e-ip·xu\left(x\right)dx,$$

in the momenta space, the last equation is equivalent to

$$\left(p02-\sum j=pj-m\right)\varphi \left(p\right)=,\mathrm{in}\Omega ,$$

where we have denoted $p=(p,p,p,p)\in \mathbb{R},$ and $x=(x,x,x,x)\in \mathbb{R}$

Observe that a general solution for this last equation is given by the wave function

$$\varphi ^\left(p\right)=\delta \left(p02-\sum j=pj-m\right)\varphi \left(p\right),$$

where $\varphi \in W,\left(\Omega \right).$

Indeed,

$$\begin{array}{ccc}\hfill \left(p02-\sum j=pj-m\right)\varphi ^\left(p\right)& =& \left(p02-\sum j=pj-m\right)\delta \left(p02-\sum j=pj-m\right)\varphi \left(p\right)\hfill \\ & =& ,\mathrm{in}\Omega .\hfill \end{array}$$

(150)

Here, we recall that generically for the Dirac delta function $\delta \left(t\right),$ we have

$$\delta \left(t\right)=\left\{\begin{array}{cc},\hfill & \mathrm{if}t\ne ,\hfill \\ +\infty ,\hfill & \mathrm{if}t=.\hfill \end{array}\right.$$

(151)

Observe that, for the scalar case in the previous section, we have

$$T00=\sum j=\left({\displaystyle \partial u\partial xj}\right)+mu.$$

Also, from

$$-{\displaystyle \partial u\partial t}+\sum j={\displaystyle \partial u\partial xj}-mu=,\mathrm{in}\Omega ,$$

we get

$$\int \Omega \left({\displaystyle \partial u\partial t}\right)dx-\sum j=\int \Omega \left({\displaystyle \partial u\partial xj}\right)dx-m\int \Omega udx=,$$

so that

$$\int \Omega \left({\displaystyle \partial u\partial t}\right)dx=\sum j=\int \Omega \left({\displaystyle \partial u\partial xj}\right)dx+m\int \Omega udx.$$

From such results, we may infer that

$$\begin{array}{ccc}\hfill \int \Omega T00dx& =& \int \Omega \left({\displaystyle \partial u\partial t}\right)dx\hfill \\ & =& ∥{\displaystyle \partial u\partial t}∥L\hfill \\ & =& \sum j=\int \Omega \left({\displaystyle \partial u\partial xj}\right)dx+m\int \Omega udx.\hfill \end{array}$$

(152)

On the other hand,

$$\begin{array}{ccc}& & \sum j=\int \Omega \left({\displaystyle \partial u\partial xj}\right)dx\hfill \\ & =& {\displaystyle \left(\pi \right)}\sum j=\int \Omega \left(\int \Omega ipj\varphi ^\left(p\right)eip·xdp\right)\left(\int \Omega ipj\prime \varphi ^(p\prime )eip\prime ·xdp\prime \right)dx\hfill \\ & =& {\displaystyle \left(\pi \right)}\sum j=\int \Omega \int \Omega \left(-pjpj\prime \varphi ^\left(p\right)\varphi ^(p\prime )\int \Omega ei(p+p\prime )·xdx\right)dpdp\prime \hfill \\ & =& {\displaystyle \left(\pi \right)/}\sum j=\int \Omega \int \Omega \left(-pjpj\prime \varphi ^\left(p\right)\varphi ^(p\prime )\delta (p+p\prime )\right)dpdp\prime \hfill \\ & =& {\displaystyle \left(\pi \right)/}\sum j=\int \Omega \left(pj\varphi ^\left(p\right)\varphi ^(-p)\right)dp.\hfill \end{array}$$

(153)

Thus, denoting $p^=(p,p,p),$$dp^=dpdpdp,$ and

$$p(p^)=\sqrt{\sum j=pj+m},$$

we may infer that

$$\begin{array}{ccc}\hfill \int \Omega T00dx& =& {\displaystyle \left(\pi \right)/}\int \Omega \left(\sum j=pj+m\right)\varphi ^\left(p\right)\varphi ^(-p))dp\hfill \\ & =& {\displaystyle \left(\pi \right)/}\int \Omega \left(\sum j=pj+m\right)\delta \left(p02-\sum j=pj-m\right)\varphi \left(p\right)\varphi (-p))dp\hfill \\ & =& {\displaystyle \left(\pi \right)/}\int \Omega \left(p(p^)\varphi (p(p^),p^)\varphi (-p(p^),-p^)\right)dp^.\hfill \end{array}$$

(154)

Summarizing we have got

$$\begin{array}{ccc}\hfill \int \Omega T00dx& =& {\displaystyle \left(\pi \right)/}\int \Omega \left(p(p^)\varphi (p(p^),p^)\varphi (-p(p^),-p^)\right)dp^\hfill \\ & =& ∥{\displaystyle \partial u\partial t}∥L,\hfill \end{array}$$

(155)

so that

$$\int \Omega T00dx=∥{\displaystyle \partial u\partial t}∥L$$

may be expressed as a kind of average expectance of $p$ related to the function $\varphi \left(p\right).$

25.3. A note on the Dirac equation

In this subsection we denote

$$\Delta =\sum j=gjjLjLj,$$

where

$$Lj=igjj{\displaystyle \partial \partial xj},\forall j\in \{,,,\}.$$

We recall that the relativistic Klein-Gordon equation may be written as

$$(\Delta -m)u=,\mathrm{in}\Omega =\mathbb{R}.$$

Moreover, for $\times$ matrices $\gamma$ indicated in the subsequent lines, we may obtain

$$\left\{Dij\right\}u=\left[-i\left(\sum j=\gamma j{\displaystyle \partial \partial xj}\right)-m\right]\left[-i\left(\sum j=\gamma j{\displaystyle \partial \partial xj}\right)+m\right]u,$$

where

$$Dii=\Delta -m$$

and

$$Dij=,\mathrm{if}i\ne j,\forall i,j\in \{,,,\}.$$

Here

$$u=(u,u,u,u)T\in V=W,(\Omega ;\mathbb{C}).$$

In such a case the fundamental Dirac equation stands for

$$\left[i\left(\sum j=\gamma j{\displaystyle \partial \partial xj}\right)-m\right]u=\mathbf{0}\in \mathbb{R},\mathrm{in}\Omega .$$

Summarizing, if $(u,u,u,u)T\in V$ is a solution of this last Dirac equation, then $u,u,u,u$ are four solutions of the Klein-Gordon equation.

In the momentum configuration space, through the Fourier transform proprieties, the Dirac equation stands for

$$(p^+m)u^\left(p\right)=\mathbf{0},\mathrm{in}\mathbb{R},$$

where

$$p^=\sum j=gjjpj\gamma j.$$

Observe that

$$u˜\left(p\right)=\delta (p^+m)u\left(p\right)$$

corresponds to a general solution of the Dirac equation.

Indeed,

$$(p^+m)u˜\left(p\right)=(p^+m)\delta (p^+m)u\left(p\right)=\mathbf{0}\in \mathbb{R},\mathrm{in}\Omega .$$

On the other hand

$$u^\left(p\right)=\delta \left(p02-\sum j=pj-m\right)u\left(p\right)$$

correspond to four solutions of the Klein-Gordon equation.

At this point, we assume such a $u^\left(p\right)$ corresponds to a solution of the Dirac equation as well.

Furthermore, here we recall that (please see the first chapter of the book [20], by N.N. Bogoliubov and D.V. Shirkov for details):

$$\gamma =\left\{\begin{array}{cccc}& & & \\ & & & \\ & & -& \\ & & & -\end{array}\right\},$$

(156)

$$\gamma =\left\{\begin{array}{cccc}& & & \\ & & & \\ & -& & \\ -& & & \end{array}\right\},$$

(157)

$$\gamma =\left\{\begin{array}{cccc}& & & -i\\ & & i& \\ & i& & \\ -i& & & \end{array}\right\},$$

(158)

$$\gamma =\left\{\begin{array}{cccc}& & & \\ & & & -\\ -& & & \\ & & & \end{array}\right\}$$

(159)

and

$$\gamma =\left\{\begin{array}{cccc}& & -i& \\ & & & -i\\ -i& & & \\ & -i& & \end{array}\right\}$$

(160)

where we also denote

$$\alpha j=\gamma \gamma j,\forall j\in \{,,\},$$

$$\sigma j=i\gamma \gamma \gamma j,\forall j\in \{,,\},$$

and

$$\beta =\gamma .$$

On the other hand, a variational formulation for the Dirac equation corresponds to the functional $A:V\to \mathbb{R}$ where

$$A\left(u\right)={\displaystyle 12}\int \Omega L(u,\nabla u)dx,$$

where

$$L(u,\nabla u)=i\sum j=\left(u*\gamma j{\displaystyle \partial u\partial xj}-{\displaystyle \partial u*\partial xj}\gamma ju\right)-mu*u,$$

where here

$$u=(u,u,u,u)T\in W,(\Omega ;\mathbb{C}).$$

From such statements and definitions, similarly as in the previous sections (please see [20] for details), we may obtain

$$Tkl={\displaystyle i}gll\left(u*\gamma k{\displaystyle \partial u\partial xl}-{\displaystyle \partial u*\partial xl}\gamma ku\right),$$

and

$$Sk,lm=-\left({\displaystyle \partial L(u,\nabla u)\partial uxk}Au,lmu-u*Au*,lm{\displaystyle \partial L(u,\nabla u)\partial uxk}\right),$$

where

$$Au,lm={\displaystyle i}\sigma ml,$$

$$Au*,lm={\displaystyle i}\sigma lm,$$

and where

$$\sigma lm={\displaystyle \gamma l\gamma k-\gamma k\gamma l},$$

so that

$$Sk,lm={\displaystyle 14}u*\left(\gamma k\sigma lm-\sigma lm\gamma k\right)u.$$

Thus,

$$\begin{array}{ccc}& & \int \Omega Sk,lmdx\hfill \\ & =& {\displaystyle 14}\int \Omega \left(u*\left(\gamma k\sigma lm-\sigma lm\gamma k\right)u\right)dx\hfill \\ & =& {\displaystyle 14}{\displaystyle \left(\pi \right)}\int \Omega \left(\int \Omega \int \Omega \left(u^\left(p\right)eip·x(\gamma k\sigma lm-\sigma lm\gamma k)u^(p\prime )eip\prime ·x\right)dpdp\prime \right)dx\hfill \\ & =& {\displaystyle 14}{\displaystyle \left(\pi \right)/}\int \Omega \int \Omega \left(u^\left(p\right)(\gamma k\sigma lm-\sigma lm\gamma k)\delta (p+p\prime )u^(p\prime )\right)dpdp\prime \hfill \\ & =& {\displaystyle 14}{\displaystyle \left(\pi \right)/}\int \Omega \left(u^\left(p\right)(\gamma k\sigma lm-\sigma lm\gamma k)u^(-p)\right)dp\hfill \\ & =& {\displaystyle 14}{\displaystyle \left(\pi \right)/}\int \Omega \left(u\left(p\right)(\gamma k\sigma lm-\sigma lm\gamma k)\delta \left(p02-\sum j=pj-m\right)u(-p)\right)dp\hfill \\ & =& {\displaystyle 14}{\displaystyle \left(\pi \right)/}\int \Omega \left(u(p(p^),p^)(\gamma k\sigma lm-\sigma lm\gamma k)u(-p(p^),-p^)\right)dp^,\hfill \end{array}$$

(161)

where

$$p(p^)=\sqrt{\sum j=pj+m}.$$

Summarizing, we have got

$$\int \Omega Sk,lmdx={\displaystyle \left(\pi \right)/}\int \Omega \left(u(p(p^),p^)(\gamma k\sigma lm-\sigma lm\gamma k)u(-p(p^),-p^)\right)dp^,$$

where $\Omega =\mathbb{R}$, $p^=(p,p,p)$ and $dp^=dpdpdp.$

26. A Note on Quantum Field Operators

This section is strongly based on the chapter 3, page 53 of the book [21], by G.B. Folland.

Therefore, here we have done a kind of review of these pages of such a book chapter indicated. In fact, we have simply opened more and clarified some calculations, in order to improve their understanding.

Let $\Omega =\Omega ^\times [,T]\subset \mathbb{R}$ where $\Omega ^\subset \mathbb{R}$ is a open, bounded and connected set with a regular boundary denoted $\partial \Omega ^.$

Define $V=W,\left(\Omega \right)$ and

$$V=W,\left(\Omega \right).$$

Consider an operator $H:V=V\cap W,\left(\Omega \right)\to Y$ where in a distributional sense,

$$H\left(u\right)=-{\displaystyle \partial u\partial t}+\nabla u-mu,$$

and where

$$Y=Y*=L\left(\Omega \right).$$

Suppose there exists operators $B:Y\to Y$ and $B:Y\to Y$ such that

$$BB\left(u\right)=H\left(u\right)+{\displaystyle 12}u$$

and

$$BB\left(u\right)=H\left(u\right)-{\displaystyle 12}u,\forall u\in V.$$

Assume also $\varphi \in V$ is such that

$$\parallel \varphi \parallel L=,$$

and $B\varphi =.$

Now define

$$\varphi k={\displaystyle Bk\left(\varphi \right)\sqrt{k!}},\forall k\in \mathbb{N}.$$

Observe that

$$\left[BB\right]=BB-BB=Id.$$

We shall prove by induction that

$$[B,Bk]=kBk-,\forall k\in \mathbb{N}.$$

(162)

Indeed, for $k=$

$$[B,B]=Id=B20,$$

so that (162) holds for $k=$.

Suppose now (162) holds for $k\in \mathbb{N},$ so that

$$[B,Bk]=kBk-.$$

In order to complete the induction, it suffices to prove that (162) holds for $k+.$

Observe that

$$\begin{array}{ccc}\hfill [B,Bk+]& =& (BBk+-Bk+B)\hfill \\ & =& \left(BBk\right)B-Bk+B\hfill \\ & =& (BkB+kBk-)B-Bk+B\hfill \\ & =& Bk\left(BB\right)+kBk-Bk+B\hfill \\ & =& Bk(BB+Id)+kB-Bk+B\hfill \\ & =& Bk+B+Bk+kBk-Bk+B\hfill \\ & =& (k+)Bk.\hfill \end{array}$$

(163)

Thus, the induction is complete, so that

$$[B,Bk]=kBk-,\forall k\in \mathbb{N}.$$

Moreover, we recall that

$$B\varphi =,$$

so that

$$\begin{array}{ccc}\hfill B\varphi k& =& B\left({\displaystyle Bk\varphi \sqrt{k!}}\right)\hfill \\ & =& {\displaystyle (BkB+kBk-)\varphi \sqrt{k!}}\hfill \\ & =& {\displaystyle k\varphi k-\sqrt{(k-)!}\sqrt{k!}}\hfill \\ & =& {\displaystyle k\varphi k-\sqrt{k}}\hfill \\ & =& \sqrt{k}\varphi k-,\forall k\in \mathbb{N}.\hfill \end{array}$$

(164)

Summarizing, we have got

$$B\varphi k=\sqrt{k}\varphi k-,\forall k\in \mathbb{N}.$$

Now, we shall prove that

$$B\varphi k=\sqrt{k+}\varphi k+,\forall k\in \mathbb{N}.$$

Observe that

$$\begin{array}{ccc}\hfill Bk+\varphi & =& \varphi k+(\sqrt{(k+)!}\hfill \\ & =& B\left(Bk\varphi \right)\hfill \\ & =& \left(B\varphi k\right)\sqrt{k!}.\hfill \end{array}$$

(165)

Summarizing, we have got

$$\left(B\varphi k\right)\sqrt{k!}=\varphi k+(\sqrt{(k+)!},$$

so that

$$\left(B\varphi k\right)=\sqrt{k+}\varphi k+,\forall k\in \mathbb{N}.$$

Finally, from such results, we may infer that

$$\begin{array}{ccc}\hfill BB\varphi k& =& B\left(\sqrt{k+}\varphi k+\right)\hfill \\ & =& \sqrt{k+}B\varphi k+\hfill \\ & =& \sqrt{k+}\sqrt{k+}\varphi k\hfill \\ & =& (k+)\varphi k,\forall k\in \mathbb{N}.\hfill \end{array}$$

(166)

Similarly,

$$\begin{array}{ccc}\hfill BB\varphi k& =& B\left(\sqrt{k}\varphi k-\right)\hfill \\ & =& \sqrt{k}B\varphi k-\hfill \\ & =& \sqrt{k}\sqrt{k}\varphi k\hfill \\ & =& k\varphi k.\hfill \end{array}$$

(167)

Therefore we have got

$$H\varphi k=BB\varphi k-{\displaystyle 12}\varphi k=(k+)\varphi k-{\displaystyle 12}\varphi k=\left(k+{\displaystyle 12}\right)\varphi k,$$

that is

$$H\varphi k=\left(k+{\displaystyle 12}\right)\varphi k,\forall k\in \mathbb{N}.$$

Thus, for each $k\in \mathbb{N}$, $k+{\displaystyle 12}$ is an eigenvalue of H with corresponding eigenvector $\varphi$.

26.1. An application concerning the harmonic oscillator operator in quantum mechanics

In this section we have the aim of representing the relativistic Klein-Gordon equation through the creation and annihilation operations related to the harmonic oscillator in quantum mechanics.

Consider first the one-dimensional Hamiltonian, corresponding to the harmonic oscillator, namely

$$H=-{\displaystyle \hslash m}{\displaystyle ddx}+K{\displaystyle x},$$

which through an appropriate re-scale results into the following related Hamiltonian $H$, where

$$H={\displaystyle 12}\left(-{\displaystyle ddx}+x\right).$$

Define now the operators

$$B=A={\displaystyle \sqrt{}}\left(x+{\displaystyle ddx}\right),$$

and

$$B=A*={\displaystyle \sqrt{}}\left(x-{\displaystyle ddx}\right).$$

Clearly,

$$H=BB-{\displaystyle Id}=BB+{\displaystyle Id},$$

so that

$$[A,A*]=[B,B]=BB-BB=Id.$$

Similarly, as in the previous sections, by induction, we may obtain

$$[B,Bk]=kBk-,\forall k\in \mathbb{N}.$$

For

$$\varphi =\pi -/e-{\displaystyle x},$$

we define

$$\varphi k={\displaystyle \sqrt{k}}Bk\varphi ,\forall k\in \mathbb{N}.$$

Also from the previous section, we may obtain

$$B\varphi k=A*\varphi k=\sqrt{k+}\varphi k+,$$

$$B\varphi k=A\varphi k=\sqrt{k}\varphi k-,\forall k\in \mathbb{N}.$$

$$BB=A*A\varphi k=k\varphi k,$$

and

$$BB\varphi k=AA*\varphi k=(k+)\varphi k,\forall k\in \mathbb{N}\cup \left\{\right\}.$$

so that

$$H\varphi k=(k+/)\varphi k,\forall k\in \mathbb{N}.$$

Here we recall that

$$B\varphi =A\varphi =,$$

and

$$\parallel \varphi \parallel L=.$$

In reference [21], page 54 it is proven that such a sequence $\left\{\varphi k\right\}$ is an ortho-normal basis for $L\left(\mathbb{R}\right).$

Finally, observe that for $\mathbb{R}$ we may define

$$\left(B\right)j=Aj={\displaystyle \sqrt{}}\left({\displaystyle \partial \partial xj}+xj\right),$$

and

$$\left(B\right)j=Aj*={\displaystyle \sqrt{}}\left(-{\displaystyle \partial \partial xj}+xj\right),\forall j\in \{,,,\}.$$

Here generically,

$$\mathbf{x}=(x,x,x,x)\in \mathbb{R}.$$

Observe that clearly

$${\displaystyle \partial \partial xj}={\displaystyle \sqrt{}}(Aj-Aj*),$$

and

$$xjId={\displaystyle \sqrt{}}(Aj+Aj*),\forall j\in \{,,,\}.$$

Denoting $x=t$ where t stands for time, consider the relativistic Klein-Gordon equation,

$$-{\displaystyle \partial \varphi \partial t}+\sum j={\displaystyle \partial \varphi \partial xj}-m\varphi =.$$

From the previous results, we may represent such an equation by

$$\left(-{\displaystyle 12}(A-A*)+\sum j={\displaystyle 12}(Aj-Aj*)-mId\right)\varphi =.$$

We highlight from the previous results we know the action of $A$ and $A$ on an appropriate basis of $L\left(\mathbb{R}\right)$ obtained though an appropriate tensorial product of the bases

$$\{\left\{\varphi k\left(xj\right)\right\},\mathrm{for}j\in \{,,,\}\}.$$

We shall call the operators $A$ and $A$ as the creation and annihilation operators concerning the original harmonic operator in quantum mechanics.

To justify such a nomenclature, we recall that $Aj*\varphi \left(xj\right)=\varphi \left(xj\right)$ and $Aj\varphi \left(xj\right)=,\forall j\in \{,,,\}.$

27. A Dual Variational Formulation for a Related Model

In this section we develop a concave dual variational formulation for a Ginzburg-Landau type equation.

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider a functional $J:V\to \mathbb{R}$ defined by

$$\begin{array}{ccc}\hfill J\left(u\right)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx\hfill \\ & & +{\displaystyle \alpha }\int \Omega (u-\beta )dx-\langle u,f\rangle L,\hfill \end{array}$$

(168)

where $\gamma >,$$\alpha >$, $\beta >$, $f\in L\left(\Omega \right),$ and

$$V=W,\left(\Omega \right).$$

We also denote $Y=Y*=L\left(\Omega \right).$

Define now

$$V=\{u\in V:\parallel u\parallel \infty \le K\},$$

for some appropriate $K>$ and, $J:V\times Y\to \mathbb{R}$ by

$$J(u,v*)=J\left(u\right)+{\displaystyle K12}\int \Omega (-\gamma \nabla u+v*u-f)dx,$$

where

$$K={\displaystyle \alpha K32+\epsilon }$$

for some small parameter $<\epsilon \ll .$

Observe that

$$\begin{array}{ccc}\hfill J(u,v*)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+\langle u,v*\rangle L\hfill \\ & & +{\displaystyle K}\int \Omega (-\gamma \nabla u+v*u-f)dx-\langle u,f\rangle L\hfill \\ & & -\langle u,v*\rangle L+{\displaystyle \alpha }\int \Omega (u-\beta )dx\hfill \\ & \ge & \underset{u\in V}{inf}\left\{{\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+\langle u,v*\rangle L\right.\hfill \\ & & \left.+{\displaystyle K}\int \Omega (-\gamma \nabla u+v*u-f)dx-\langle u,f\rangle L\right\}\hfill \\ & & +\underset{v\in Y}{inf}\left\{-\langle v,v*\rangle L+{\displaystyle \alpha }\int \Omega (v-\beta )dx\right\}\hfill \\ & =& -F*(v*)-G*(v*)\hfill \\ & \equiv & J*(v*),\forall u\in V,v*\in Y*,\hfill \end{array}$$

(169)

where we have denoted

$$F*(v*)=\underset{u\in V}{sup}\{-\langle u,v*\rangle L-F(u,v*)\},$$

$$F(u,v*)={\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle K12}\int \Omega (-\gamma \nabla u+v*u-f)dx-\langle u,f\rangle L,$$

and

$$G\left(v\right)={\displaystyle \alpha }\int \Omega (v-\beta )dx,$$

$$\begin{array}{ccc}\hfill G*(v*)& =& \underset{v\in Y}{sup}\{\langle v,v*\rangle L-G\left(v\right)\}\hfill \\ & =& {\displaystyle \alpha }\int \Omega (v*)dx+\beta \int \Omega v*dx.\hfill \end{array}$$

(170)

Observe that

$${\displaystyle \partial F(u,v*)\partial u}=-\gamma \nabla +v*+K(-\nabla +v*),$$

so that we define

$$B*=\{v*\in Y*:-\gamma \nabla +v*+K(-\nabla +v*)>\mathbf{0}\}.$$

With such assumptions and definitions in mind, we may prove the following theorem:

Theorem 4.

For $J*(v*)=-F*(v*)-G*(v*),$ suppose $v^*\in B*$ is such that

$$\delta J*(v^*)=\mathbf{0}.$$

Let $u\in Y$ be such that

$${\displaystyle \partial H(u,v^*)\partial u}=\mathbf{0},$$

where

$$H(u,v*)=F(u,v*)+\langle u,v*\rangle L.$$

Suppose

$$u\in V.$$

Under such hypotheses,

$$F*(v^*)=H(u,v^*),$$

$$\delta J\left(u\right)=\mathbf{0},$$

and

$$\begin{array}{ccc}\hfill J\left(u\right)& =& J(u,v^*)\hfill \\ & =& \underset{u\in V}{inf}J(u,v^*)\hfill \\ & =& \underset{v*\in Y*}{sup}J*(v*)\hfill \\ & =& J*(v^*).\hfill \end{array}$$

(171)

Proof. 

The proof that

$$F*(v^*)=H(u,v^*),$$

is immediate from $v^\in B*.$

Moreover, the proof that

$$\delta J\left(u\right)=\mathbf{0},$$

and

$$J\left(u\right)=J(u,v^*)=J*(v^*)$$

may be done similarly as in the previous sections.

Observe that

$$J*(v*)=-F*(v*)-G*(v*)=\underset{u\in V}{inf}\{H(u,v*)-G*(v*)\},$$

so that $J$ is concave in $v$ as the infimum of a family of concave functionals in $v$.

From this and $\delta J*(v^*)=\mathbf{0}$ we get

$$J*(v^*)=\underset{v*\in Y*}{sup}J*(v*).$$

Furthermore observe that

$$\begin{array}{ccc}\hfill J\left(u\right)& =& J(u,v^*)\hfill \\ & \le & J(u,v*)\hfill \\ & =& F(u,v^*)+\langle u,v^*\rangle L-G*(v^*)\hfill \\ & \le & F(u,v^*)+\underset{v*\in Y*}{sup}\left\{\langle u,v^*\rangle L-G*(v^*)\right\}\hfill \\ & =& F(u,v^*)+G\left(u\right)\hfill \\ & =& J(u,v^*),\forall u\in V.\hfill \end{array}$$

(172)

Hence

$$J\left(u\right)=J(u,v^*)=\underset{u\in V}{inf}J(u,v^*).$$

Joining the pieces, we have got

$$\begin{array}{ccc}\hfill J\left(u\right)& =& J(u,v^*)\hfill \\ & =& \underset{u\in V}{inf}J(u,v^*)\hfill \\ & =& \underset{v*\in Y*}{sup}J*(v*)\hfill \\ & =& J*(v^*).\hfill \end{array}$$

(173)

The proof is complete. □

28. The generalized method of lines applied to fourth order differential equations

In this sections we develop an application of the generalized method of lines to a fourth order equation.

We start by addressing the following ordinary differential equation (ode):

$$\epsilon {\displaystyle du\left(x\right)dx}-f=,\mathrm{in}[,],$$

with the boundary conditions

$$u\left(\right)=u\prime \left(\right)=$$

and

$$u\left(\right)=u\prime \left(\right)=.$$

In terms of linear elasticity, such a boundary conditions corresponds to a bi-clamped beam.

In a finite difference context, this last equation corresponds to

$$\epsilon \left({\displaystyle un+-un++un-un-+un-d}\right)-fn=,\forall n\in \{,\dots N-\},$$

where N is the number of nodes and $d=/N$.

Considering that, from the boundary conditions, $u-=u=,$ for $n=$ we get

$$u-u+u={\displaystyle fd\epsilon },$$

so that

$$u=au+bu+c,$$

where

$$a=/,b-/\mathrm{and}c={\displaystyle fd\epsilon }.$$

Similarly, for $n=$, we obtain

$$-u+u-u+u={\displaystyle fd\epsilon }.$$

Hence, replacing the value of $u$ previously obtained in this last equation, we have

$$-(au+bu+c)+u-u+u={\displaystyle fd\epsilon },$$

so that

$$u=au+bu+c,$$

where defining $m12=(-a)$, we have also

$$a={\displaystyle b+m12},$$

$$b=-{\displaystyle m12},$$

$$c={\displaystyle m12}\left({\displaystyle fd\epsilon }+c\right).$$

Now reasoning inductively, for n, having

$$un-=an-un+bn-un++cn-,$$

and

$$un-=an-un-+bn-un+cn-$$

we obtain

$$un-=an-(an-un+bn-un++cn-)+bn-un+cn-,$$

so that from this and

$$un+-un++un-un-+un-={\displaystyle fnd\epsilon },$$

we obtain

$$\begin{array}{ccc}& & an-(an-un+bn-un++cn-)+bn-un+cn-\hfill \\ & & -(an-un+bn-un++cn-)+un-un++un+={\displaystyle fnd\epsilon },\hfill \end{array}$$

(174)

so that

$$un=anun++bnun++cn$$

where defining

$$m12=(an-\left(an-\right)+bn--an-+)$$

we obtain

$$an=-{\displaystyle m12}(an-bn--bn--)$$

$$bn=-{\displaystyle m12},$$

and

$$cn={\displaystyle m12}\left(an-cn-+cn--cn--{\displaystyle fnd\epsilon }\right).$$

Summarizing, we have got

$$un=anun++bnun++cn,\forall n\in \{,·N-\}.$$

Observe now that from the boundary conditions,

$$uN-=uN=.$$

From these last two equations, we may obtain

$$uN-=cN,$$

and

$$uN-=aN-uN-+bN-uN-+cN-,$$

and so on up to obtaining

$$u=au+bu+c.$$

The problem is then solved.

28.1. A numerical example

We develop a numerical example considering

$$\epsilon =,$$

and

$$f\equiv ,\mathrm{in}[,].$$

Thus, we have solved the equation

$$\epsilon {\displaystyle du\left(x\right)dx}-f=,\mathrm{in}[,],$$

with the boundary conditions

$$u\left(\right)=u\prime \left(\right)=$$

and

$$u\left(\right)=u\prime \left(\right)=.$$

In a finite differences context, we have used $N=100$ nodes and $d=/N.$

For a solution $u\left(x\right)$, please see Figure 19.

In the next lines, we present the concerning software in MAT-LAB

**************

clear all

m8=100;

d=1/m8;

e1=1.0;

for i=1:m8

f(i,1)=1.0;

end;

a(1)=2/3;

b(1)=-1/6;

c(1)=f(1,1)*$d$/(6e1);

m12=(6-4*a(1));

a(2)=(4*b(1)+4)/m12;

b(2)=-1/m12;

c(2)=1/m12*(4*c(1)+f(2,1)*$d$/e1);

for i=3:m8-2

m12=(a(i-2)*a(i-1)+b(i-2)-4*a(i-1)+6);

a(i)=-1/m12*(a(i-2)*b(i-1)-4*b(i-1)-4);

b(i)=-1/m12;

c(i)=1/m12*(f(i,1)*$d$/e1-c(i-2)-a(i-2)*c(i-1)+4*c(i-1));

end;

u(m8,1)=0;

u(m8-1,1)=0;

for i=2:m8-1;

u(m8-i,1)=a(m8-i)*u(m8-i+1,1)+b(m8-i)*u(m8-i+2,1)+c(m8-i);

end;

for i=1:m8

x(i)=i*d;

end;

plot(x,u)

******************

29. A Note on Hyper-Finite Differences for the Generalized Method of Lines

In this section we develop an application of the hyper finite differences method through an approximation of the generalized method of lines.

Consider the equation

$$\left\{\begin{array}{cc}-\epsilon u\prime \prime \left(x\right)+\alpha u-\beta u-f=,& \mathrm{in}\Omega =[,],\\ u\left(\right)=,& u\left(\right)=\end{array}\right.$$

(175)

As $\epsilon >$ is small, in order to decrease the error concerning the approximations used we propose to divide the domain $\Omega =[,]$ into $N$ sub-intervals of same measure. Thus we define

$$xk={\displaystyle kN},\forall k\in \{,,\dots ,N\}.$$

For each sub-interval $Ik=[xk-,xk]$ we are going to obtain an approximate solution of the equation in question with the general boundary conditions

$$u((k-)/N)=U[k-],$$

and

$$u(k/N)=U\left[k\right].$$

Denoting such a solution by

$$\left\{u\right[i,k\left]\right\}$$

where

$$xi={\displaystyle k-N}+id,$$

and

$$d={\displaystyle mN},$$

where $m$ is the fixed number of nodes in each interval $I$.

Observe that in a finite differences context, linearizing it about a initial solution $\left\{u[i,k]\right\},$ the equation in question stands for:

$$\begin{array}{ccc}& & -\epsilon {\displaystyle \left(u\right[i+,k]-u[i,k]+u[i-,k\left]\right)d}+\alpha u[i,k]u[i,k]-\alpha u[i,k]\hfill \\ & & -\beta u[i,k]-f[i,k]=,\forall i\in \{,\dots ,m-\}.\hfill \end{array}$$

(176)

In particular, for $i=$, we obtain

$$\begin{array}{ccc}& & -\epsilon {\displaystyle \left(u\right[,k]-u[,k]+u[,k\left]\right)d}+\alpha u[,k]u[,k]-\alpha u[,k]\hfill \\ & & -\beta u[,k]-f[,k]=,\hfill \end{array}$$

(177)

so that

$$\begin{array}{ccc}\hfill u[,k]& =& a[,k]u[,k]+b[,k]u[,k]+c[,k]T[,k]\hfill \\ & & +e[,k]+Er[,k],\hfill \end{array}$$

(178)

where

$$a[,k]=/,$$

$$b[,k]=/,$$

$$c[,k]=/,$$

$$e[,k]=f[,k]{\displaystyle d\epsilon },$$

$$T[,k]=(-\alpha u[,k]u[i,k]+\alpha u[,k]-\beta u[,k]){\displaystyle d\epsilon },$$

and

$$Er[,k]=.$$

Now reasoning inductively, having

$$\begin{array}{ccc}\hfill u[i-,k]& =& a[i-,k]u[i,k]+b[i-,k]u[,k]+c[i-,k]T[i-,k]\hfill \\ & & +e[i-,k]+Er[i-,k],\hfill \end{array}$$

(179)

and

$$\begin{array}{ccc}& & -\epsilon {\displaystyle \left(u\right[i+,k]-u[i,k]+u[i-,k\left]\right)d}+\alpha u[i,k]u[i,k]-\alpha u[i,k]\hfill \\ & & -\beta u[i,k]-f[i,k]=,\hfill \end{array}$$

(180)

so that

$$(u[i+,k]-u[i,k]+u[i-,k])+T[i,k]+f[i,k]{\displaystyle d\epsilon }=,$$

where,

$$T[i,k]=(-\alpha u[i,k]u[i,k]+\alpha u[i,k]+\beta u[i,k]){\displaystyle d\epsilon },$$

we obtain

$$\begin{array}{ccc}\hfill u[i,k]& =& a[i,k]u[i,k]+b[i,k]u[,k]+c[i,k]T[i,k]\hfill \\ & & +e[i,k]+Er[i,k],\hfill \end{array}$$

(181)

where,

$$a[i,k]=(-a[i-,k])-,$$

$$b[i,k]=a[i,k]b[i-,k],$$

$$c[i,k]=a[i,k]\left(c\right[i-,k]+),$$

$$e[i,k]=a[i,k]\left(e[i-,k]+{\displaystyle f[i,k]d\epsilon }\right),$$

and

$$Er[i,k]=a[i,k]\left(Er[i-,k]\right)+c[i,k](T[i-,k]-T[i,k]).$$

Observe that in particular for $i=m-$, we have $u[m,k]=U\left[k\right]$ and $u[,k]=U[k-],$ so that from above, neglecting $Er[,k]$, we also obtain

$$\begin{array}{ccc}& & u[m-,k]\approx a[m-]u[m,k]+b[m-,k]u[,k]\hfill \\ & & +c[m-,k]T[m-,k](u[m,k],u[,k])+e[m-,k]\hfill \\ & =& Hm-(U\left[k\right],U[k-]).\hfill \end{array}$$

(182)

Similarly, for $i=m-$ we may obtain

$$\begin{array}{ccc}& & u[m-,k]\approx a[m-]u[m-,k]+b[m-,k]u[,k]\hfill \\ & & +c[m-,k]T[m-,k](u[m-,k],u[,k])+e[m-,k]\hfill \\ & =& Hm-(U\left[k\right],U[k-]),\hfill \end{array}$$

(183)

and so on, up to finding

$$u[,k]=H(U\left[k\right],U[k-]),\forall k\in \{,\dots ,N\}.$$

At this point we connect the sub-intervals by setting

$$U\left[\right]=U\left[N\right]=$$

and obtaining $\{U\left[\right],\dots ,U[N-]\},$ by solving the equations

$$\begin{array}{ccc}& & -\epsilon {\displaystyle (u[m-,k]-U\left[k\right]+u[,k+])d}+\alpha u[m,k]U\left[k\right]-\alpha u[m,k]\hfill \\ & & -\beta U\left[k\right]-f[m,k]=,\forall k\in \{,\dots ,N-\}.\hfill \end{array}$$

(184)

Having obtained $\{U\left[k\right],\forall k\in \{,\dots ,N-\}\}$ we may obtain the solution $\left\{u\right[i,k\left]\right\}$ where $i\in \{,\dots ,m\}$ and $k\in \{,\dots ,N\}.$

The next step is to replace $\left\{u[i,k]\right\}$ by $\left\{u\right[i,k\left]\right\}$ and then to repeat the process until an appropriate convergence criterion is satisfied.

The problem is then approximately solved.

We have obtained numerical results for $\epsilon =.001$, $f\equiv ,$ on $\Omega$, $N=10$, $m=100$ and $\alpha =\beta =.$

For the related software in MATHEMATICA we have obtained $U\left[\right],\dots ,U\left[\right],$

Here the software and results:

**************************

Clear[u, U, z, N1];

m8 = 100;

N1 = 10;

d = 1/m8/N1;

e1 = 0.001;

For[k = 1, k < N1 + 1, k++,

For[i = 0, i < m8 + 1, i++,

uo[i, k] = 1.01]];

A = 1.0;

B = 1.0;

a[1] = 1.0/2;

b[1] = 1.0/2;

c[1] = 1/2.0;

e[1] = $d/e/.$;

For[i = 2, i < m8, i++,

a[i] = 1/(2.0 - a[i - 1]);

b[i] = b[i - 1]*a[i];

c[i] = a[i]*(c[i - 1] + 1.0);

e[i] = $a\left[i\right]*(e[i-]+d/e)$;

];

For[k1 = 1, k1 < 10, k1++,

Print[k1];

Clear[U, z];

For[k = 1, k < N1 + 1, k++,

u[0, k] = U[k - 1];

u[m8, k] = U[k];

For[i = 1, i < m8, i++,

z = a[m8 - i]*u[m8 - i + 1, k] + b[m8 - i]*u[0, k] +

c[m8 - i]*(-3*A*uo[m8 - i + 1, k]*u[m8 - i + 1, k] +

2*A*uo[m8 - i + 1, k] + B*u[m8 - i + 1, k])*$d/e$ +

e[m8 - i];

u[m8 - i, k] = Expand[z]]];

U[0] = 0.0;

U[N1] = 0.0;

S = 0;

For[k = 1, k < N1, k++,

S = S + (e1*(-u[m8 - 1, k] + 2*U[k] - u[1, k + 1])/$d$ +

3*A*U[k]*uo[m8, k] - 2*A*uo[m8, k] - B*U[k] - 1)];

Sol = NMinimize[S, U[1], U[2], U[3], U[4], U[5], U[6], U[7], U[8], U[9]];

For[k = 1, k < N1, k++,

w4[k] = U[k] Ṡol[[2, k]]];

For[k = 1, k < N1, k++,

U[k] = w4[k]];

For[k = 1, k < N1 + 1, k++,

For[i = 0, i < m8 + 1, i++,

uo[i, k] = u[i, k]]];

Print[U[5]]];

For[k = 0, k < N1 + 1, k++,

Print["U[", k, "]=", U[k]]]

U[0]=0.

U[1]=1.27567

U[2]=1.32297

U[3]=1.32466

U[4]=1.32472

U[5]=1.32472

U[6]=1.32472

U[7]=1.32472

U[8]=1.32472

U[9]=1.32471

U[10]=0.

**********************

Remark 13.

Observe that along the domain we have obtained approximately the constant value $u=.32472$. This is expected since $\epsilon =.001$ is small and such a value u is approximately the solution of equation

$$\alpha u-\beta u-=.$$

30. Applications to the optimal shape design for a beam model

In this section, we present a numerical procedure for the shape optimization concerning the Bernoulli beam model.

Let $\Omega =[,]\subset \mathbb{R}$ corresponds to the horizontal axis of a straight beam with rectangular cross section $b\times h\left(x\right)$, that is, the beam has a variable thickness $h\left(x\right)$ distributed along such a horizontal axis x, where $x\in [,].$

Define now

$$V=\{w\in W,\left(\Omega \right):w\left(\right)=w\left(\right)=\},$$

which corresponds to a simply supported beam.

Consider the problem of minimizing in $V\times B$ the functional

$$J(w,h)={\displaystyle 12}\int \Omega H\left(x\right)wxx\left(x\right)dx$$

subject to

$$\left(H\left(x\right)wxx\left(x\right)\right)xx-P\left(x\right)=,\mathrm{in}\Omega ,$$

where

$$H\left(x\right)={\displaystyle h\left(x\right)b12}E,$$

$h\left(x\right)$ is variable beam thickness, $A\left(x\right)=bh\left(x\right)$ corresponds to a rectangular cross section perpendicular to the x axis, and E is the young elasticity model.

Also, we define

$$B=\left\{h:[,]\to \mathbb{R}\mathrm{measurable}:hmin\le h\left(x\right)\le hmax\mathrm{and}\int 01h\left(x\right)\le chmax\right\},$$

where $<c<$ and

$$C*=\{w\in V:\left(H\left(x\right)wxx\left(x\right)\right)xx-P\left(x\right)=,\mathrm{in}\Omega \}.$$

Observe that

$$\begin{array}{ccc}& & \underset{(w,h)\in C*\times B}{inf}J(w,h)\hfill \\ & =& \underset{h\in B}{inf}\left\{\underset{w\in C*}{inf}J(w,h)\right\}\hfill \\ & =& \underset{h\in B}{inf}\left\{\underset{w^\in V}{sup}\left\{\underset{w\in V}{inf}\left\{{\displaystyle 12}\int \Omega H\left(x\right)wxx\left(x\right)dx-\langle w^,\left(H\left(x\right)wxx\left(x\right)\right)xx-P\left(x\right)\rangle L\right\}\right\}\right\}\hfill \\ & =& \underset{h\in B}{inf}\left\{\underset{w^\in V}{sup}\left\{-{\displaystyle 12}\int \Omega H\left(x\right)w^xxdx+\langle w^,P\rangle L\right\}\right\}\hfill \\ & =& \underset{h\in B}{inf}\left\{\underset{M\in D*}{inf}\left\{{\displaystyle 12}\int \Omega {\displaystyle MH\left(x\right)}dx\right\}\right\}.\hfill \end{array}$$

(185)

where

$$D*=\{M\in Y*:Mxx-P=,\mathrm{in}\Omega ,\mathrm{and}M\left(\right)=M\left(\right)=\}.$$

Summarizing, we have got

$$\underset{(w,h)\in C*\times B}{inf}J(w,h)=\underset{(M,h)\in D*\times B}{inf}\left\{{\displaystyle 12}\int \Omega {\displaystyle MH\left(x\right)}dx\right\}.$$

In order to obtain numerical results, we suggest the following primal dual procedure:

Set $n=$ and

$$hn\left(x\right)=chmax.$$

Calculate $wn\in V$ solution of equation

$$\left(Hn\left(x\right)\left(wn\right)xx\right)xx=P\left(x\right),$$

where

$$Hn\left(x\right)={\displaystyle Ebhn\left(x\right)12}.$$

Calculate $hn+\left(x\right)\in B$ such that

$$J*(Mn,hn+)=\underset{h\in B}{inf}J*(Mn,h),$$

where

$$Mn=Hn\left(wn\right)xx,$$

$$J*(M,h)={\displaystyle 12}\int \Omega {\displaystyle MH\left(x\right)}dx.$$

Set $n:=n+$ and go to step 2 until an appropriate convergence criterion is satisfied.

We have developed numerical results for $c=.65$, $E=210107$, $b=.m$, $P\left(x\right)=36102N$, $hmin=.072m$ and $hmax=.18m.$

We have also defined

$$h\left(x\right)=t\left(x\right)hmax,$$

where

$$.\le t\left(x\right)\le ,\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega .$$

For the optimal solution $w=w\left(x\right)$, please see Figure 20.

For a corresponding optimal solution $t=t\left(x\right)$, please see Figure 21.

Remark 14.

For such a simply-supported beam model, for the numerical solution of equation

$$\left(H\left(x\right)wxx\right)xx=P,$$

with the boundary conditions

$$w\left(\right)=w\left(\right)=w\prime \prime \left(\right)=w\prime \prime \left(\right)=$$

firstly we have solved the equation

$$vxx-P=$$

with the boundary conditions

$$v\left(\right)=v\left(\right)=.$$

Subsequently, we have solved the equation

$$H\left(x\right)wxx=v$$

with the boundary conditions

$$w\left(\right)=w\left(\right)=.$$

Here we present the software developed in MAT-LAB.

******************

clear all

global m8 d d2wo H e1 ho h1 xo b5

m8=100;

d=1.0/m8;

b5=0.1;

e1=210*$107$;

ho=0.18;

A=zeros(m8-1,m8-1);

for i=1:m8-1

A(1,i)=1.0;

xo(i,1)=0.55;

x3(i,1)=0.55;

end;

lb=0.4*ones(m8-1,1);

ub=ones(m8-1,1);

b=zeros(m8-1,1);

b(1,1)=0.65*(m8-1);

for i=1:m8

f(i,1)=1.0;

L(i,1)=1/2;

P(i,1)=36.0*$102$;

end;

i=1;

m12=2;

m50(i)=1/m12;

z(i)=1/m50(i)*(-P(i,1)*$d$);

for i=2:m8-1

m12=2-m50(i-1);

m50(i)=1/m12;

z(i)=m50(i)*(-P(i,1)*$d$+z(i-1));

end;

v(m8,1)=0;

for i=1:m8-1

v(m8-i,1)=m50(m8-i)*v(m8-i+1,1)+z(m8-i);

end;

k=1;

b12=1.0;

while $(b12>10-)$ and $(k<10)$

k

k=k+1;

for i=1:m8-1

H(i,1)=b5*$L(i,)*ho$/12*e1;

f1(i,1)=v(i,1)/H(i,1);

end;

i=1;

m12=2;

m70(i)=1/m12;

z1(i)=m70(i)*(-f1(i,1)*$d$);

for i=2:m8-1

m12=2-m70(i-1);

m70(i)=1/m12;

z1(i)=m70(i)*(-f1(i,1)*$d$+z1(i-1));

end;

w(m8,1)=0;

for i=1:m8-1

w(m8-i,1)=m70(m8-i)*w(m8-i+1,1)+z1(m8-i);

end;

d2wo(1,1)=(-2*w(1,1)+w(2,1))/$d$;

for i=2:m8-1

d2wo(i,1)=(w(i+1,1)-2*w(i,1)+w(i-1,1))/$d$;

end;

k9=1;

b14=1.0;

while $(b14>10-)\mathrm{and}(k<120)$

k9

k9=k9+1;

X=fmincon(’beamNov2023’,xo,A,b,$\left[\right],\left[\right]$,lb,ub);

b14=max(abs(xo-X))

xo=X;

end;

b12=max(abs(xo-x3))

x3=xo;

for i=1:m8-1

L(i,1)=xo(i,1);

end;

end;

***************

With the auxiliary function "beamNov2023":

********************

function S=beamNov2023(x)

global m8 d d2wo H e1 ho h1 xo b5

S=0;

for i=1:m8-1

S=S+1/$\left(x(i,)\right)/ho$/b5/e1*(H(i,1)*$dwo(i,))$*12;

end;

*****************************

We develop numerical results also for

$$V=W,\left(\Omega \right)=\{w\in W,\left(\Omega \right)\mathrm{such}\mathrm{that}w\left(\right)=w\left(\right)=w\prime \left(\right)=w\prime \left(\right)=\}.$$

Such boundary conditions corresponds to bi-clamped beam. The remaining data is equal to the previous example

For the optimal solution $w=w\left(x\right)$, please see Figure 22.

For a corresponding optimal solution $t=t\left(x\right)$, please see Figure 23.

Remark 15.

For such a bi-clamped beam model, for the numerical solution of equation

$$\left(H\left(x\right)wxx\right)xx=P,$$

with the boundary conditions

$$w\left(\right)=w\left(\right)=w\prime \left(\right)=w\prime \left(\right)=,$$

firstly we have solved the equation

$$vxx-P=$$

with the boundary conditions

$$v\left(\right)=v\left(\right)=.$$

Subsequently, we solved the equation

$$H\left(x\right)wxx=v+ax+b$$

with the boundary conditions

$$w\left(\right)=w\left(\right)=,$$

obtaining $a,b\in \mathbb{R}$ such that the boundary conditions

$$w\prime \left(\right)=w\prime \left(\right)=$$

are also satisfied.

Here we present the software developed in MAT-LAB.

*************************

clear all

global m8 d d2wo H e1 ho h1 xo b5

m8=100;

d=1.0/m8;

b5=0.1;

e1=210*$107$;

ho=0.18;

A=zeros(m8-1,m8-1);

for i=1:m8-1

A(1,i)=1.0;

xo(i,1)=0.55;

x3(i,1)=0.55;

end;

lb=0.4*ones(m8-1,1);

ub=ones(m8-1,1);

b=zeros(m8-1,1);

b(1,1)=0.65*(m8-1);

for i=1:m8

f(i,1)=1.0;

L(i,1)=1/2;

P(i,1)=36.0*$102$;

end;

i=1;

m12=2;

m50(i)=1/m12;

z(i)=1/m50(i)*(-P(i,1)*$d$);

for i=2:m8-1

m12=2-m50(i-1);

m50(i)=1/m12;

z(i)=m50(i)*(-P(i,1)*$d$+z(i-1));

end;

v(m8,1)=0;

for i=1:m8-1

v(m8-i,1)=m50(m8-i)*v(m8-i+1,1)+z(m8-i);

end;

k=1;

b12=1.0;

while $(b12>10-)\mathrm{and}(k<10)$

k

k=k+1;

for i=1:m8-1

H(i,1)=b5*$L(i,)*ho$/12*e1;

f1(i,1)=v(i,1)/H(i,1);

f2(i,1)=i*d/H(i,1);

f3(i,1)=1/H(i,1);

end;

i=1;

m12=2;

m70(i)=1/m12;

z1(i)=m70(i)*(-f1(i,1)*$d$);

z2(i)=m70(i)*(-f2(i,1)*$d$);

z3(i)=m70(i)*(-f3(i,1)*$d$);

for i=2:m8-1

m12=2-m70(i-1);

m70(i)=1/m12;

z1(i)=m70(i)*(-f1(i,1)*$d$+z1(i-1));

z2(i)=m70(i)*(-f2(i,1)*$d$+z2(i-1));

z3(i)=m70(i)*(-f3(i,1)*$d$+z3(i-1));

end;

w1(m8,1)=0;

w2(m8,1)=0;

w3(m8,1)=0;

for i=1:m8-1

w1(m8-i,1)=m70(m8-i)*w1(m8-i+1,1)+z1(m8-i);

w2(m8-i,1)=m70(m8-i)*w2(m8-i+1,1)+z2(m8-i);

w3(m8-i,1)=m70(m8-i)*w3(m8-i+1,1)+z3(m8-i);

end;

m3(1,1)=w2(1,1);

m3(1,2)=w3(1,1);

m3(2,1)=w2(m8-1,1);

m3(2,2)=w3(m8-1,1);

h3(1,1)=-w1(1,1);

h3(2,1)=-w1(m8-1,1);

h5(:,1)=inv(m3)*h3;

for i=1:m8

wo(i,1)=w1(i,1)+h5(1,1)*w2(i,1)+h5(2,1)*w3(i,1);

end;

d2wo(1,1)=(-2*wo(1,1)+wo(2,1))/$d$;

for i=2:m8-1

d2wo(i,1)=(wo(i+1,1)-2*wo(i,1)+wo(i-1,1))/$d$;

end;

k9=1;

b14=1.0;

while $(b14>10-)\mathrm{and}(k<120)$

k9

k9=k9+1;

X=fmincon(’beamNov2023’,xo,A,b,$\left[\right],\left[\right]$,lb,ub);

b14=max(abs(xo-X))

xo=X;

end;

b12=max(abs(xo-x3))

x3=xo;

for i=1:m8-1

L(i,1)=xo(i,1);

end;

end;

*****************************

Remark 16.

About the numerical results obtained for these two beam models, a final word of caution is necessary.

Indeed, the full convergence in such cases is hard to obtain so that we have obtained just approximations of critical points with the functionals close to their optimal values. It is also worth emphasizing we have fixed the number of iterations so that the solutions and shapes obtained are just approximate ones.

31. Applications to the Optimal Shape Design for a Plate Model

In this section, we present a numerical procedure for the shape optimization concerning a thin plate model.

Let $\Omega =[,]\times [,]\subset \mathbb{R}$ corresponds to the middle surface of a thin plate with a variable thickness $h(x,y)$.

Define now

$$V=\{w\in W,\left(\Omega \right):w=\mathrm{on}\partial \Omega \},$$

which corresponds to a simply supported plate.

Consider the problem of minimizing in $V\times B$ the functional

$$J(w,h)={\displaystyle 12}\int \Omega H(x,y)(\nabla w(x,y))dx$$

subject to

$$\nabla \left[(H(x,y)\nabla w(x,y))\right]-P(x,y)=,\mathrm{in}\Omega ,$$

where

$$H(x,y)={\displaystyle h(x,y)12}E/(-w52),$$

$h=h(x,y)$ is variable plate thickness, E is the young elasticity model and $w=.$.

Also, we define

$$B=\left\{h:\Omega \to \mathbb{R}\mathrm{measurable}:hmin\le h(x,y)\le hmax\mathrm{and}\int \Omega h(x,y)\le chmax\right\},$$

where $<c<$ and

$$C*=\{w\in V:\nabla [H(x,y)\nabla w(x,y))]-P(x,y)=,\mathrm{in}\Omega \}.$$

Observe that

$$\begin{array}{ccc}& & \underset{(w,h)\in C*\times B}{inf}J(w,h)\hfill \\ & =& \underset{h\in B}{inf}\left\{\underset{w\in C*}{inf}J(w,h)\right\}\hfill \\ & =& \underset{h\in B}{inf}\left\{\underset{w^\in V}{sup}\left\{\underset{w\in V}{inf}\left\{{\displaystyle 12}\int \Omega H(x,y)[\nabla w(x,y)]dx-\langle w^,\nabla [H(x,y)\nabla w(x,y)]-P(x,y)\rangle L\right\}\right\}\right\}\hfill \\ & =& \underset{h\in B}{inf}\left\{\underset{w^\in V}{sup}\left\{-{\displaystyle 12}\int \Omega H(x,y)[\nabla w^(x,y)]dx+\langle w^,P\rangle L\right\}\right\}\hfill \\ & =& \underset{h\in B}{inf}\left\{\underset{M˜\in D*}{inf}\left\{{\displaystyle 12}\int \Omega {\displaystyle M˜H(x,y)}dx\right\}\right\}.\hfill \end{array}$$

(186)

where

$$D*=\{M˜\in Y*\nabla M˜-P=,\mathrm{in}\Omega ,\mathrm{and}M˜=,\mathrm{on}\Omega \}.$$

Summarizing, we have got

$$\underset{(w,h)\in C*\times B}{inf}J(w,h)=\underset{(M˜,h)\in D*\times B}{inf}\left\{{\displaystyle 12}\int \Omega {\displaystyle M˜H(x,y)}dx\right\}.$$

In order to obtain numerical results, we suggest the following primal dual procedure:

Set $n=$ and

$$hn\left(x\right)=chmax.$$

Calculate $wn\in V$ solution of equation

$$\nabla (Hn(x,y)\nabla wn(x,y))=P(x,y),$$

where

$$Hn(x,y)={\displaystyle Ehn\left(x\right)12(-w52)}.$$

Calculate $hn+\in B$ such that

$$J*(M˜n,hn+)=\underset{h\in B}{inf}J*(M˜n,h),$$

where

$$M˜n=Hn(x,y)\nabla wn,$$

$$J*(M˜,h)={\displaystyle 12}\int \Omega {\displaystyle M˜H(x,y)}dx.$$

Set $n:=n+$ and go to step 2 until an appropriate convergence criterion is satisfied.

We have developed numerical results for $c=.75$, $E=200105$, $P(x,y)=102N$, $hmin=.45*(.12)m$ and $hmax=.12m.$

We have also defined

$$h(x,y)=t(x,y)hmax,$$

where

$$.45\le t(x,y)\le ,\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega .$$

For the optimal solution $w=w(x,y)$, please see Figure 24.

For a corresponding optimal solution $t=t(x,y)$, please see Figure 25.

Remark 17.

For such a simply-supported plate model, for the numerical solution of equation

$$\nabla [H(x,y)\nabla w(x,y)]=P,$$

with the boundary conditions

$$w=\mathrm{on}\partial \Omega ,$$

firstly we have solved the equation

$$\nabla v-P=$$

with the boundary conditions

$$v=\mathrm{on}\partial \Omega .$$

Subsequently, we have solved the equation

$$H(x,y)\nabla w(x,y)=v(x,y)$$

with the boundary conditions

$$w=\mathrm{on}\partial \Omega .$$

Here we present the software developed in MAT-LAB.

*********************

clear all

global m8 d d2xwo d2ywo H e1 ho xo b5

m8=40;

d=1.0/m8;

w5=0.3;

e1=200*$105$/$(-w52)$;

ho=0.12;

A=zeros($(m-),(m-)$);

for i=1:$(m-)$

A(1,i)=1.0;

xo(i,1)=0.55;

x3(i,1)=0.55;

end;

lb=0.45*ones($(m-)$,1);

ub=ones($(m-)$,1);

b=zeros($(m-)$,1);

b(1,1)=0.75*$(m-);$

for i=1:(m8-1)

for j=1:m8-1

f(i,j,1)=1.0;

L(i,j,1)=1/2;

P(i,j,1)=2*$102$; end;

end;

for i=1:m8

wo(:,i)=0.001*ones(m8-1,1);

end;

m2=zeros(m8-1,m8-1);

for i=2:m8-2

m2(i,i)=-2.0;

m2(i,i-1)=1.0;

m2(i,i+1)=1.0;

end;

m2(1,1)=-2.0;

m2(1,2)=1.0;

m2(m8-1,m8-1)=-2.0;

m2(m8-1,m8-2)=1.0;

Id=eye(m8-1);

i=1;

m12=2*Id-m2*$d/d$; m50(:,:,i)=inv(m12);

z(:,i)=m50(:,:,i)*(-P(:,i,1)*$d$);

for i=2:m8-1

m12=2*Id-m2*$d/d$-m50(:,:,i-1);

m50(:,:,i)=inv(m12);

z(:,i)=m50(:,:,i)*(-P(:,i,1)*$d$+z(:,i-1));

end; v(:,m8)=zeros(m8-1,1);

for i=1:m8-1

v(:,m8-i)=m50(:,:,m8-i)*v(:,m8-i+1)+z(:,m8-i);

end;

k=1;

b12=1.0;

while ($b12>10-$) and ($k<12$)

k

k=k+1;

for i=1:m8-1

for j=1:m8-1

H(j,i,1)=$L(j,i,)*ho$/12*e1;

f1(j,i,1)=v(j,i)/H(j,i,1);

end;

end;

i=1;

m12=2*Id-m2*$d/d$;

m70(:,:,i)=inv(m12);

z1(:,i)=m70(:,:,i)*(-f1(:,i,1)*$d$);

for i=2:m8-1

m12=2*Id-m2*$d/d$-m70(:,:,i-1);

m70(:,:,i)=inv(m12);

z1(:,i)=m70(:,:,i)*(-f1(:,i,1)*$d$+z1(:,i-1));

end;

w(:,m8)=zeros(m8-1,1);

for i=1:m8-1

w(:,m8-i)=m70(:,:,m8-i)*w(:,m8-i+1)+z1(:,m8-i);

end;

d2xwo(:,1)=(-2*w(:,1)+w(:,2))/$d$;

for i=2:m8-1

d2xwo(:,i)=(w(:,i+1)-2*w(:,i)+w(:,i-1))/$d$;

end;

for i=1:m8-1

d2ywo(:,i)=m2*w(:,i)/$d$;

end;

k9=1; b14=1.0;

while ($b14>10-$) and ($k<30$)

k9

k9=k9+1;

X=fmincon(’beamNov2023A3’,xo,A,b,$\left[\right],\left[\right]$,lb,ub);

b14=max(abs(xo-X))

xo=X;

end;

b12=max(max(abs(w-wo)))

wo=w;

x3=xo;

for i=1:m8-1

for j=1:m8-1

L(j,i,1)=xo((i-1)*(m8-1)+j,1);

end;

end;

end;

for i=1:m8-1

x8(i,1)=i*d;

end;

mesh(x8,x8,L);

*********************

With the auxiliary function "beamNov2023A3’, where

****************************

function S=beamNov2023A3(x)

global m8 d d2xwo d2ywo H e1 ho xo b5

S=0;

for i=1:m8-1

for j=1:m8-1

x1(j,i)=x((m8-1)*(i-1)+j,1);

end;

end;

for i=1:m8-1

for j=1:m8-1

S=S+$/\left(\left(x(j,i)\right)\right)/ho/e*\left(H(j,i,)\right)*(dxwo(j,i)+dywo(j,i))*12$;

end;

end;

********************************

Remark 18.

About the numerical results obtained for this plate model, a final word of caution is necessary.

Indeed, the full convergence in such a case is hard to obtain so that we have obtained just approximations of critical points with the functional close to its optimal value. It is also worth emphasizing we have fixed the number of iterations so that the solution and shape obtained are just approximate ones.

32. A Note on the First Maxwell Equation of Electromagnetism

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega$.

Suppose $\mathbf{E}:\Omega \to \mathbb{R}$ is an electric field of resulting from a punctual charge q localized at $(,,)\in \Omega .$

Let $\Omega \subset \Omega$ be also an open, bounded and connected set with a regular ($C$ class) boundary denoted by $S=\partial \Omega$, such that $(,,)\in \Omega$, where, in polar coordinates $(r,\theta ,\varphi )\in \mathbb{R}$, we have

$$S=\left\{\mathbf{r}\right(\theta ,\varphi ):\le \theta \le \pi \mathrm{and}\le \varphi \le \pi \},$$

and where,

$$\mathbf{r}(\theta ,\varphi )=X(\theta ,\varphi )\mathbf{i}+Y(\theta ,\varphi )\mathbf{j}+Z(\theta ,\varphi )\mathbf{k}.$$

and

$$\mathbf{i}=(.,),\mathbf{j}=(,,),\mathbf{k}=(,,)\in \mathbb{R}.$$

Observe that denoting

$$\mathbf{e}r=sin\left(\theta \right)cos\left(\varphi \right)\mathbf{i}+sin\left(\theta \right)sin\left(\varphi \right)\mathbf{j}+cos\left(\theta \right)\mathbf{k},$$

we have

$$\mathbf{E}=K^q{\displaystyle r}\mathbf{e}r,$$

for an appropriate real constant $K^.$

We highlight the integral

$$In=\int S\mathbf{E}·\mathbf{n}dS=c$$

where such a real constant $c$ does not depend on the $C$ class function

$$\mathbf{r}:[,\pi ]\times [,\pi ]\to \mathbb{R}.$$

Here $\mathbf{n}$ denotes the normal outward field to S.

In this section, we develop such an integral $I$, in details.

Denoting

$$R(\theta ,\varphi )=\parallel \mathbf{r}(\theta ,\varphi )\parallel \mathbb{R},$$

we have

$$\mathbf{r}(\theta ,\varphi )=R(\theta ,\varphi )\mathbf{e}r.$$

Moreover, we recall that

$$dS=\parallel \mathbf{r}\theta \times \mathbf{r}\varphi \parallel \mathbb{R}d\theta d\varphi ,$$

where

$$N=\mathbf{r}\theta \times \mathbf{r}\varphi$$

is such that

$$\mathbf{n}={\displaystyle N\parallel N\parallel \mathbb{R}}.$$

Thus

$$\int S\mathbf{E}·\mathbf{n}dS=K^q\int \varphi =\varphi =\pi \int \theta =\theta =\pi {\displaystyle R(\theta ,\varphi )}\mathbf{e}r·{\displaystyle N\parallel N\parallel \mathbb{R}}\parallel N\parallel \mathbb{R}d\theta d\varphi .$$

In summary,

$$\int S\mathbf{E}·\mathbf{n}dS=K^q\int \varphi =\varphi =\pi \int \theta =\theta =\pi {\displaystyle R(\theta ,\varphi )}\mathbf{e}r·Nd\theta d\varphi .$$

Observe that

$$\begin{array}{ccc}\hfill N& =& \left(R(\theta ,\varphi )\mathbf{e}r\right)\theta \times \left(R(\theta ,\varphi )\mathbf{e}r\right)\varphi \hfill \\ & =& (R\theta \mathbf{e}r+R\left(\mathbf{e}r\right)\theta )\times (R\varphi \mathbf{e}r+R\left(\mathbf{e}r\right)\varphi )\hfill \\ & =& R\theta R\varphi \mathbf{e}r\times \mathbf{e}r+R\theta \mathbf{e}r\times R\left(\mathbf{e}r\right)\varphi \hfill \\ & & +R\left(\mathbf{e}r\right)\theta \times \left(R\varphi \mathbf{e}r\right)+R\left(\mathbf{e}r\right)\theta \times \left(\mathbf{e}r\right)\varphi .\hfill \end{array}$$

(187)

Consequently, from such results, we obtain

$$\mathbf{e}r·N=R\mathbf{e}r·\left(\mathbf{e}r\right)\theta \times \left(\mathbf{e}r\right)\varphi .$$

Thus, we have got

$$\begin{array}{ccc}\hfill \int S\mathbf{E}·\mathbf{n}dS& =& K^q\int \varphi =\varphi =\pi \int \theta =\theta =\pi {\displaystyle R(\theta ,\varphi )}\mathbf{e}r·Nd\theta d\varphi \hfill \\ & =& K^q\int \varphi =\varphi =\pi \int \theta =\theta =\pi {\displaystyle R(\theta ,\varphi )}R(\theta ,\varphi )\mathbf{e}r·\left(\mathbf{e}r\right)\theta \times \left(\mathbf{e}r\right)\varphi d\theta d\varphi \hfill \\ & =& K^q\pi \hfill \\ & \equiv & c.\hfill \end{array}$$

(188)

Summarizing, we have got

$$\int S\mathbf{E}·\mathbf{n}dS=c.$$

where such a real constant does not depend on function $\mathbf{r}:[,\pi ]\times [,\pi ]\to \mathbb{R}.$

Consider now a charge $q$ localized at the center of a sphere $\Omega$ of radius $R>$ and boundary $S=\partial \Omega .$

The electric field on the sphere surface generated by $q$ is given by

$$\mathbf{E}={\displaystyle \pi \epsilon }{\displaystyle qR}\mathbf{n},$$

where $\mathbf{n}$ is the normal outward field to $S$.

Clearly

$$\int S\mathbf{E}·\mathbf{n}dS={\displaystyle \pi \epsilon }{\displaystyle qR}\left(\pi R\right)={\displaystyle q\epsilon }.$$

Consider again the set $\Omega$ but now with a charge $q$ localized at a point $\mathbf{x}$ inside the interior of $\Omega$, which is denoted by $\Omega$.

At first the electric field $\mathbf{E}$ generated by $q$ is not of $C$ class on $\Omega .$

However, there exists $R>$ such that

$$BR\left(\mathbf{x}\right)\subset \Omega =\Omega .$$

Define $\Omega =\Omega \setminus BR\left(\mathbf{x}\right)$.

Therefore, $\mathbf{E}$ is of $C$ class on $\Omega .$

Denoting the boundary of $\Omega$ by $S=\partial \Omega \cup \partial BR\left(\mathbf{x}\right)$, from the previous results and denoting again $S=\partial \Omega$, we may infer that

$$\int S\mathbf{E}·\mathbf{n}dS=\int S\mathbf{E}·\mathbf{n}dS-\int \partial BR\left(\mathbf{x}\right)\mathbf{E}·\mathbf{n}dS=c-c=,$$

for a not relabeled real constant $c$, so that

$$\begin{array}{ccc}\hfill \int S\mathbf{E}·\mathbf{n}dS& =& \int S\mathbf{E}·\mathbf{n}dS-\int \partial BR\left(\mathbf{x}\right)\mathbf{E}·\mathbf{n}dS\hfill \\ & =& \int S\mathbf{E}·\mathbf{n}dS-{\displaystyle q\epsilon }\hfill \\ & =& .\hfill \end{array}$$

(189)

Therefore, we have got

$$\int S\mathbf{E}·\mathbf{n}dS={\displaystyle q\epsilon }.$$

Assume now on $\Omega$ we have a density of charges $\rho \left(\mathbf{x}\right)$.

For a small volume $\Delta V˜$ consider a punctual charge $q$ localized in $\mathbf{x}\in \Omega$ such that

$$q\approx \rho \mathbf{x})\Delta V˜.$$

Denoting by $\Delta \mathbf{E}$ the electric field generated by $q$, from the previous results we may infer that

$$\int S\Delta \mathbf{E}·\mathbf{n}dS={\displaystyle q\epsilon }\approx {\displaystyle \rho \left(\mathbf{x}\right)\Delta V˜\epsilon }.$$

Such an equation in its differential form, stands for:

$$\int Sd\mathbf{E}·\mathbf{n}dS={\displaystyle \rho \left(\mathbf{x}\right)dV˜\epsilon }.$$

Integrating in $\Omega$ we may obtain

$$\begin{array}{ccc}\hfill \int S\mathbf{E}·\mathbf{n}dS& =& \int S\int \Omega d\mathbf{E}·\mathbf{n}dS\hfill \\ & =& \int \Omega {\displaystyle \rho \left(\mathbf{x}\right)\epsilon }dV˜,\hfill \end{array}$$

(190)

so that

$$\int S\mathbf{E}·\mathbf{n}dS=\int \Omega {\displaystyle \rho \left(\mathbf{x}\right)\epsilon }dV˜.$$

From this and the Divergence Theorem, we have

$$\int S\mathbf{E}·\mathbf{n}dS=\int \Omega \mathrm{div}\mathbf{E}dV˜=\int \Omega {\displaystyle \rho \left(\mathbf{x}\right)\epsilon }dV˜.$$

Summarizing, we have got

$$\int \Omega \mathrm{div}\mathbf{E}dV˜=\int \Omega {\displaystyle \rho \left(\mathbf{x}\right)\epsilon }dV˜.$$

This is the integral form of the first Maxwell equation of electromagnetism.

For this last equation, the set $\Omega \subset \Omega$ is rather arbitrary so that for $\Omega$ as a ball of small radius $r>$ with center at a point $\mathbf{x}\in \Omega$, from the Mean Value Theorem fot integrals and letting $r\to +$, we obtain

$$\mathrm{div}\mathbf{E}={\displaystyle \rho \epsilon },\mathrm{in}\Omega .$$

This last equation stands for the differential form of the first Maxwell equation of electromagnetism.

Remark 19.

Summarizing, in this section we have formally obtained a mathematical deduction of the first Maxwell equation of electromagnetism.

33. A note on relaxation for a general model in the vectorial calculus of variations

Let $\Omega \subset \mathbb{R}n$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider a function $g:\mathbb{R}N\times n\to \mathbb{R}$ twice differentiable and such that

$$g\left(y\right)\to +\infty ,\mathrm{as}\left|y\right|\to +\infty .$$

Define a functional $G:V\to \mathbb{R}$ by

$$G(\nabla u)={\displaystyle 12}\int \Omega g(\nabla u)dx,$$

where

$$V=\{W,(\Omega ;\mathbb{R}N):u=u\mathrm{on}\partial \Omega \}.$$

Moreover, for $f\in L(\Omega ;\mathbb{R}N)$, define also

$$J\left(u\right)=G(\nabla u)-\langle u,f\rangle L.$$

We assume there exists $\alpha \in \mathbb{R}$ such that

$$\alpha =\underset{u\in V}{inf}J\left(u\right).$$

Observe that from the convex analysis basic theory, we have that

$$\begin{array}{ccc}\hfill \alpha & =& \underset{u\in V}{inf}J\left(u\right)\hfill \\ & =& \underset{u\in V}{inf}J**\left(u\right)\hfill \\ & =& \underset{u\in V}{inf}\{(G\circ \nabla )**\left(u\right)-\langle u,f\rangle L\}.\hfill \end{array}$$

(191)

On the other hand

$$\begin{array}{ccc}\hfill (G\circ \nabla )**\left(u\right)& \le & H\left(u\right)\hfill \\ & \equiv & \underset{(\lambda ,(v,w\left)\right)\in [,]\times B(u,\lambda )}{inf}\{\lambda G(\nabla w)+(-\lambda )G(\nabla v)\}\hfill \\ & \le & G(\nabla u),\hfill \end{array}$$

(192)

where

$$B(u,\lambda )=\left\{\right(v,w)\in V:\lambda w+(-\lambda )v=u\}.$$

From such results, we may infer that

$$\underset{u\in V}{inf}J**\left(u\right)=\underset{u\in V}{inf}\{H\left(u\right)-\langle u,f\rangle L\}=\underset{u\in V}{inf}J\left(u\right).$$

Furthermore, observe that

$$\lambda \nabla w+(-\lambda )\nabla v=\nabla u,$$

so that

$$\begin{array}{ccc}\hfill \nabla v& =& \nabla u+\lambda (\nabla v-\nabla w)\hfill \\ & =& \nabla u+\lambda \nabla \varphi ,\hfill \end{array}$$

(193)

where $\varphi =v-w\in W,(\Omega ;\mathbb{R}N)$ so that

$$\nabla \varphi =\nabla v-\nabla w,$$

and

$$\nabla w=\nabla v-\nabla \varphi .$$

Therefore,

$$\nabla w=\nabla v-\nabla \varphi =\nabla u+\lambda \nabla \varphi -\nabla \varphi =\nabla u-(-\lambda )\nabla \varphi .$$

Replacing such results into the expression of H, we have

$$H\left(u\right)=\underset{(\lambda ,\varphi )\in [,]\times V}{inf}\{\lambda G(\nabla u-(-\lambda )\nabla \varphi )+(-\lambda )G(\nabla u+\lambda \nabla \varphi )\},$$

where

$$V=W,(\Omega ;\mathbb{R}N).$$

Joining the pieces, we have got

$$\begin{array}{ccc}\hfill \underset{u\in V}{inf}J\left(u\right)& =& \underset{u\in V}{inf}J**\left(u\right)\hfill \\ & =& \underset{u\in V}{inf}\{H\left(u\right)-\langle u,f\rangle L\}\hfill \\ & =& \underset{(\lambda ,\varphi ,u)\in [,]\times V\times V}{inf}\{\lambda G(\nabla u-(-\lambda )\nabla \varphi )+(-\lambda )G(\nabla u+\lambda \nabla \varphi )-\langle u,f\rangle L\}.\hfill \end{array}$$

This last functional corresponds to a relaxation for the original non-convex functional.

The note is complete.

33.1. Some related numerical results

In this subsection we present numerical results for an one-dimensional model and related relaxed formulation.

For $\Omega =[,]\subset \mathbb{R}$, consider the functional $J:V\to \mathbb{R}$ where

$$J\left(u\right)={\displaystyle 12}\int \Omega ((u\prime )-)dx+{\displaystyle 12}\int \Omega (u-f)dx,$$

$$V=\{u\in W,\left(\Omega \right):u\left(\right)=\mathrm{and}u\left(\right)=/\},$$

$f\in Y=Y*=L\left(\Omega \right).$

Based on the results of the previous section, denoting $V=W,\left(\Omega \right),$ we define the following relaxed functional $J:[,]\times V\times V\to \mathbb{R},$ where

$$J(\lambda ,u,\varphi )={\displaystyle \lambda }\int \Omega ((u\prime -(-\lambda )\varphi \prime )-)dx+{\displaystyle -\lambda }\int \Omega ((u\prime +\lambda \varphi \prime )-)dx+{\displaystyle 12}\int \Omega (u-f)dx.$$

Indeed, we have developed an algorithm for minimizing the following regularized functional $J:[,]\times V\times V\to \mathbb{R},$ where

$$J(\lambda ,u,\varphi )=J(\lambda ,u,\varphi )+{\displaystyle \epsilon }\int \Omega \left(u\prime \prime \right)dx,$$

for a small parameter $\epsilon >$.

For the case in which $f\left(x\right)=sin\left(\pi x\right)/$, for the optimal solution u, please see Figure 26.

For the case in which $f\left(x\right)=cos\left(\pi x\right)/$, for the optimal solution u, please see Figure 27.

For the case in which $f\left(x\right)=$, for the optimal solution u, please see Figure 28.

We highlight to obtain the solution for this last case which $f=$ is harder. A good solution was possible only using

$$x=$$

as the initial solution concerning the iterative process.

Here we present the software in MAT-LAB developed.

*****************

clear all

global m8 d u e3

m8=100;

d=1/m8;

e3=0.0005;

for i=1:2*m8+1

xo(i,1)=0.36;

end;

b12=1.0;

k=1;

while $(b12>10-)\mathrm{and}(k<60)$

k

k=k+1;

X=fminunc(’funDecember2023’,xo);

b12=max(abs(xo-X))

xo=X;

u(m8/2)

end;

for i=1:m8

x(i,1)=i*d;

end;

plot(x,u);

***********************

With the main function "funDecember2023"

*********************

function S=funDecember2023(x)

global m8 d u e3

for i=1:m8

u(i,1)=x(i,1);

v(i,1)=x(i+m8,1);

yo(i,1)=sin(pi*i*d)/2;

end;

L=(1+sin(x(2*m8+1,1)))/2;

u(m8,1)=1/2;

v(m8,1)=0.0;

du(1,1)=u(1,1)/d;

dv(1,1)=v(1,1)/d;

for i=2:m8

du(i,1)=(u(i,1)-u(i-1,1))/d;

dv(i,1)=(v(i,1)-v(i-1,1))/d;

end;

d2u(1,1)=(-2*u(1,1)+u(2,1))/$d$;

for i=2:m8-1

d2u(i,1)=(u(i-1,1)-2*u(i,1)+u(i+1,1))/$d$;

end;

S=0;

for i=1:m8

S=S+$/*L*((du(i,)-(-L)*dv(i,))-)$;

S=S+$/*(-L)*((du(i,)+L*dv(i,))-)$;

S=S+$(u(i,)-yo(i,))$;

end;

for i=1:m8-1

S=S+e3*$du(i,)$;

end;

*******************

33.2. A related duality principle and concerning convex dual formulation

With the notation and statements of the previous sections in mind, consider the functionals $J:V\to \mathbb{R}$ and $J:[,]\times V\times V\to \mathbb{R}$ where

$$J\left(u\right)=G(\nabla u)+{\displaystyle 12}\int \Omega u·udx-\langle u,f\rangle L,$$

and

$$\begin{array}{ccc}\hfill J(\lambda ,u,\varphi )& =& \lambda G(\nabla u-(-\lambda )\nabla \varphi )+(-\lambda )G(\nabla u+\lambda \nabla \varphi )\hfill \\ & & +{\displaystyle \lambda }\int \Omega (u-(-\lambda )\varphi )·(u-(-\lambda )\varphi )dx\hfill \\ & & +{\displaystyle (-\lambda )}\int \Omega (u+\lambda \varphi )·(u+\lambda \varphi )dx\hfill \\ & & -\lambda \langle u-(-\lambda )\varphi ,f\rangle L-(-\lambda )\langle u+\lambda \varphi ,f\rangle L.\hfill \end{array}$$

(194)

Here we have denoted

$$V=\{u\in W,(\Omega ;\mathbb{R}N):u=u\mathrm{on}\partial \Omega =S\},$$

$$V=W,(\Omega ;\mathbb{R}N),$$

$$Y=Y*=L(\Omega ;\mathbb{R}N\times n)$$

and

$$Y=Y*=L(\Omega ;\mathbb{R}N).$$

Observe that

$$J**\left(u\right)\le \underset{(\lambda ,\varphi )\in [,]\times V}{min}J(\lambda ,u,\varphi ).$$

Moreover,

$$\begin{array}{ccc}\hfill J(\lambda ,u,\varphi )& =& -\langle \nabla u-(-\lambda )\nabla \varphi ,v*\rangle L+\lambda G(\nabla u-(-\lambda )\nabla \varphi )\hfill \\ & & -\langle \nabla u-(-\lambda )\nabla \varphi ,v*\rangle L+(-\lambda )G(\nabla u+\lambda \nabla \varphi )\hfill \\ & & -\langle u-(-\lambda )\varphi ,v*\rangle L+{\displaystyle \lambda }\int \Omega (u-(-\lambda )\varphi )·(u-(-\lambda )\varphi )dx\hfill \\ & & -\langle u+\lambda \varphi ,v*\rangle L+{\displaystyle (-\lambda )}\int \Omega (u+\lambda \varphi )·(u+\lambda \varphi )dx\hfill \\ & & +\langle \nabla u-(-\lambda )\nabla \varphi ,v*\rangle L+\langle \nabla u-(-\lambda )\nabla \varphi ,v*\rangle L\hfill \\ & & +\langle u-(-\lambda )\varphi ,v*\rangle L+\langle u+\lambda \varphi ,v*\rangle L\hfill \\ & & -\lambda \langle u-(-\lambda )\varphi ,f\rangle L-(-\lambda )\langle u+\lambda \varphi ,f\rangle L.\hfill \end{array}$$

(195)

Therefore,

$$\begin{array}{ccc}\hfill J(\lambda ,u,\varphi )& \ge & \underset{v\in Y}{inf}\{-\langle v,v*\rangle L+\lambda G\left(v\right)\}\hfill \\ & & +\underset{v\in Y}{inf}\{-\langle v,v*\rangle L+(-\lambda )G\left(v\right)\}\hfill \\ & & +\underset{v\in Y}{inf}\left\{-\langle v,v*\rangle L+{\displaystyle \lambda }\int \Omega \left(v\right)·\left(v\right)dx\right\}\hfill \\ & & +\underset{v\in Y}{inf}\left\{-\langle v,v*\rangle L+{\displaystyle (-\lambda )}\int \Omega \left(v\right)·\left(v\right)dx\right\}\hfill \\ & & +\underset{(u,\varphi )\in V\times V}{inf}\{\langle \nabla u-(-\lambda )\nabla \varphi ,v*\rangle L+\langle \nabla u-(-\lambda )\nabla \varphi ,v*\rangle L\hfill \\ & & +\langle u-(-\lambda )\varphi ,v*\rangle L+\langle u+\lambda \varphi ,v*\rangle L\hfill \\ & & -\lambda \langle u-(-\lambda )\varphi ,f\rangle L-(-\lambda )\langle u+\lambda \varphi ,f\rangle L\}\hfill \\ & =& -\lambda G*\left({\displaystyle v*\lambda }\right)-(-\lambda )G*\left({\displaystyle v*(-\lambda )}\right)\hfill \\ & & -F*(v*,\lambda )-F*(v*,\lambda )\hfill \\ & & +\int S(v*)ijnj\left(u\right)idS+\int S(v*)ijnj\left(u\right)idS,\hfill \\ & & \forall \lambda \in (,),u\in V,\varphi \in V,v*\in A*,\hfill \end{array}$$

(196)

where

$$G*(v*)=\underset{v\in Y}{sup}\{\langle v,v*\rangle L-G\left(v\right)\},$$

$$\begin{array}{ccc}\hfill F*(v*,\lambda )& =& \underset{v\in Y}{sup}\left\{\langle v,v*\rangle L-{\displaystyle \lambda }\int \Omega v·vdx\right\}\hfill \\ & =& {\displaystyle \lambda }\int \Omega v*·v*dx,\hfill \end{array}$$

(197)

$$\begin{array}{ccc}\hfill F*(v*,\lambda )& =& \underset{v\in Y}{sup}\left\{\langle v,v*\rangle L-{\displaystyle (-\lambda )}\int \Omega v·vdx\right\}\hfill \\ & =& {\displaystyle (-\lambda )}\int \Omega v*·v*dx.\hfill \end{array}$$

(198)

Furthermore, $A*=A*\cap A*$ where

$$A*=\{v*=(v*,v*,v*,v*)\in [Y*]\times [Y*]:-\mathrm{div}(v*)i-\mathrm{div}(v*)i+(v*)i+(v*)i-fi=,\mathrm{in}\Omega \},$$

and

$$\begin{array}{ccc}\hfill A*& =& \{v*=(v*,v*,v*,v*)\in [Y*]\times [Y*]:\hfill \\ & & -(-+\lambda )\mathrm{div}(v*)i-\lambda \mathrm{div}(v*)i+(-+\lambda )(v*)i+\lambda (v*)i=,\mathrm{in}\Omega \}.\hfill \end{array}$$

(199)

Summarizing, we have got

$$\begin{array}{ccc}& & \underset{(\lambda ,u\varphi )\in (,)\times V\times V}{inf}J(\lambda ,u,\varphi )\hfill \\ & \ge & \underset{v*\in A*}{sup}\left\{\underset{\lambda \in (,)}{inf}\left\{-\lambda G*\left({\displaystyle v*\lambda }\right)-(-\lambda )G*\left({\displaystyle v*(-\lambda )}\right)\right.\right.\hfill \\ & & \left.\left.-F*(v*,\lambda )-F*(v*,\lambda )+\int \partial \Omega (v*)ijnj\left(u\right)idS+\int \partial \Omega (v*)ijnj\left(u\right)idS\right\}\right\}.\hfill \end{array}$$

(200)

Remark 20.

We highlight this last dual function in $v$ is convex (in fact concave) on the convex set $A$.

33.3. A numerical example

For $\Omega =[,]\subset \mathbb{R}$ consider a functional $J:V\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J\left(u\right)& =& {\displaystyle 12}\int \Omega min\{(u\prime \left(x\right)-),(u\prime \left(x\right)+)\}dx+{\displaystyle 12}\int \Omega (u-f)dx\hfill \\ & =& {\displaystyle 12}\int \Omega (u\prime )dx-\int \Omega |u\prime |dx+{\displaystyle 12}\int \Omega (u-f)dx,\hfill \end{array}$$

(201)

where

$$V=\{u\in W,\left(\Omega \right):u\left(\right)=\mathrm{and}u\left(\right)=/\},$$

$Y=Y*=L\left(\Omega \right)$ and $f\in Y.$

Define $G:Y\to \mathbb{R}$ and $F:V\to \mathbb{R}$ by

$$G(u\prime )={\displaystyle 12}\int \Omega (u\prime )dx-\int \Omega |u\prime |dx,$$

and

$$F\left(u\right)={\displaystyle 12}\int \Omega udx,$$

respectively.

Denoting $V=W,\left(\Omega \right),$ define also $J:V\times V\times (,)\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill J(u,\varphi ,\lambda )& =& \lambda G(u\prime -(-\lambda )\varphi \prime )+(-\lambda )G(u\prime +\lambda \varphi \prime )\hfill \\ & & +\lambda F(u-(-\lambda \left)\varphi \right)+(-\lambda )F(u+\lambda \varphi )\hfill \\ & & -\langle u,f\rangle L.\hfill \end{array}$$

(202)

Observe that

$$\begin{array}{ccc}\hfill \left(\lambda G\right)*(v*)& =& \underset{v\in Y}{sup}\{\langle v,v*\rangle L-\lambda G\left(v\right)\}\hfill \\ & =& \lambda G*\left({\displaystyle v*\lambda }\right)\hfill \\ & =& {\displaystyle \lambda }\int \Omega (v*)dx+\int \Omega |v*|dx,\hfill \end{array}$$

(203)

$$\begin{array}{ccc}\hfill \left((-\lambda )G\right)*(v*)& =& \underset{v\in Y}{sup}\{\langle v,v*\rangle L-(-\lambda )G\left(v\right)\}\hfill \\ & =& (-\lambda )G*\left({\displaystyle v*(-\lambda )}\right)\hfill \\ & =& {\displaystyle (-\lambda )}\int \Omega (v*)dx+\int \Omega |v*|dx,\hfill \end{array}$$

(204)

$$\begin{array}{ccc}\hfill \left(\lambda F\right)*(v*)& =& \underset{v\in Y}{sup}\{\langle v,v*\rangle L-\lambda F\left(v\right)\}\hfill \\ & =& \lambda F*\left({\displaystyle v*\lambda }\right)\hfill \\ & =& {\displaystyle \lambda }\int \Omega (v*)dx,\hfill \end{array}$$

(205)

and

$$\begin{array}{ccc}\hfill \left((-\lambda )F\right)*(v*)& =& \underset{v\in Y}{sup}\{\langle v,v*\rangle L-(-\lambda )F\left(v\right)\}\hfill \\ & =& (-\lambda )F*\left({\displaystyle v*(-\lambda )}\right)\hfill \\ & =& {\displaystyle (-\lambda )}\int \Omega (v*)dx.\hfill \end{array}$$

(206)

Denoting $v*=(v*,\dots ,v*)\in [Y*]$, define $J*:[Y*]\times (,)\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill J*(v*,\lambda )& =& -\lambda G*\left({\displaystyle v*\lambda }\right)-(-\lambda )G*\left({\displaystyle v*(-\lambda )}\right)\hfill \\ & & -\lambda F*\left({\displaystyle v*\lambda }\right)-(-\lambda )F*\left({\displaystyle v*(-\lambda )}\right)\hfill \\ & & +v*\left(\right)u\left(\right)+v*\left(\right)u\left(\right).\hfill \end{array}$$

(207)

Similarly as in the previous section, we may obtain

$$\underset{u\in V}{inf}J\left(u\right)\ge \underset{\lambda \in (,)}{inf}\left\{\underset{v*\in A*}{sup}J*(v*,\lambda )\right\},$$

where $A*=A*\cap A*$,

$$A*=\{v*\in Y*:(v*)\prime +(v*)\prime -v*-v*+f=,\mathrm{in}\Omega \},$$

and

$$A*=\{(v*,\lambda )\in [Y*]\times (,):-(-\lambda )(v*)\prime +\lambda (v*)\prime +(-\lambda )v*-\lambda v*=,\mathrm{in}\Omega \}.$$

From such expressions of $A$ and $A$ we may obtain

$$v*=(v*)\prime +\lambda f,$$

and

$$v*=(v*)\prime +(-\lambda )f.$$

Replacing such expressions for $v$ and $v$ into the expression of $J$, and from now and on denoting $v*=(v*,v*)\in [Y*]$, we may obtain $J*:[Y*]\times (,]\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J*(v*,\lambda )& =& -{\displaystyle \lambda }\int \Omega (v*)dx-\int \Omega |v*|dx\hfill \\ & & -{\displaystyle (-\lambda )}\int \Omega (v*)dx-\int \Omega |v*|dx\hfill \\ & & -{\displaystyle \lambda }\int \Omega ((v*)\prime +\lambda f)dx\hfill \\ & & -{\displaystyle \left(\lambda \right)}\int \Omega ((v*)\prime +(-\lambda )f)dx\hfill \\ & & +v*\left(\right)u\left(\right)+v*\left(\right)u\left(\right).\hfill \end{array}$$

(208)

Consequently, we have got

$$\underset{u\in V}{inf}J\left(u\right)\ge \underset{v*\in [Y*]}{sup}\left\{\underset{\lambda \in (,)}{inf}J*(v*,\lambda )\right\}.$$

In order to obtain numerical results we have designed the following algorithm:

Set $n=$ and $\lambda n=/$.

Calculate $(v*)n\in [Y*]$ such that

$$J*((v*)n,\lambda n)=\underset{v*\in [Y*]}{sup}J*(v*,\lambda n).$$

Calculate $\lambda n+\in (,)$ such that

$$J*((v*)n,\lambda n+)=\underset{\lambda \in (,)}{inf}J*((v*)n,\lambda ).$$

Set $n:=n+$ and go to item (2) until the satisfaction of an appropriate convergence criterion.

We have developed numerical results for the following cases

$$f\left(x\right)=sin\left(\pi x\right)/,$$

$$f\left(x\right)=cos\left(\pi x\right)/,$$

$$f\left(x\right)=.$$

Observe that for the optimal point we have

$$v*=u-(-\lambda )\varphi ,$$

and

$$v*=u+\lambda \varphi ,$$

so that

$$u=\lambda v*+(-\lambda )v*.$$

For the optimal solution $u\left(x\right)$ found for the cases (1), (2) and (3), please see the Figure 29, Figure 30 and Figure 31, respectively.

Here we present the concerning software in MAT-LAB.

************************

clear all

global m8 d L v1 v2 v3 v4 yo dv1 dv2 e1

m8=140;

d=1/m8;

e1=0.0001;

L=1/2;

for i=1:2*m8

xo(i,1)=0.01;

end;

for i=1:m8

yo(i,1)=sin(pi*i*d)/2;

end;

x1=1/2;

k=1;

b12=1;

while $(b12>10-)$ and $(k<100)$

k

k=k+1;

X1=fminunc(’funFeb24’,xo);

b12=max(abs(X1-xo))

xo=X1;

X2=fminunc(’funFeb24A’,x1);

x1=X2;

L=(sin(x1)+1)/2;

L

end;

u(m8,1)=1/2;

for i=1:m8-1

u(i,1)=L*v3(i,1)+(1-L)*v4(i,1);

end;

for i=1:m8

x(i,1)=i*d;

end;

plot(x,u);

***********************************

Here the auxiliary function "funFeb24"

********************************

function S=funFeb24(x)

global m8 d L v1 v2 v3 v4 yo dv1 dv2 e1

for i=1:m8

v1(i,1)=x(i,1);

v2(i,1)=x(m8+i,1);

end;

for i=1:m8-1

dv1(i,1)=(v1(i+1,1)-v1(i,1))/d;

dv2(i,1)=(v2(i+1,1)-v2(i,1))/d;

end;

S=0;

for i=1:m8

S=S+$//sqrt(L+e)*v(i,)+sqrt(v(i,)+e)$;

S=S+$//sqrt((-L)+e)*v(i,)+sqrt(v(i,)+e)$;

end;

for i=1:m8-1

v3(i,1)=dv1(i,1)+L*yo(i,1);

v4(i,1)=dv2(i,1)-(L-1)*yo(i,1);

S=S+$//sqrt(L+e)*v(i,)+//sqrt((-L)+e)*v(i,)$;

end;

S=S-(v1(m8,1)+v2(m8,1))/d/2;

*********************************

Finally, the auxiliary function "funFeb24A"

**********************************************

function S1=funFeb24A(y)

global m8 d L v1 v2 v3 v4 yo e1

L=(sin(y)+1)/2;

for i=1:m8-1

dv1(i,1)=(v1(i+1,1)-v1(i,1))/d;

dv2(i,1)=(v2(i+1,1)-v2(i,1))/d;

end;

S=0;

for i=1:m8

S=S+$//sqrt(L+e)*v(i,)+sqrt(v(i,)+e)$;

S=S+$//sqrt((-L)+e)*v(i,)+sqrt(v(i,)+e)$;

end;

for i=1:m8-1

v3(i,1)=dv1(i,1)+L*yo(i,1);

v4(i,1)=dv2(i,1)-(L-1)*yo(i,1);

S=S+$//sqrt(L+e)*v(i,)+//sqrt((-L)+e)*v(i,)$;

end;

S=S-(v1(m8,1)+v2(m8,1))/d/2;

S1=-S;

*************************

34. One More Note on Relaxation for a General Model in the Vectorial Calculus of Variations

Let $\Omega \subset \mathbb{R}n$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider a function $g:\mathbb{R}N\times n\to \mathbb{R}$ twice differentiable and such that

$$g\left(y\right)\to +\infty ,\mathrm{as}\left|y\right|\to +\infty .$$

Define a functional $G:V\to \mathbb{R}$ by

$$G(\nabla u)={\displaystyle 12}\int \Omega g(\nabla u)dx,$$

where

$$V=\{W,(\Omega ;\mathbb{R}N):u=u\mathrm{on}\partial \Omega \}.$$

Moreover, for $f\in L(\Omega ;\mathbb{R}N)$, define also

$$J\left(u\right)=G(\nabla u)-\langle u,f\rangle L.$$

We assume there exists $\alpha \in \mathbb{R}$ such that

$$\alpha =\underset{u\in V}{inf}J\left(u\right).$$

Observe that from the convex analysis basic theory, we have that

$$\begin{array}{ccc}\hfill \alpha & =& \underset{u\in V}{inf}J\left(u\right)\hfill \\ & =& \underset{u\in V}{inf}J**\left(u\right)\hfill \\ & =& \underset{u\in V}{inf}\{(G\circ \nabla )**\left(u\right)-\langle u,f\rangle L\}.\hfill \end{array}$$

(209)

On the other hand

$$\begin{array}{ccc}\hfill (G\circ \nabla )**\left(u\right)& \le & H\left(u\right)\hfill \\ & \equiv & \underset{(\lambda ,(v,\dots ,vm))\in B\times B(u,\lambda )}{inf}\left\{\sum j=m\lambda jG(\nabla vj)\right\}\hfill \\ & \le & G(\nabla u),\hfill \end{array}$$

(210)

where

$$B=\left\{\lambda =(\lambda ,\dots ,\lambda m)\in \mathbb{R}m:\lambda j\ge ,\forall j\in \{,\dots ,m\},\mathrm{and}\sum j=m\lambda j=\right\},$$

and

$$B(u,\lambda )=\left\{v=(v,\dots ,vm)\in \left[V\right]m:\sum j=m\lambda jvj=u\right\}.$$

From such results, we may infer that

$$\underset{u\in V}{inf}J**\left(u\right)=\underset{u\in V}{inf}\{H\left(u\right)-\langle u,f\rangle L\}=\underset{u\in V}{inf}J\left(u\right).$$

Furthermore, observe that

$$\sum j=m\lambda j\nabla vj=\nabla u,$$

and

$$\lambda m=-\sum j=m-\lambda j,$$

so that

$$\begin{array}{ccc}\hfill \nabla vm& =& \nabla u-\sum j=m-\lambda j(\nabla vj-\nabla vm)\hfill \\ & =& \nabla u+\sum j=m-\lambda j\nabla \varphi j,\hfill \end{array}$$

(211)

where $\varphi j=-vj+vm\in W,(\Omega ;\mathbb{R}N)$ so that

$$\nabla \varphi j=-\nabla vj+\nabla vm,$$

and

$$\nabla vm=\nabla vj+\nabla \varphi j,\forall j\in \{,\dots ,m\}.$$

Therefore,

$$\nabla vj=\nabla vm-\nabla \varphi j=\nabla u+\sum k=m-\lambda k\nabla \varphi k-\nabla \varphi j.$$

Replacing such results into the expression of H, we have

$$H\left(u\right)=\underset{(\lambda ,\varphi )\in B\times \left(V\right)m-}{inf}\left\{\sum j=m-\lambda jG\left(\nabla u+\sum k=m-\nabla \varphi k-\nabla \varphi j\right)+\lambda mG\left(\nabla u+\sum k=m-\lambda k\nabla \varphi k\right)\right\},$$

where we recall that

$$V=W,(\Omega ;\mathbb{R}N).$$

Joining the pieces, we have got

$$\begin{array}{ccc}\hfill \underset{u\in V}{inf}J\left(u\right)& =& \underset{u\in V}{inf}J**\left(u\right)\hfill \\ & =& \underset{u\in V}{inf}\{H\left(u\right)-\langle u,f\rangle L\}\hfill \\ & =& \underset{(u,\lambda ,\varphi )\in V\times B\times \left(V\right)m-}{inf}\left\{\sum j=m-\lambda jG\left(\nabla u+\sum k=m-\lambda k\nabla \varphi k-\nabla \varphi j\right)\right.\hfill \\ & & +\lambda mG\left(\nabla u+\sum k=m\lambda k\nabla \varphi k\right)\hfill \\ & & \left.-\langle u,f\rangle L\right\}.\hfill \end{array}$$

This last functional corresponds to a relaxation for the original non-convex functional.

The note is complete.

34.1. A related duality principle and concerning convex dual formulation

With the notation and statements of the previous sections in mind, consider the functionals $J:V\to \mathbb{R}$ and $J:B\times V\times \left[V\right]m\to \mathbb{R}$ where

$$J\left(u\right)=G(\nabla u)+{\displaystyle 12}\int \Omega u·udx-\langle u,f\rangle L,$$

and

$$\begin{array}{ccc}\hfill J(\lambda ,u,\varphi )& =& \sum j=m\lambda G\left(\nabla u+\sum k=m-\lambda k\varphi k-\nabla \varphi j\right)\hfill \\ & & +\lambda mG\left(\nabla u+\sum k=m-\lambda k\nabla \varphi k\right)\hfill \\ & & +\sum j=m-{\displaystyle \lambda j}\int \Omega \left(u+\sum k=m-\lambda k\varphi k-\nabla \varphi j\right)·\left(u+\sum k=m-\lambda k\varphi k-\nabla \varphi j\right)dx\hfill \\ & & +{\displaystyle \left(\lambda m\right)}\int \Omega \left(u+\sum k=m-\lambda k\varphi k\right)·\left(u+\sum k=m-\lambda k\varphi k\right)dx\hfill \\ & & -\sum j=m-\lambda j〈u+\sum k=m-\lambda k\varphi k-\varphi j,f〉L\hfill \\ & & -\left(\lambda m\right)〈u+\sum k=m-\lambda k\varphi k,f〉L.\hfill \end{array}$$

(212)

Here we have denoted

$$V=\{u\in W,(\Omega ;\mathbb{R}N):u=u\mathrm{on}\partial \Omega =S\},$$

$$V=W,(\Omega ;\mathbb{R}N),$$

$$Y=Y*=L(\Omega ;\mathbb{R}N\times n)$$

and

$$Y=Y*=L(\Omega ;\mathbb{R}N).$$

Observe that

$$J**\left(u\right)\le \underset{(\lambda ,\varphi )\in B\times \left(V\right)m-}{min}J(\lambda ,u,\varphi ).$$

Moreover,

$$\begin{array}{ccc}\hfill J(\lambda ,u,\varphi )& =& -\sum j=m-〈\nabla u+\sum k=m-\lambda k\nabla \varphi k-\nabla \varphi j,(v*)j〉L\hfill \\ & & +\sum j=m-\lambda jG\left(\nabla u+\sum k=m-\lambda k\nabla \varphi k-\nabla \varphi j\right)\hfill \\ & & -〈\nabla u+\sum k=m-\lambda k\nabla \varphi k,(v*)m〉L+\lambda mG\left(\nabla u+\sum k=m-\lambda k\nabla \varphi k\right)\hfill \\ & & -\sum j=m-〈u+\sum k=m-\lambda k\varphi k-\varphi j,(v*)j〉L\hfill \\ & & +\sum j=m-{\displaystyle \lambda j}\int \Omega \left(u+\sum k=m-\lambda k\varphi k-\varphi j\right)·\left(u+\sum k=m-\lambda k\varphi k-\varphi j\right)dx\hfill \\ & & -〈u+\sum k=m-\lambda k\varphi k,(v*)m〉L\hfill \\ & & +{\displaystyle \lambda m}\int \Omega \left(u+\sum k=m-\lambda k\varphi k\right)·\left(u+\sum k=m-\lambda k\varphi k\right)dx\hfill \\ & & +\sum j=m-〈\nabla u+\sum k=m-\lambda k\nabla \varphi k-\nabla \varphi j,(v*)j〉L\hfill \\ & & +〈\nabla u+\sum k=m-\lambda k\nabla \varphi k,(v*)m〉L\hfill \\ & & +\sum j=m-〈u+\sum k=m-\lambda k\varphi k-\varphi j,(v*)j〉L\hfill \\ & & +〈u+\sum k=m-\lambda k\varphi k,(v*)m〉L-\langle u,f\rangle L\hfill \end{array}$$

(213)

Therefore,

$$\begin{array}{ccc}\hfill J(\lambda ,u,\varphi )& \ge & \underset{v\in \left[Y\right]m-}{inf}\left\{\sum j=m-\left(-\langle \left(v\right)j,(v*)j\rangle L+\lambda jG\left(\left(v\right)j\right)\right)\right\}\hfill \\ & & +\underset{\left(v\right)m\in Y}{inf}\{-\langle \left(v\right)m,(v*)m\rangle L+\lambda mG\left(\left(v\right)m\right)\}\hfill \\ & & +\underset{v\in \left[Y\right]m-}{inf}\left\{\sum j=m-\left(-\langle \left(v\right)j,(v*)j\rangle L+{\displaystyle \lambda j}\int \Omega \left(v\right)j·\left(v\right)jdx\right)\right\}\hfill \\ & & +\underset{\left(v\right)m\in Y}{inf}\left\{-\langle \left(v\right)m,(v*)m\rangle L+{\displaystyle \lambda m}\int \Omega \left(v\right)j·\left(v\right)jdx\right\}\hfill \\ & & +\underset{(u,\varphi )\in V\times \left(V\right)m-}{inf}\left\{\sum j=m-〈\nabla u+\sum k=m-\lambda k\nabla \varphi k-\varphi j,(v*)j〉L\right.\hfill \\ & & +〈\nabla u+\sum k=m-\lambda k\nabla \varphi k,(v*)m〉L\hfill \\ & & +\sum j=m-〈u+\sum k=m-\nabla \varphi k-\varphi j,v*〉L\hfill \\ & & \left.+〈u+\sum k=m-\lambda k\varphi k,(v*)m〉L-\langle u,f\rangle L\right\}\hfill \\ & =& -\sum j=m-\lambda jG*\left({\displaystyle (v*)j\lambda j}\right)-\lambda mG*\left({\displaystyle (v*)m\lambda m}\right)\hfill \\ & & -\sum j=m-\left(F\right)j*((v*)j,\lambda j)-\left(F\right)m*((v*)m,\lambda m)\hfill \\ & & +\sum k=m\int S\left((v*)k\right)ijnj\left(u\right)idS,\hfill \\ & & \forall \lambda \in B,u\in V,\varphi \in \left(V\right)m-,v*\in A*,\hfill \end{array}$$

(214)

where

$$G*(v*)=\underset{v\in Y}{sup}\{\langle v,v*\rangle L-G\left(v\right)\},$$

$$\begin{array}{ccc}\hfill \left(F\right)j*((v*)j,\lambda j)& =& \underset{v\in Y}{sup}\left\{\langle \left(v\right)j,(v*)j\rangle L-{\displaystyle \lambda j}\int \Omega \left(v\right)j·\left(v\right)jdx\right\}\hfill \\ & =& {\displaystyle \lambda j}\int \Omega \left(v\right)j*·\left(v\right)j*dx,\forall j\in \{,\dots ,m\}.\hfill \end{array}$$

(215)

Furthermore, $A*=A*\cap A*\left(\lambda \right)$ where

$$A*=\left\{v*=(v*,v*)\in [Y*]m\times [Y*]m:-\sum j=m\left(\mathrm{div}\left((v*)j\right)i+\left((v*)j\right)i\right)-fi=,\mathrm{in}\Omega \right\},$$

and

$$\begin{array}{ccc}\hfill A*\left(\lambda \right)& =& \left\{v*=(v*,v*)\in [Y*]m\times [Y*]m:\right.\hfill \\ & & \lambda k\sum j=m\mathrm{div}\left((v*)j\right)i-\mathrm{div}\left((v*)k\right)i-\lambda k\sum j=m\left((v*)j\right)i+\left((v*)k\right)i=,\hfill \\ & & \left.\mathrm{in}\Omega ,\forall k\in \{,\dots ,m-\},\forall i\in \{,\dots ,N\}\right\}.\hfill \end{array}$$

(216)

Summarizing, we have got

$$\begin{array}{ccc}& & \underset{(\lambda ,u\varphi )\in B\times V\times \left(V\right)m-}{inf}J(\lambda ,u,\varphi )\hfill \\ & \ge & \underset{\lambda \in B}{inf}\left\{\underset{v*\in A*}{sup}\left\{-\sum j=m\lambda jG*\left({\displaystyle (v*)j\lambda j}\right)\right.\right.\hfill \\ & & \left.\left.-\sum j=m(F*)j((v*)j,\lambda j)+\sum k=m\int \partial \Omega \left((v*)k\right)ijnj\left(u\right)idS\right\}\right\}.\hfill \end{array}$$

(217)

Remark 21.

We highlight this last dual function in $v$ is convex (in fact concave) on the convex set $A$.

35. A General Convex Primal Dual Formulation with a Restriction for an Originally Non-Convex Primal One

Let $\Omega \subset \mathbb{R}$ be an open bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider the functional $J:V\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J\left(u\right)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle \alpha }\int \Omega (u-\beta )dx\hfill \\ & & -\langle u,f\rangle L,\hfill \end{array}$$

(218)

where $\alpha >,\beta >,\gamma >$, $V=W,\left(\Omega \right)$ and $Y=Y*=L\left(\Omega \right).$

Define $F:V\to \mathbb{R}$ and $F:V\times Y*\to \mathbb{R}$ by

$$F\left(u\right)={\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle K}\int \Omega udx-\langle u,f\rangle L,$$

and

$$\begin{array}{ccc}\hfill F(u,v*)& =& -\langle u,v*\rangle L+{\displaystyle K}\int \Omega udx\hfill \\ & & +{\displaystyle \alpha }\int \Omega (v*)dx+\beta \int \Omega v*dx.\hfill \end{array}$$

(219)

Define also $F*:Y*\to \mathbb{R}$ and $F*:Y*\times Y*\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill F*(v*)& =& \underset{u\in V}{sup}\{\langle u,v*\rangle L-F\left(u\right)\}\hfill \\ & =& {\displaystyle 12}\int \Omega {\displaystyle (v*+f)-\gamma \nabla +K}dx,\hfill \end{array}$$

(220)

and

$$\begin{array}{ccc}\hfill F*(v*,v*)& =& \underset{u\in V}{sup}\{-\langle u,v*\rangle L-F(u,v*)\}\hfill \\ & =& -{\displaystyle 12}\int \Omega {\displaystyle (v*)v*-K}dx\hfill \\ & & -{\displaystyle \alpha }\int \Omega (v*)dx-\beta \int \Omega v*dx.\hfill \end{array}$$

(221)

if $v*\in B*$, where

$$B*=\{v*\in Y*:\parallel v*\parallel \infty \le K/\},$$

for some appropriate $K>$ to be specified.

At this point we define

$$V=\{u\in V:\parallel u\parallel \infty \le K\},$$

$$A+=\{u\in V:uf\ge ,\mathrm{in}\Omega \},$$

$$V=V\cap A+,$$

$$D*=\{v*\in Y*:\parallel v*\parallel \infty \le /K\},$$

for appropriate $K>$ to be specified, and $J*:D*\times B*\to \mathbb{R}$ by

$$J*(v*,v*)=-F*(v*)+F*(v*,v*).$$

Moreover, we define $J*:V\times D*\times B*\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill J*(u,v*,v*)& =& J*(v*,v*)+{\displaystyle K}\parallel v*-(-\gamma \nabla +K)u\parallel 22\hfill \\ & & +{\displaystyle 10\alpha K32}\parallel v*-(-v*+K)u\parallel 22\hfill \end{array}$$

(222)

Observe that

$${\displaystyle \partial J*(u,v*,v*)\partial (v*)}=-{\displaystyle -\gamma \nabla +K}-{\displaystyle v*-K}+K+{\displaystyle \alpha K32},$$

$${\displaystyle \partial J*(u,v*,v*)\partial u}=K(-\gamma \nabla +K)+{\displaystyle \alpha K32}(-v*+K),$$

and

$${\displaystyle \partial J*(u,v*,v*)\partial u\partial v*}=-K(-\gamma \nabla +K)-{\displaystyle \alpha K32}(-v*+K).$$

Now we set $K,K,K$ such that

$$K\gg max\{K,K,,\alpha ,\beta ,\gamma ,/\alpha ,/\gamma ,/\beta \},$$

$$K\gg max\{K,,\alpha ,\beta ,\gamma ,/\alpha ,/\gamma ,/\beta \},$$

and $K\approx .$

From such results and constant choices, we may obtain

$$\begin{array}{ccc}\hfill det\{\delta u,v*J*(u,v*,v*)\}& =& {\displaystyle \partial J*(u,v*,v*)\partial (v*)}{\displaystyle \partial J*(u,v*,v*)\partial u}-\left({\displaystyle \partial J*(u,v*,v*)\partial u\partial v*}\right)\hfill \\ & =& \mathcal{O}\left({\displaystyle K(-\gamma \nabla +v*)\alpha K32}+K(-\gamma \nabla +v*)\right)+\mathcal{O}\left({\displaystyle KK}\right),\hfill \\ & & \mathrm{in}V\times D*\times B*.\hfill \end{array}$$

(223)

Define now

$$C*=\left\{v*\in Y*:{\displaystyle (-\gamma \nabla +v*)\alpha K32}+(-\gamma \nabla +v*)>{\displaystyle cK}Id\right\},$$

where we assume that $c>$ is such that if $v*\in C*$, then

$$det\{\delta u,v*J*(u,v*,v*)\}>\mathbf{0},\mathrm{in}B*\cap C*.$$

Finally, we also suppose the concerning constants are such that $B*\cap C*$ is convex.

With such statements, definitions and results in mind, we may prove the following theorem.

Theorem 5.

Let $(u,v^*,v^*)\in V\times D*\times (B*\cap C*)$ be such that

$$\delta J*(u,v^*,v^*)=\mathbf{0}.$$

Under such hypotheses,

$$\delta J\left(u\right)=\mathbf{0},$$

and

$$\begin{array}{ccc}\hfill J\left(u\right)& =& J\left(u\right)+{\displaystyle K12}\parallel -\gamma \nabla u+v^*u-f\parallel 22\hfill \\ & =& \underset{u\in V}{inf}\left\{J\left(u\right)+{\displaystyle K12}\parallel -\gamma \nabla u+v^*u-f\parallel 22\right\}\hfill \\ & =& \underset{v*\in B*}{sup}\left\{\underset{(u,v*)\in V\times D*}{inf}J*(u,v*,v*)\right\}\hfill \\ & =& J*(u,v^*,v^*).\hfill \end{array}$$

(224)

Proof. 

The proof that

$$\delta J\left(u\right)=-\gamma \nabla u+v^*u-f=\mathbf{0}$$

and

$$J\left(u\right)=J*(u,v^*,v^*),$$

may be done similarly as in the previous sections and will not be repeated.

Furthermore, since

$$\delta J*(u,v^*,v^*)=\mathbf{0},$$

$v*\in B*\times C*$ and $J$ is concave in $v$ on $V\times D*\times B*$, we have

$$J*(u,v^*,v^*)=\underset{(u,v*)\in V\times D*}{inf}J*(u,v*,v^*),$$

and

$$J*(u,v^*,v^*)=\underset{v*\in B*}{sup}J*(u,v^*,v*).$$

From such results and the Saddle Point Theorem we may infer that

$$\begin{array}{ccc}\hfill J\left(u\right)& =& J\left(u\right)+{\displaystyle K12}\parallel -\gamma \nabla u+v^*u-f\parallel 22\hfill \\ & =& \underset{v*\in B*}{sup}\left\{\underset{(u,v*)\in V\times D*}{inf}J*(u,v*,v*)\right\}\hfill \\ & =& J*(u,v^*,v^*).\hfill \end{array}$$

(225)

Finally, from evident convexity,

$$\begin{array}{ccc}\hfill J\left(u\right)& =& J\left(u\right)+{\displaystyle K12}\parallel -\gamma \nabla u+v^*u-f\parallel 22\hfill \\ & =& \underset{u\in V}{inf}\left\{J\left(u\right)+{\displaystyle K12}\parallel -\gamma \nabla u+v^*u-f\parallel 22\right\}.\hfill \end{array}$$

(226)

Joining the pieces, we have got

$$\begin{array}{ccc}\hfill J\left(u\right)& =& J\left(u\right)+{\displaystyle K12}\parallel -\gamma \nabla u+v^*u-f\parallel 22\hfill \\ & =& \underset{u\in V}{inf}\left\{J\left(u\right)+{\displaystyle K12}\parallel -\gamma \nabla u+v^*u-f\parallel 22\right\}\hfill \\ & =& \underset{v*\in B*}{sup}\left\{\underset{(u,v*)\in V\times D*}{inf}J*(u,v*,v*)\right\}\hfill \\ & =& J*(u,v^*,v^*).\hfill \end{array}$$

(227)

The proof is complete.

□

36. A General Convex Dual Formulation for an Originally Non-Convex Primal One

In this section we develop a convex dual formulation for an originally non-convex primal formulation.

Let $\Omega \subset \mathbb{R}$ be an open bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider the functional $J:V\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J\left(u\right)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle \alpha }\int \Omega (u-\beta )dx\hfill \\ & & -\langle u,f\rangle L,\hfill \end{array}$$

(228)

where $\alpha >,\beta >,\gamma >$, $V=W,\left(\Omega \right)$ and $Y=Y*=L\left(\Omega \right).$

At the moment, fix a matrix $K>$ and $K>$ to be specified.

Define $F:V\to \mathbb{R}$, $F:V\to \mathbb{R}$ and $F:V\times Y*\to \mathbb{R},$ by

$$\begin{array}{ccc}\hfill F\left(u\right)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle K}\int \Omega udx\hfill \\ & & -\langle u,f\rangle L,\hfill \end{array}$$

(229)

$$\begin{array}{ccc}\hfill F\left(u\right)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle K}\int \Omega udx,\hfill \end{array}$$

(230)

$$F(u,v*)=-\langle u,v*\rangle L+K\int \Omega udx+{\displaystyle \alpha }\int \Omega (v*)dx+\beta \int \Omega v*dx+\langle u,f\rangle L.$$

Define also $F*:Y*\to \mathbb{R}$ and $F*:Y*\to \mathbb{R},$

$$\begin{array}{ccc}\hfill F*(v*)& =& \underset{u\in V}{sup}\{\langle u,v*\rangle L-F\left(u\right)\}\hfill \\ & =& {\displaystyle 12}\int \Omega {\displaystyle (v*)-\gamma \nabla +K}dx,\hfill \end{array}$$

(231)

$$\begin{array}{ccc}\hfill F*(v*)& =& \underset{u\in V}{sup}\{\langle u,v*\rangle L-F\left(u\right)\}\hfill \\ & =& {\displaystyle 12}\int \Omega {\displaystyle (v*)-\gamma \nabla +K}dx,\hfill \end{array}$$

(232)

At this point we also define

$$B*=\{v*\in Y*:\parallel v*\parallel \infty \le K/\},$$

$$V=\{u\in V:\parallel u\parallel \infty \le K\},$$

$$A+=\{u\in V:uf\ge ,\mathrm{in}\Omega \},$$

$$V=V\cap A+,$$

$$D*=\{v*\in Y*:\parallel v*\parallel \infty \le /K\},$$

for an appropriate $K>$ to be specified.

Furthermore, we define $F*:D*\times D*\times B*\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill F*(v*,v*,v*)& =& \underset{u\in V}{sup}\{-\langle u,v*+v*\rangle L-F(u,v*)\}\hfill \\ & =& -{\displaystyle 12}\int \Omega {\displaystyle (v*+v*-f)v*-K}dx\hfill \\ & & -{\displaystyle \alpha }\int \Omega (v*)dx-\beta \int \Omega v*dx.\hfill \end{array}$$

(233)

Moreover, we define $J*:D*\times D*\times B*\to \mathbb{R}$ by

$$J*(u,v*,v*)=-F*(v*)-F(v*)+F*(v*,v*,v*)$$

and $J*:D*\times D*\times B*\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill J*(v*,v*,v*)& =& J*(v*,v*,v*)\hfill \\ & & +{\displaystyle K}\int \Omega (v*-v*)dx\hfill \\ & & +{\displaystyle K}\int \Omega \left({\displaystyle v*-\gamma \nabla +K}-{\displaystyle v*+v*-f-v*+K}\right)dx.\hfill \end{array}$$

(234)

Now observe that

$${\displaystyle \partial J*(v*,v*,v*)\partial (v*)}=-{\displaystyle -\gamma \nabla +K}+K+K\left({\displaystyle -\gamma \nabla +K}-{\displaystyle K-v*}\right)-{\displaystyle -K+v*},$$

and

$${\displaystyle \partial J*(v*,v*,v*)\partial (v*)}=-{\displaystyle -\gamma \nabla +K}+K+{\displaystyle K(-K+v*)}-{\displaystyle -K+v*},$$

and

$${\displaystyle \partial J*(v*,v*,v*)\partial v*\partial v*}=-K-K{\displaystyle \left(-\gamma \nabla +K-K-v*\right)K-v*}-{\displaystyle -K+v*}.$$

We set $K\gg K$,

$$K\gg K,$$

and $K\approx \sqrt{}$. Moreover, after a re-scale if necessary, we assume $\alpha \approx .15.$

From such results and constant choices, with the help of the software MATHEMATICA, we may obtain

$$\begin{array}{ccc}\hfill det\{\delta v*,v*J*(v*,v*,v*)\}& =& {\displaystyle \partial J*(v*,v*,v*)\partial (v*)}{\displaystyle \partial J*(v*,v*,v*)\partial u}-\left({\displaystyle \partial J*(v*,v*,v*)\partial u\partial v*}\right)\hfill \\ & =& \mathcal{O}\left(K((-\gamma \nabla +v*)+(-\gamma \nabla +v*))\right).\hfill \end{array}$$

(235)

Define now

$$H(v*)\equiv ((-\gamma \nabla +v*)+(-\gamma \nabla +v*)),$$

Observe that we may obtain $c>$ such that if $v*\in (C*\times B*)$, then

$$det\{\delta v*,v*J*(v*,v*,v*)\}>\mathbf{0},$$

where

$$C*=\{v*\in Y*:H(v*)\ge cId\}.$$

Furthermore, we assume $K>$ and $c>$ are such that $C*\cap B*$ is convex.

With such statements, definitions and results in mind, we may prove the following theorem.

Theorem 6.

Let $(v^*,v^*,v^*)\in D*\times D*\times (B*\cap C*)$ be such that

$$\delta J*(v^*,v^*,v^*)=\mathbf{0}.$$

Under such hypotheses,

$$\delta J\left(u\right)=\mathbf{0},$$

and

$$\begin{array}{ccc}& & J\left(u\right)\hfill \\ & =& \underset{v*\in B*}{sup}\left\{\underset{(v*,v*)\in D*\times D*}{inf}J*(v*,v*,v*)\right\}\hfill \\ & =& J*(v^*,v^*,v^*).\hfill \end{array}$$

(236)

Proof. 

The proof that

$$J\left(u\right)=\mathbf{0},$$

$$-\gamma \nabla u+v^*u-f=\mathbf{0},$$

and

$$J\left(u\right)=J*(v^*,v^*,v^*),$$

may be done similarly as in the previous sections and will not be repeated.

Furthermore, since

$$\delta J*(v^*,v^*,v^*)=\mathbf{0},$$

$v*\in B*\cap C*$ and $J$ is concave in $v$ on $D\times D*\times B*$, we have

$$J*(v^*,v^*v^*)=\underset{(v*,v*)\in D*\times D*}{inf}J*(v*,v*,v^*),$$

and

$$J*(v^*,v^*,v^*)=\underset{v*\in B*}{sup}J*(v^*,v^*,v*).$$

From such results and the Saddle Point Theorem we may infer that

$$\begin{array}{ccc}\hfill J\left(u\right)& =& J*(v^*,v^*,v^*)\hfill \\ & =& \underset{v*\in B*}{sup}\left\{\underset{(v*,v*)\in D*\times D*}{inf}J*(v*,v*,v*)\right\}\hfill \\ & =& J*v^*,v^*,v^*).\hfill \end{array}$$

(237)

The proof is complete.

□

37. A Note on the Special Relativistic Physics

Consider in $\mathbb{R}$ two observers O and $O$ and related referential Cartesian frames $O(x,y,z)$ and $O\prime (x\prime ,y\prime ,z\prime )$ respectively.

Suppose a particle moves from a point $(x,y,z)$ to a point $(x+\Delta x,y+\Delta y,z+\Delta z)$ related to $O(x,y,z)$ on a time interval $\Delta t$.

Denote

$$I=\Delta x+\Delta y+\Delta z,$$

and $I=\Delta t$.

In a Newtonian physics context, we have

$$I=\Delta x+\Delta y+\Delta z=\Delta x\prime +\Delta y\prime +\Delta z\prime ,$$

and

$$I=\Delta t=\Delta t\prime ,$$

that is, $I$ and $I$ remain invariant.

However, through experiments in higher energy physics, it was discovered that in fact is $I$ which remains invariant (this had been previously proposed in the Einstein special relativity theory in 1905), where

$$I=-c\Delta t+\Delta x+\Delta y+\Delta z,$$

so that

$$-c\Delta t+\Delta x+\Delta y+\Delta z=-c\Delta t\prime +\Delta x\prime +\Delta y\prime +\Delta z\prime =I,$$

for any pair of observers O and $O\prime .$ Here c denotes the speed of light, and in the case in which $v,v\prime \ll c$ we have the Newtonian approximation

$$\Delta t\prime \approx \Delta t.$$

From the expression of $I$ we obtain

$$\begin{array}{ccc}& & -c{\displaystyle \Delta t\prime \Delta t}+{\displaystyle \Delta x\prime \Delta t}+{\displaystyle \Delta y\prime \Delta t}+{\displaystyle \Delta z\prime \Delta t}\hfill \\ & =& -c{\displaystyle \Delta t\Delta t}+{\displaystyle \Delta x\Delta t}+{\displaystyle \Delta y\Delta t}+{\displaystyle \Delta z\Delta t}.\hfill \end{array}$$

(238)

Thus,

$$\begin{array}{ccc}& & -c{\displaystyle \Delta t\prime \Delta t}+\left({\displaystyle \Delta x\prime \Delta t\prime }+{\displaystyle \Delta y\prime \Delta t\prime }+{\displaystyle \Delta z\prime \Delta t\prime }\right){\displaystyle \Delta t\prime \Delta t}\hfill \\ & =& -c+{\displaystyle \Delta x\Delta t}+{\displaystyle \Delta y\Delta t}+{\displaystyle \Delta z\Delta t}\hfill \end{array}$$

(239)

so that

$$\left({\displaystyle \Delta t\prime \Delta t}\right)={\displaystyle c-\left(\Delta x\Delta t+\Delta y\Delta t+\Delta z\Delta t\right)c-\left(\Delta x\prime \Delta t\prime +\Delta y\prime \Delta t\prime +\Delta z\prime \Delta t\prime \right)}.$$

Letting $\Delta t,\Delta t\prime \to$, we obtain

$$\left({\displaystyle \partial t\prime \partial t}\right)={\displaystyle -vc-(v\prime )c}.$$

In particular for constant v and $v\prime =$ we have

$$\left({\displaystyle \Delta t\prime \Delta t}\right)=-{\displaystyle vc},$$

so that

$$\Delta t\prime =\sqrt{-{\displaystyle vc}}\Delta t.$$

Consider now that O is at rest and $O$ has a constant velocity

$$ve$$

where $\{e,e,e\}$ is the canonical basis for $\mathbb{R}$ related to $O.$

Consider $O(x,y,z)$ and $O\prime (x,y,z)$ such that the axis $x$ coincide with the axis x, axis $y$ is parallel to axis y and axis $z$ is parallel to $z.$

Since v is constant, we have

$$v={\displaystyle \Delta x\Delta t},$$

and

$$v\prime =.$$

Assuming $x\left(\right)=,$ and the initial time $t=$, we have $\Delta x=x,$ and $\Delta t=t$ so that

$$t\prime =\sqrt{-{\displaystyle vc}}t,$$

so that

$$t\prime ={\displaystyle -vc\sqrt{-vc}}t={\displaystyle \left(t-vvtc\right)\sqrt{-vc}},$$

and thus

$$t\prime ={\displaystyle \left(t-vxc\right)\sqrt{-vc}}.$$

On the other hand we have $v\prime =$.

We may easily check that the solution

$$x\prime ={\displaystyle x-vt\sqrt{-vc}},$$

lead us to $v\prime =$.

Indeed,

$${\displaystyle \Delta x\prime \sqrt{-vc}\Delta t\prime }={\displaystyle \Delta x\prime \Delta t},$$

so that, considering that v is constant, we obtain

$${\displaystyle dx\prime dt}={\displaystyle d(x-vt)dt\sqrt{-vc}}={\displaystyle dxdt-v\sqrt{-vc}}={\displaystyle v-v\sqrt{-vc}}=,$$

that is,

$${\displaystyle dx\prime dt}=.$$

Thus,

$${\displaystyle d\left(x\prime \sqrt{-vc}\right)dt\prime }=,$$

so that

$$x\prime \sqrt{-{\displaystyle vc}}=c$$

for some constant $c\in \mathbb{R}$ so that

$$x\prime =c,$$

for some $c\in \mathbb{R}.$

Therefore

$$v\prime ={\displaystyle dx\prime dt\prime }=.$$

Summarizing, for the Newton mechanics we have

$$t\prime =t$$

,

$$x\prime =x-vt,$$

$$y\prime =y,$$

and

$$z\prime =z.$$

On the other hand, for the special relativity context, we have the following Lorentz relations

$$t\prime ={\displaystyle \left(t-vxc\right)\sqrt{-vc}}.$$

$$x\prime ={\displaystyle x-vt\sqrt{-vc}},$$

$$y\prime =y,$$

and

$$z\prime =z.$$

37.1. The Kinetics energy for the special relativity context

Consider the motion of a particle system described by the position field

$$\mathbf{r}:\Omega \times [,T]\to \mathbb{R},$$

where $\Omega \subset \mathbb{R}$, $[,T]$ is a time interval and

$$\mathbf{r}(x,y,z,t)=(ct,X(x,y,z,t),X(x,y,z,t),X(x,y,z,t)).$$

In my understanding, this is the special relativity theory context.

The related density field is denoted by

$$\rho :\Omega \times [,T]\to \mathbb{R}+,$$

where

$$\rho (x,y,z,t)=m\left|\varphi (x,y,z,t)\right|,$$

$m$ is total system mass at rest, and $\varphi :\Omega \times [,T]\to \mathbb{C}$ is a wave function such that

$$\int \Omega \left|\varphi (x,y,z,t)\right|dx=,\forall t\in [,T].$$

The Kinetics energy differential is given by

$$dEc=-dm{\displaystyle \partial \mathbf{r}\partial t}·{\displaystyle \partial \mathbf{r}\partial t},$$

where

$$\begin{array}{ccc}\hfill {\displaystyle \partial \mathbf{r}\partial t}·{\displaystyle \partial \mathbf{r}\partial t}& =& \left(c,{\displaystyle \partial X\partial t},{\displaystyle \partial X\partial t},{\displaystyle \partial X\partial t}\right)·\left(c,{\displaystyle \partial X\partial t},{\displaystyle \partial X\partial t},{\displaystyle \partial X\partial t}\right)\hfill \\ & =& -c+\left({\displaystyle \partial X\partial t}\right)+\left({\displaystyle \partial X\partial t}\right)+\left({\displaystyle \partial X\partial t}\right)\hfill \\ & =& -c+v,\hfill \end{array}$$

(240)

where

$$v=\left({\displaystyle \partial X\partial t}\right)+\left({\displaystyle \partial X\partial t}\right)+\left({\displaystyle \partial X\partial t}\right).$$

Moreover,

$$dm={\displaystyle m\sqrt{-vc}}\left|\varphi (x,y,z,t)\right|dxdydz,$$

so that

$$\begin{array}{ccc}\hfill dEc& =& {\displaystyle m\sqrt{-vc}}(c-v)\left|\varphi (x,y,z,t)\right|dxdydz\hfill \\ & =& mc\sqrt{c-v}\left|\varphi \right|dxdydz.\hfill \end{array}$$

(241)

Thus,

$$Ec\left(t\right)=\int \Omega dEc=\int \Omega mc\sqrt{c-v}\left|\varphi \right|dxdydz.$$

In particular for a constant v (not varying in $(x,y,z,t)$), we obtain

$$Ec\left(t\right)=mc\sqrt{c-v}.$$

Hence if $v\ll c,$ we have

$$Ec\left(t\right)\approx mc.$$

This is the most famous Einstein equation previously published in his article of 1905.

37.2. The Kinetics energy for the general relativity context

In a general relativity theory context, the motion of a particle system will be specified by a field

$$(\mathbf{r}\circ u^):\Omega \times [,T]\to \mathbb{R}$$

where

$$(\mathbf{r}\circ u^)(x,t)=(ct,X(u^(x,t)),X(u^(x,t)),X(u^(x,t))),$$

where

$$u^(x,t)=(u\left(t\right),u(x,t),u(x,t),u(x,t)),$$

$$u\left(t\right)=t,$$

$$x=(x,x,x)\in \Omega \subset \mathbb{R},$$

and $t\in [,T]$, where $[,T]$ is a time interval.

The corresponding density is represented by

$$(\rho \circ u^):\Omega \times [,T]\to \mathbb{R}+,$$

where

$$(\rho \circ u^)(x,t)=m\left|\varphi (u^(x,t))\right|,$$

$m$ is total system mass at rest and $\varphi :\Omega \times [,T]\to \mathbb{C}$ is a complex wave function such that

$$\int \Omega |\varphi (u^(x,t))|\sqrt{-g}|det\{u^\prime (x,t)\}|dx=,\forall t\in [,T]$$

where

$$dx=dxdxdx,$$

$$\mathbf{g}j={\displaystyle \partial \mathbf{r}\partial uj}$$

$$gjk=\mathbf{g}j·\mathbf{g}k,\forall j,k\in \{,,,\}.$$

and $g=det\left\{gjk\right\}$.

Now observe that

$$\begin{array}{ccc}\hfill {\displaystyle \partial \mathbf{r}\partial t}·{\displaystyle \partial \mathbf{r}\partial t}& =& {\displaystyle \partial \mathbf{r}\partial uj}{\displaystyle \partial uj\partial t}·{\displaystyle \partial \mathbf{r}\partial uk}{\displaystyle \partial uk\partial t}\hfill \\ & =& {\displaystyle \partial \mathbf{r}\partial uj}·{\displaystyle \partial \mathbf{r}\partial uk}{\displaystyle \partial uj\partial t}{\displaystyle \partial uk\partial t}\hfill \\ & =& gjk{\displaystyle \partial uj\partial t}{\displaystyle \partial uk\partial t}.\hfill \end{array}$$

(242)

Observe that

$${\displaystyle \partial \mathbf{r}\partial t}·{\displaystyle \partial \mathbf{r}\partial t}=gjk{\displaystyle \partial uj\partial t}{\displaystyle \partial uk\partial t}=-c+v.$$

Moreover, the Kinetics energy differential is given by

$$dEc=-dm{\displaystyle \partial \mathbf{r}\partial t}·{\displaystyle \partial \mathbf{r}\partial t},$$

where

$$dm={\displaystyle m\sqrt{-vc}}|\varphi (u^(x,t))|\sqrt{-g}|det\{u^\prime (x,t)\}|dx,$$

so that the total Kinetics energy is expressed by

$$Ec=\int T\int \Omega dEcdt,$$

that is,

$$\begin{array}{ccc}\hfill Ec& =& \int T\int \Omega {\displaystyle m\sqrt{-vc}}(c-v)\left|\varphi \right(u^(x,t)|\sqrt{-g}|det\{u^\prime (x,t)\}|dxdt\hfill \\ & =& \int T\int \Omega mc\sqrt{c-v}|\varphi (u^(x,t))|\sqrt{-g}|det\{u^\prime (x,t)\}|dxdt\hfill \\ & =& \int T\int \Omega mc\sqrt{-gjk{\displaystyle \partial uj\partial t}{\displaystyle \partial uk\partial t}}|\varphi (u^(x,t))|\sqrt{-g}|det\{u^\prime (x,t)\}|dxdt.\hfill \end{array}$$

(243)

Summarizing, for the general relativity theory context

$$Ec=\int T\int \Omega mc\sqrt{-gjk{\displaystyle \partial uj\partial t}{\displaystyle \partial uk\partial t}}|\varphi (u^(x,t))|\sqrt{-g}|det\{u^\prime (x,t)\}|dxdt.$$

38. About an Energy Term Related to the Manifold Curvature Variation

In this section we consider a particle system motion represented by a field

$$\mathbf{r}:\Omega \to \mathbb{R}$$

of $C$ class where here $\Omega =\Omega ^\times [,T],$$\Omega ^\subset \mathbb{R}$ is an open, bounded and connected set, and $[,T]$ is a time interval.

More specifically, point-wise we denote

$$\mathbf{r}\left(\mathbf{u}\right)=(ct,X\left(\mathbf{u}\right),X\left(\mathbf{u}\right),X\left(\mathbf{u}\right)),$$

where $u=t,$ and $\mathbf{u}=(u,u,u,u)\in \Omega .$

Now, define

$$\mathbf{g}j={\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uj},$$

and

$$gjk=\mathbf{g}j·\mathbf{g}k,\forall j,k\in \{,,,\}.$$

Moreover

$$\left\{gjk\right\}=\left\{gjk\right\}-,$$

and

$$g=det\left\{gjk\right\}.$$

We assume

$$\left\{{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uj},\mathrm{for}j\in \{,,,\}\right\}$$

is a basis for $\mathbb{R}$, $\forall \mathbf{u}\in \Omega .$

At this point we define the Christofel symbols, denoted by $\Gamma$, by

$$\Gamma jkl={\displaystyle 12}glp\left\{{\displaystyle \partial gkp\partial uj}+{\displaystyle \partial gjp\partial uk}-{\displaystyle \partial gjk\partial up}\right\},\forall j,k,l\in \{,,,\}.$$

Theorem 7.

Considering these last previous statements and definitions, we have that

$${\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uj\partial uk}=\Gamma jkl{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial ul},\forall j,k\in \{,,,\},\forall \mathbf{u}\in \Omega .$$

Proof. 

Fix $\mathbf{u}\in \Omega$ and $j,k,m\in \{,,,\}.$

Observe that

$$\begin{array}{ccc}\hfill \Gamma jklglm& =& {\displaystyle 12}gmlglp\left\{{\displaystyle \partial gkp\partial uj}+{\displaystyle \partial gjp\partial uk}-{\displaystyle \partial gjk\partial up}\right\}\hfill \\ & =& {\displaystyle 12}\delta mp\left\{{\displaystyle \partial gkp\partial uj}+{\displaystyle \partial gjp\partial uk}-{\displaystyle \partial gjk\partial up}\right\}\hfill \\ & =& {\displaystyle 12}\left\{{\displaystyle \partial gkm\partial uj}+{\displaystyle \partial gjm\partial uk}-{\displaystyle \partial gjk\partial um}\right\}\hfill \\ & =& {\displaystyle 12}\left\{{\displaystyle \partial \partial uj}\left({\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uk}·{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial um}\right)+{\displaystyle \partial \partial uk}\left({\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uj}·{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial um}\right)-{\displaystyle \partial \partial um}\left({\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uj}·{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uk}\right)\right\}\hfill \\ & =& {\displaystyle 12}\left\{{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uk\partial uj}·{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial um}+{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial um\partial uj}·{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uk}\right.\hfill \\ & & +{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uj\partial uk}·{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial um}+{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial um\partial uk}·{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uj}\hfill \\ & & \left.-{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial um\partial uj}·{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uk}-{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial um\partial uk}·{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uj}\right\}\hfill \\ & =& {\displaystyle 12}\left\{{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uj\partial uk}·{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial um}+{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uj\partial uk}·{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial um}\right\}\hfill \\ & =& {\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uj\partial uk}·{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial um}.\hfill \end{array}$$

(244)

Summarizing, we have got

$$\Gamma jkl{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial ul}·{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial um}=\Gamma jklglm={\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uj\partial uk}·{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial um}.$$

Since

$$\left\{{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uj},\mathrm{for}j\in \{,,,\}\right\},$$

is a basis for $\mathbb{R}$, we may infer that

$${\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uj\partial uk}=\Gamma jkl{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial ul},\forall j,k\in \{,,,\},\forall \mathbf{u}\in \Omega .$$

The proof is complete.

□

38.1. The energy term related to curvature variation

We define such an energy term, denoted by $E$, as

$$Eq(\varphi ,\mathbf{r})={\displaystyle 12}\int \Omega gjkglp{\displaystyle \partial \partial uj}\left(\varphi {\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uk}\right)·{\displaystyle \partial \partial ul}\left(\varphi *{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial up}\right)\sqrt{-g}du,$$

where $du=dudududu.$

Here $\varphi :\Omega \to \mathbb{C}$ is a complex wave function representing the scalar density field.

Now observe that

$$\begin{array}{ccc}& & {\displaystyle \partial \partial uj}\left(\varphi {\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uk}\right)·{\displaystyle \partial \partial ul}\left(\varphi *{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial up}\right)\hfill \\ & =& \left({\displaystyle \partial \varphi \partial uj}{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uk}+\varphi {\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uj\partial uk}\right)·\left({\displaystyle \partial \varphi *\partial ul}{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial up}+\varphi *{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial ul\partial up}\right)\hfill \\ & =& {\displaystyle \partial \varphi \partial uj}{\displaystyle \partial \varphi *\partial uj}gkp+\left|\varphi \right|{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uj\partial uk}·{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial ul\partial up}\hfill \\ & & +\varphi {\displaystyle \partial \varphi *\partial ul}{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uj\partial uk}·{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial up}\hfill \\ & & +\varphi *{\displaystyle \partial \varphi \partial uj}{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial ul\partial up}·{\displaystyle \partial \mathbf{r}\left(\mathbf{u}\right)\partial uk}\hfill \\ & =& {\displaystyle \partial \varphi \partial uj}{\displaystyle \partial \varphi *\partial uj}gkp+\left|\varphi \right|\Gamma jkm\Gamma lpogmo\hfill \\ & & +\varphi {\displaystyle \partial \varphi *\partial ul}\Gamma jksgsp+\varphi *{\displaystyle \partial \varphi \partial uj}\Gamma lprgrk.\hfill \end{array}$$

(245)

From such results, we may infer that

$$\begin{array}{ccc}\hfill Eq(\varphi ,\mathbf{r})& =& {\displaystyle 12}\int \Omega gjk{\displaystyle \partial \varphi \partial uj}{\displaystyle \partial \varphi *\partial uk}\sqrt{-g}du\hfill \\ & & +{\displaystyle 12}\int \Omega gjkglp\Gamma jkr\Gamma lpsgrs\left|\varphi \right|\sqrt{-g}du\hfill \\ & & +{\displaystyle 12}\int \Omega gjk\Gamma jkl\left(\varphi {\displaystyle \partial \varphi *\partial ul}+\varphi *{\displaystyle \partial \varphi \partial ul}\right)\sqrt{-g}du.\hfill \end{array}$$

(246)

39. A Note on the Definition of Temperature

The main results in this section may be found in similar form in the book [16], page 261.

Consider a system with $N=\sum j=NNj$ and suppose each set of $N$ particles has a set of $C$ possible states.

Therefore, the number of states of such $N$ particles is given by

$$\Delta \Gamma j={\displaystyle \left(Cj\right)NjNj!},$$

where we have considered simple permutations as equivalent states.

Define

$$Sj=ln\left(\Delta \Gamma j\right),$$

and define the system entropy, denoted by S, as

$$S=A\left(\sum j=NSj\right),$$

where $A>$ is a normalizing constant.

Thus,

$$S=A\sum j=Nln\left({\displaystyle \left(Cj\right)NjNj!}\right),$$

so that

$$S=A\left(\sum j=N\left(Njln\left(Cj\right)-ln(Nj!)\right)\right).$$

If $N$ is large enough, we have the following approximation

$$ln(Nj!)\approx Njln\left(Nj\right).$$

In particular for $Cj=,\forall j\in \{,\dots ,N\}$ we obtain

$$S=A\left(\sum j=NSj\right)\approx -A\left(\sum j=NNjln\left(Nj\right)\right),$$

At this point we define the following local density $N$ where

$$N^j(x,t)={\displaystyle |\varphi j(x,t)|\left|\varphi (x,t)\right|}N,$$

where

$$\left|\varphi (x,t)\right|=\sum j=N\left|\varphi j(x,t)\right|.$$

Here, $\varphi j:\Omega \to \mathbb{C}$ denotes the wave function of the particles corresponding to the system part $N$.

The final definition of Entropy is given by

$$S(x,t)=A\left(\sum j=NSj(x,t)\right)$$

where

$$\begin{array}{ccc}\hfill Sj(x,t)& =& -N^j(x,t)ln(N^j(x,t))\hfill \\ & =& -{\displaystyle |\varphi j(x,t)|\varphi (x,t)|}Nln\left({\displaystyle |\varphi j(x,t)|\varphi (x,t)|}N\right).\hfill \end{array}$$

(247)

Here we highlight the position field for each particle system part $N$ is given by

$$\mathbf{r}^j(x,t)=\mathbf{x}+\mathbf{r}j(x,t),$$

where $\mathbf{r}$ is related to the internal energy, that is, related to the atomic/electronic vibrational motion linked with the concept of temperature, as specified in the next lines.

The total kinetics energy is given by

$$E(x,t)=-{\displaystyle 12}\sum j=Nmpj\left|\varphi j(x,t)\right|{\displaystyle \partial \mathbf{r}j(x,t)\partial t}·{\displaystyle \partial \mathbf{r}j(x,t)\partial t}.$$

At this point, we define the scalar field of temperature, denoted by $T(x,t)$, such as symbolically

$${\displaystyle \partial S\partial E}={\displaystyle T(x,t)}.$$

More specifically, we define

$$T(x,t)=\sum j=N{\displaystyle \partial E\partial \varphi j\partial S\partial \varphi j},$$

so that

$$T(x,t)={\displaystyle -12\sum j=Nmpj\varphi j(x,t)\partial \mathbf{r}j(x,t)\partial t·\partial \mathbf{r}j(x,t)\partial t-A\varphi jN\left|\varphi \right|ln\left(|\varphi j|N\left|\varphi \right|+\right)}.$$

39.1. A note on basic Thermodynamics

Consider a solid $\Omega \subset \mathbb{R}$ where such a $\Omega$ is an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Denoting by $[,T]$ a time interval, consider a particle system where the field of displacements is given by

$$\mathbf{r}j(x,t)=\mathbf{r}(x,t)+\mathbf{u}(x,t)+\left(\mathbf{r}\right)j(x,t),$$

where $\mathbf{r}:\Omega \times [,T]\to \mathbb{R}$ is a macroscopic displacement field, $\mathbf{u}:\Omega \times [,T]\to \mathbb{R}$ is the elastic displacement field and $\left(\mathbf{r}\right)j:\Omega \times [,T]\to \mathbb{R}$ denotes the displacement field related to the atomic and electronic vibration motion concerning the concept of temperature, as specified in the previous section.

In particular for the case in which

$$\mathbf{r}(x,t)=x,$$

we define the heat functional, denoted by W, as

$$\begin{array}{ccc}\hfill W& =& {\displaystyle 12}\int T\int \Omega \rho (x,t){\displaystyle \partial \mathbf{u}(x,t)\partial t}·{\displaystyle \partial \mathbf{u}(x,t)\partial t}dxdt\hfill \\ & & -\int T\int \Omega \mathbf{F}·\mathbf{u}dxdt\hfill \\ & & +{\displaystyle 12}\int T\int \Omega Hijkleij\left(\mathbf{u}\right)ekl\left(\mathbf{u}\right)dxdt\hfill \\ & & +{\displaystyle 12}\sum j=N\int T\int \Omega mpj\left|\varphi j(x,t)\right|{\displaystyle \partial \left(\mathbf{r}\right)j(x,t)\partial t}·{\displaystyle \partial \left(\mathbf{r}\right)j(x,t)\partial t}dxdt,\hfill \end{array}$$

(248)

where

$$\rho (x,t)=\sum j=Nmpj\left|\varphi j(x,t)\right|$$

is the point wise total density,

$${\displaystyle 12}\int T\int \Omega Hijkleij\left(\mathbf{u}\right)ekl\left(\mathbf{u}\right)dxdt$$

is a standard elastic inner energy for small displacements $\mathbf{u},$$\mathbf{F}(x,t)$ is the resulting field of external forces acting point wise on $\Omega$, and for the term

$${\displaystyle 12}\sum j=N\int T\int \Omega mpj\left|\varphi j(x,t)\right|{\displaystyle \partial \left(\mathbf{r}\right)j(x,t)\partial t}·{\displaystyle \partial \left(\mathbf{r}\right)j(x,t)\partial t}dxdt$$

we are refereing to the definitions and notations of the previous section.

At this point we denote

$$Ein={\displaystyle 12}\sum j=N\int T\int \Omega mpj\left|\varphi j(x,t)\right|{\displaystyle \partial \left(\mathbf{r}\right)j(x,t)\partial t}·{\displaystyle \partial \left(\mathbf{r}\right)j(x,t)\partial t}dxdt,$$

and

$$\begin{array}{ccc}\hfill ET& =& {\displaystyle 12}\int T\int \Omega \rho (x,t){\displaystyle \partial \mathbf{u}(x,t)\partial t}·{\displaystyle \partial \mathbf{u}(x,t)\partial t}dxdt\hfill \\ & & -\int T\int \Omega \mathbf{F}·\mathbf{u}dxdt\hfill \\ & & +{\displaystyle 12}\int T\int \Omega Hijkleij\left(\mathbf{u}\right)ekl\left(\mathbf{u}\right)dxdt.\hfill \end{array}$$

(249)

Hence $W=ET+Ein$ and from the previous section we may generically denote

$$\delta Ein=T\delta S,$$

Therefore

$$\delta W=\delta ET+\delta Ein=\delta ET+T\delta S.$$

For a standard reversible process we must have $\delta ET=\mathbf{0}$.

so that

$$\delta W=T\delta S.$$

For a general case in which other types of internal energy (such as $E$ indicated in the previous sections and even $E$) are partially and irreversibly converted into a $E$ type of energy, in which

$$\delta ET\ne \mathbf{0},$$

we may have

$$\delta W<T\delta S.$$

Remark 22.

Indeed, in general the vibrational motion related to $E$ is of relativistic nature so that in fact we would need to consider

$$Ein={\displaystyle 12}\sum j=N\int T\int \Omega mpjc\left|\varphi j(x,t)\right|\sqrt{c-{\displaystyle \partial \left(\mathbf{r}\right)j(x,t)\partial t}·{\displaystyle \partial \left(\mathbf{r}\right)j(x,t)\partial t}}\sqrt{-gj}dxdt.$$

40. A formal proof of Castigliano Theorem

In this section we present the mathematical formalism of a result in elasticity theory known as the Castigliano’s Theorem.

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipischitzian) boundary denoted by $\partial \Omega .$

In a context of linear elasticity, consider the functional $J:V\to \mathbb{R}$ where

$$J\left(u\right)=Ein-\langle ui,fi\rangle L-\sum j=Nui\left(xj\right)Pij,$$

$u=(u,u,u)\in W,(\Omega ;\mathbb{R})\equiv V,$$f=(f,f,f)\in L(\Omega ;\mathbb{R})$, $Y=Y*=L(\Omega ;\mathbb{R}),$ and

$$Pij\in \mathbb{R},\forall i\in \{,,\},j\in \{,\dots ,N\}$$

for some $N\in \mathbb{N}.$

Here we have denoted

$$Ein={\displaystyle 12}\int \Omega Hijkleij\left(u\right)ekl\left(u\right)dx,$$

$$eij\left(u\right)={\displaystyle 12}\left({\displaystyle \partial ui\partial xj}+{\displaystyle \partial uj\partial xi}\right).$$

Moreover $H$ is a fourth order positive definite and constant tensor.

Observe that the variation of J in $u$ give us the following Euler-Lagrange equation

$$-\left(Hijklekl\left(u\right)\right),j-fi-\sum j=NPij\delta \left(xj\right)=\mathbf{0},\mathrm{in}\Omega .$$

(250)

Symbolically such a system stands for

$${\displaystyle \partial J\left(u\right)\partial ui}=\mathbf{0},\forall i\in \{,,\},$$

so that

$${\displaystyle \partial (Ein-\langle ui,fi\rangle L-\sum j=Nui\left(xj\right)Pij)\partial ui}=\mathbf{0},\forall i\in \{,,\}.$$

(251)

We denote $u\in V$ solution of (250) by $u=u(f,P),$ so that multiplying the concerning extremal equation by $u$ and integrating by parts, we get

$$\begin{array}{ccc}\hfill H(u(f,P),f,P)& =& Ein\left(u(f,P)\right)-\langle ui(f,P),fi\rangle L-\sum j=Nui(xj,f,P)Pij\hfill \\ & =& ,\forall f\in Y*,P\in \mathbb{R}N.\hfill \end{array}$$

(252)

Therefore

$${\displaystyle ddPij}\left(H(u(f,P),f,P)\right)=,$$

so that

$${\displaystyle dEindPij}-{\displaystyle ddPij}\left(\langle ui(f,P),fi\rangle L+\sum j=Nui(xj,f,p)Pij\rangle L\right)=,$$

that is

$$\begin{array}{ccc}& & {\displaystyle dEindPij}+〈{\displaystyle \partial (Ein-\langle ui,fi\rangle L-\sum j=Nui\left(xj\right)Pij)\partial uk},{\displaystyle \partial uk\partial Pij}〉L\hfill \\ & & -{\displaystyle \partial \partial Pij}\left(\langle ui,fi\rangle L+\sum j=Nui\left(xj\right)Pij\right)\hfill \\ & =& .\hfill \end{array}$$

(253)

From this and (250) we obtain

$${\displaystyle dEindPij}-ui\left(xj\right)=,$$

so that

$$ui\left(xj\right)={\displaystyle dEindPij}={\displaystyle ddPij}\left({\displaystyle 12}\int \Omega Hijkleij\left(u(f,P)\right)ekl\left(u(f,P)\right)dx\right),$$

$\forall i\in \{,,\},\forall j\in \{,\dots ,N\}.$

With such results in mind, we have proven the following theorem.

Theorem 8

(Castigliano). Considering the notations and definitions in this section, we have

$$ui\left(xj\right)={\displaystyle dEindPij}={\displaystyle ddPij}\left({\displaystyle 12}\int \Omega Hijkleij\left(u(f,P)\right)ekl\left(u(f,P)\right)dx\right),$$

$\forall i\in \{,,\},\forall j\in \{,\dots ,N\}.$

40.1. A generalization of Castigliano theorem

In this subsection we present a more general version of the Castigliano theorem.

Considering the context of last section, we recall that

$$\begin{array}{ccc}\hfill H(u(f,P),f,P)& =& Ein\left(u(f,P)\right)-\langle ui(f,P),fi\rangle L-\sum j=Nui(xj,f,P)Pij\hfill \\ & =& ,\forall f\in Y*,P\in \mathbb{R}N.\hfill \end{array}$$

(254)

Therefore, for $xk\in \Omega$ such that

$$xk\ne xj,\forall j\in \{,\dots ,N\},$$

we have

$$〈{\displaystyle ddfi}\left(H(u(f,P),f,P)\right),\delta \left(xk\right)〉L=,$$

so that

$$\begin{array}{ccc}& & 〈{\displaystyle ddfi}\left(Ein\left(u(f,P)\right)\right),\delta \left(xk\right)〉L\hfill \\ & & -〈{\displaystyle ddfi}\left(\langle ui(f,P),fi\rangle L+\sum j=Nui(xj,f,p)Pij\rangle L\right),\delta \left(xk\right)〉L\hfill \\ & =& ,\hfill \end{array}$$

(255)

that is

$$\begin{array}{ccc}& & 〈{\displaystyle ddfi}\left(Ein\left(u(f,P)\right)\right),\delta \left(xk\right)〉L\hfill \\ & & +〈{\displaystyle dduk}\left(Ein\left(u(f,P)\right)-\langle ui(f,P),fi\rangle L-\sum j=Nui(xj,f,p)Pij\right){\displaystyle dukdfi},\delta \left(xk\right)〉L\hfill \\ & & -〈{\displaystyle \partial \partial fi}\left(\langle ui(f,P),fi\rangle L-\sum j=Nui(xj,f,p)Pij\right),\delta x\left(xk\right)〉L\hfill \\ & =& .\hfill \end{array}$$

(256)

From such results, we may obtain

$$〈{\displaystyle ddfi}\left(Ein\left(u(f,P)\right)\right),\delta \left(xk\right)〉L-\langle ui\left(x\right),\delta \left(xk\right)\rangle L=,$$

so that

$$〈{\displaystyle ddfi}\left(Ein\left(u(f,P)\right)\right),\delta \left(xk\right)〉L-ui\left(xk\right)=,$$

that is

$$ui\left(xk\right)=〈{\displaystyle ddfi}\left(Ein\left(u(f,P)\right)\right),\delta \left(xk\right)〉L,$$

$\forall i\in \{,,\},\forall xk\in \Omega \mathrm{such}\mathrm{that}xk\ne xj,\forall j\in \{,\dots ,N\}.$

With such results in mind, we have proven the following theorem.

Theorem 9

(The Generalized Castigliano Theorem). Considering the notations and definitions in this section, we have

$$ui\left(xk\right)=〈{\displaystyle ddfi}\left(Ein\left(u(f,P)\right)\right),\delta \left(xk\right)〉L,$$

$\forall i\in \{,,\},\forall xk\in \Omega \mathrm{such}\mathrm{that}xk\ne xj,\forall j\in \{,\dots ,N\}.$

40.2. The virtual work principle

Considering the definitions, results and statements of the previous section and subsection, we may easily prove the following theorem.

Theorem 10

(The virtual work principle). Let $xl\in \Omega$ such that $xl\ne xj,\forall j\in \{,\dots ,N\}.$

For a virtual constant load $Plk\in \mathbb{R}$ on $x$ at the direction of $uk\left(xl\right),$ define now $J:V\to \mathbb{R}$ where

$$J\left(u\right)=Ein-\langle ui,fi\rangle L-\sum j=Nui\left(xj\right)Pij-Plkuk\left(xl\right).$$

Under such hypotheses,

$$uk\left(xl\right)=\left({\displaystyle dEin\left(u(f,P,Plk)\right)dPlk}\right)Plk=,$$

$\forall k\in \{,,\},\forall xl\in \Omega \mathrm{such}\mathrm{that}xl\ne xj,\forall j\in \{,\dots ,N\}.$

Proof. 

The proof is exactly the same as in the Castigliano Theorem in the previous section except by setting the virtual load $Plk=$ in the end of this calculation and will not be repeated. □

40.3. A numerical example related to the Castigliano Theorem

Let $\omega =[,]\subset \mathbb{R}$ be the axis of a straight beam with a rectangular cross section of dimensions $B\times h$, where units in this subsections refer to the international system.

Let

$$I={\displaystyle bh12}$$

and denote by $E>$ the Young modulus for a steel beam.

Assume such a beam is subject to a vertical load $P>$ uniformly distributed on $\Omega .$

Assume also the beam is clamped at $x=$ and simply supported at $x=.$

Denoting by $w\in V=W,\left(\Omega \right)$ the vertical field of displacements results fro the action of P, the related boundary conditions are given by

$$w\left(\right)=w\left(\right)=,w,x\left(\right)=,w,xx\left(\right)=.$$

The total beam energy is defined by $J:V\to \mathbb{R}$, where

$$J\left(w\right)={\displaystyle EI}\int \Omega w,xxdx-\int \Omega Pwdx.$$

In order to apply the results of the previous section, we free the rotations of the beam at $x=$, considering a moment load M on $x=$ with general work

$$Mw,x\left(\right).$$

Hence, we define

$$V=\{w\in W,\left(\Omega \right):w\left(\right)=w\left(\right)=w,xx\left(\right)=\},$$

and define $J:V\to \mathbb{R}$, by

$$J\left(w\right)={\displaystyle EI}\int \Omega w,xxdx-\int \Omega Pwdx-Mw,x\left(\right).$$

We emphasize again, through the methods of the previous section, we intend to obtain the value of M which corresponds To

$$w,x\left(\right)=.$$

Let $\phi \in W,.$ The variation

$$\delta J(w;\phi ),$$

stands for

$$\delta J(w;\phi )=EI\int \Omega w,xx\phi ,xxdx-\int \Omega P\phi dx-M\phi ,x\left(\right).$$

Here assuming, w is smooth enough, integrating by parts and recalling that $\phi \in V,$ we obtain

$$\delta J(w;\phi )=EI\int \Omega w,xxxx\phi dx-EIwxx\left(\right)\phi x\left(\right)-\int \Omega P\phi dx-M\phi ,x\left(\right),$$

so that the extremal condition

$$\delta J(w;\phi )=,\forall \phi \in V,$$

provide us the following natural boundary condition

$$M=-EIw,xx\left(\right)$$

and the equation

$$EIwxxxx-P=,\mathrm{in}\Omega .$$

Here we recall the remaining essential boundary conditions,

$$w\left(\right)=w\left(\right)=w,xx\left(\right)=.$$

A particular solution of such an equation

$$EIwxxxx-P=,\mathrm{in}\Omega$$

stands for

$$wp\left(x\right)=ax,$$

where

$$a={\displaystyle !EI}.$$

The concerning general solution stands for

$$w\left(x\right)=wp\left(x\right)+bx+cx+dx+e.$$

The boundary condition $w\left(\right)=$ implies that $e=$.

The boundary condition

$$M=-EIw,xx\left(\right),$$

stands for

$$M=-EI\left(c\right),$$

so that

$$c=-{\displaystyle MEI}.$$

Moreover, from

$$wxx\left(\right)=,$$

we obtain

$$12a+b=c=.$$

From

$$w\left(\right)=$$

we have

$$a+b+c+d=.$$

From these last two equations, we obtain

$$b=-{\displaystyle c}-a\equiv b\left(c\right),$$

and

$$d=a-{\displaystyle 23}c.$$

From the Castigliano Theorem, $M\in \mathbb{R}$ must be such that

$$wx\left(\right)={\displaystyle dEindM}=,$$

where

$$Ein={\displaystyle EI}\int \Omega w,xxdx.$$

Observe that

$${\displaystyle dEindM}={\displaystyle dEindc}{\displaystyle dcdM}=,$$

so that in fact, it suffices to obtain,

$${\displaystyle dEindc}=.$$

Moreover, observe that

$$\begin{array}{ccc}\hfill Ein& =& {\displaystyle EI}\int \Omega w,xxdx\hfill \\ & =& {\displaystyle EI}\int 01(12ax+b\left(c\right)x+c)dx\hfill \\ & =& {\displaystyle EI}\left({\displaystyle 144a}+{\displaystyle 36b\left(c\right)}\right.\hfill \\ & & \left.+c+{\displaystyle 144ab\left(c\right)}+{\displaystyle 48ac}+{\displaystyle 24cb\left(c\right)}\right).\hfill \end{array}$$

(257)

Hence, we must have

$$\begin{array}{ccc}\hfill {\displaystyle dEindc}& =& {\displaystyle \partial Ein\partial b}{\displaystyle dbdc}+{\displaystyle \partial Ein\partial c}\hfill \\ & =& ,\hfill \end{array}$$

(258)

which has a solution

$$c={\displaystyle a}.$$

Thus,

$$c={\displaystyle \left(\right)!EI},$$

so that

$$M=-cEI=-{\displaystyle P!}.$$

40.4. Checking this last result for M by solving the concerning ordinary differential equation

In this section we check the result obtained for M by solving the following ODE,

$$EIw,xxxx-P=,\mathrm{in}\Omega =[,],$$

with the boundary conditions,

$$w\left(\right)=w\left(\right)=w,x\left(\right)=w,xx\left(\right)=.$$

We recall the general solution stands for

$$w\left(x\right)=ax+bx+cx+dx+e,$$

where

$$a={\displaystyle !EI}.$$

From $w\left(\right)=$, we obtain $e=.$

From $w,x\left(\right)=,$ we obtain $d=.$

From $w\left(\right)=,$ we have

$$a+b+c=.$$

From $wxx\left(\right)=,$ we have

$$12a+b+c=.$$

From such results, we obtain

$$b=-{\displaystyle a},$$

and

$$c={\displaystyle a}$$

Observe that

$$M\left(x\right)=-EIw,xx,$$

so that in particular

$$M=M\left(\right)=-EIw,xx\left(\right)=-EI\left(c\right)=-{\displaystyle P!}.$$

This value for M here obtained coincide with the one obtained in the previous subsection, as expected.

The objective of this section is complete.

41. Duality for a general relaxed primal variational formulation

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular boundary denoted by $\partial \Omega .$

Consider a functional $J:V\to \mathbb{R}$ where

$$J\left(u\right)={\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle \alpha }\int \Omega (u-\beta )dx-\langle u,f\rangle L,$$

where $V=W,\left(\Omega \right),$$\gamma >,$$\alpha >$, $\beta >$, $Y=Y*=L\left(\Omega \right)$, $Y=Y*=L(\Omega ;\mathbb{R})$, and $f\in L\left(\Omega \right).$

We define the associated relaxed functional $J:V\times V\times (,),$ by

$$\begin{array}{ccc}\hfill J(u,\varphi ,\lambda )& =& {\displaystyle \lambda \gamma }\int \Omega (\nabla u-(-\lambda )\nabla \varphi )·(\nabla u-(-\lambda )\nabla \varphi )dx\hfill \\ & & +{\displaystyle (-\lambda )\gamma }\int \Omega (\nabla u+\lambda \nabla \varphi )·(\nabla u+\lambda \nabla \varphi )dx\hfill \\ & & +{\displaystyle \lambda \alpha }\int \Omega ((u-(-\lambda )\varphi )-\beta )dx+{\displaystyle (-\lambda )\alpha }\int \Omega ((u+\lambda \varphi )-\beta )dx\hfill \\ & & -\lambda \langle u-(-\lambda )\varphi ,f\rangle L-(-\lambda )\langle u+\lambda \varphi ,f\rangle L.\hfill \end{array}$$

(259)

Moreover, we define, $F:V\times V\times (,)\to \mathbb{R},$$F:V\times V\times (,)\to \mathbb{R},$$F:V\times V\times (,)\to \mathbb{R},$$F:V\times V\times (,)\to \mathbb{R},$$F:V\times V\times (,)\to \mathbb{R},$ and $F:V\times V\times (,)\to \mathbb{R},$ by

$$F(u,\varphi ,\lambda )={\displaystyle \lambda \gamma }\int \Omega (\nabla u-(-\lambda )\nabla \varphi )·(\nabla u-(-\lambda )\nabla \varphi )dx,$$

$$F(u,\varphi ,\lambda )={\displaystyle (-\lambda )\gamma }\int \Omega (\nabla u+\lambda \nabla \varphi )·(\nabla u+\lambda \nabla \varphi )dx,$$

$$F(u,\varphi ,\lambda )={\displaystyle \lambda \alpha }\int \Omega ((u-(-\lambda )\varphi )-\beta )dx,$$

$$F(u,\varphi ,\lambda )={\displaystyle (-\lambda )\alpha }\int \Omega ((u+\lambda \varphi )-\beta )dx,$$

$$F(u,\varphi ,\lambda )=-\lambda \langle u-(-\lambda )\varphi ,f\rangle L,$$

$$F(u,\varphi ,\lambda )=-(-\lambda )\langle u+\lambda \varphi ,f\rangle L,$$

respectively.

Observe that

$$\begin{array}{ccc}\hfill J(u,\varphi ,u)& =& F(u,\varphi ,\lambda )+F(u,\varphi ,\lambda )\hfill \\ & & F(u,\varphi ,\lambda )+F(u,\varphi ,\lambda )\hfill \\ & & F(u,\varphi ,\lambda )+F(u,\varphi ,\lambda ),\hfill \end{array}$$

(260)

Thus,

$$\begin{array}{ccc}\hfill J(u,\varphi ,u)& \ge & F(u,\varphi ,\lambda )+F(u,\varphi ,\lambda )\hfill \\ & & +\langle (u-(-\lambda )\varphi )-\beta ,v*\rangle L\hfill \\ & & +\langle (u+\lambda \varphi )-\beta ,v*\rangle L\hfill \\ & & F(u,\varphi ,\lambda )+F(u,\varphi ,\lambda )\hfill \\ & & F(u,\varphi ,\lambda )+F(u,\varphi ,\lambda )\hfill \\ & & +\underset{v\in Y}{inf}\{-\langle v,v*\rangle L+F˜(v,\lambda )\}\hfill \\ & & +\underset{v\in Y}{inf}\{-\langle v,v*\rangle L+F˜(v,\lambda )\}\hfill \end{array}$$

(261)

where

$$F˜(v,\lambda )={\displaystyle \lambda \alpha }\int \Omega v32dx,$$

$$F˜(v,\lambda )={\displaystyle (-\lambda )\alpha }\int \Omega v42dx,$$

Therefore, defining $F˜*:Y*\times (,)\to \mathbb{R}$ and $F˜*:Y*\times (,)\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill F˜*(v*,\lambda )& =& \underset{v\in Y}{sup}\{\langle v.v*\rangle L-F(v,\lambda )\hfill \\ & =& {\displaystyle \alpha \lambda }\int \Omega (v*)dx,\hfill \end{array}$$

(262)

and

$$\begin{array}{ccc}\hfill F˜*(v*,\lambda )& =& \underset{v\in Y}{sup}\{\langle v.v*\rangle L-F(v,\lambda )\hfill \\ & =& {\displaystyle \alpha (-\lambda )}\int \Omega (v*)dx,\hfill \end{array}$$

(263)

we may also infer that

$$\begin{array}{ccc}\hfill J(u,\varphi ,\lambda )& \ge & \underset{v\in Y}{inf}\{\langle v,v*\rangle L+F˜(v,\lambda )\}\hfill \\ & & +\underset{v\in Y}{inf}\{\langle v,v*\rangle L+F˜(v,\lambda )\}\hfill \\ & & +\underset{v\in Y}{inf}\left\{-\langle v,\mathrm{div}v*\rangle L+\int \Omega (v52-\beta )v*dx-\lambda \langle v,f\rangle L\right\}\hfill \\ & & +\underset{v\in Y}{inf}\left\{-\langle v,\mathrm{div}v*\rangle L+\int \Omega (v62-\beta )v*dx-(-\lambda )\langle v,f\rangle L\right\}\hfill \\ & & -F˜*(v*,\lambda )-F˜*(v*,\lambda )\hfill \\ & =& -F˜*(v*,\lambda )-F˜*(v*,\lambda )\hfill \\ & & -F*(v*,v*,\lambda )-F*(v*,v*,\lambda )\hfill \\ & & -F˜*(v*,\lambda )-F˜*(v*,\lambda ),\hfill \end{array}$$

(264)

if $v*=(v*,\dots ,v*)\in A*$ where,

$$A*=\{v*\in [Y*]\times [Y*]:v*>\mathrm{and}v*>,\mathrm{in}\Omega \},$$

$$F˜(v,\lambda )={\displaystyle \lambda \gamma }\int \Omega v·vdx,$$

$$F˜(v,\lambda )={\displaystyle \lambda \gamma }\int \Omega v·vdx,$$

$$F˜(v,v*,\lambda )=\int \Omega (v52-\beta )v*dx-\lambda \langle v,f\rangle L,$$

$$F˜(v,v*,\lambda )=\int \Omega (v62-\beta )v*dx-(-\lambda )\langle v,f\rangle L,$$

and

$$\begin{array}{ccc}\hfill F˜*(v*,\lambda )& =& \underset{v\in Y}{sup}\{\langle v,v*\rangle L-F˜(v,\lambda )\}\hfill \\ & =& {\displaystyle \gamma \lambda }\int \Omega v*·v*dx,\hfill \end{array}$$

(265)

$$\begin{array}{ccc}\hfill F˜*(v*,\lambda )& =& \underset{v\in Y}{sup}\{\langle v,v*\rangle L-F˜(v,\lambda )\}\hfill \\ & =& {\displaystyle \gamma (-\lambda )}\int \Omega v*·v*dx,\hfill \end{array}$$

(266)

$$\begin{array}{ccc}\hfill F˜*(v*,v*,\lambda )& =& \underset{v\in Y}{sup}\{\langle v,v*\rangle L-F˜(v,v*,\lambda )\}\hfill \\ & =& {\displaystyle 12}\int \Omega {\displaystyle (\mathrm{div}v*+\lambda f)v*}dx+\beta \int \Omega v*dx,\hfill \end{array}$$

(267)

and

$$\begin{array}{ccc}\hfill F˜*(v*,v*,\lambda )& =& \underset{v\in Y}{sup}\{\langle v,v*\rangle L-F˜(v,v*,\lambda )\}\hfill \\ & =& {\displaystyle 12}\int \Omega {\displaystyle (\mathrm{div}v*+(-\lambda )f)v*}dx+\beta \int \Omega v*dx.\hfill \end{array}$$

(268)

Denoting, as above indicated, $v*=(v*,v*,v*,v*)\in [Y*]\times [Y*]$, we define $J*:[Y*]\times [Y*]\times (,)\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill J*(v*,\lambda )& =& -F˜*(v*,\lambda )-F˜*(v*,\lambda )\hfill \\ & & -F*(v*,v*,\lambda )-F*(v*,v*,\lambda )\hfill \\ & & -F˜*(v*,\lambda )-F˜*(v*,\lambda ),\hfill \end{array}$$

(269)

Observe that we have got

$$\begin{array}{ccc}\hfill \underset{u\in V}{inf}J\left(u\right)& \ge & \underset{(u,\varphi ,\lambda )\in V\times V\times [,]}{inf}J(u,\varphi ,\lambda )\hfill \\ & \ge & \underset{\lambda \in (,)}{inf}\left\{\underset{v*\in A*}{sup}J*(v*,\lambda )\right\}.\hfill \end{array}$$

(270)

41.1. A numerical example

We have obtained numerical results for $\gamma =.$, $\alpha =.$, $\beta =.$ and $f\equiv 10,\mathrm{in}\Omega ,$ for the special case in which $\Omega =[,]\subset \mathbb{R}.$

Such results have been performed through the following algorithm:

Set $n=$ and $\lambda n=/$.

Calculate $vn*\in A*$ such that

$$J*(vn*,\lambda n)=\underset{v*\in A*}{sup}J*(v*,\lambda n),$$

Calculate $\lambda n+\in (,)$ such that

$$J*(vn*,\lambda n+)=\underset{\lambda \in (,)}{inf}J*(vn*,\lambda ),$$

Set $n:=n+$ and go to step 2 until the satisfaction of an appropriate convergence criterion.

Here, we recall that for the optimal points

$${\displaystyle \mathrm{div}v*+\lambda fv*}=u-(-\lambda )\varphi ,$$

and

$${\displaystyle \mathrm{div}v*+(-\lambda )fv*}=u+\lambda \varphi ,$$

so that

$$u=\lambda \left({\displaystyle \mathrm{div}v*+\lambda fv*}\right)+(-\lambda )\left({\displaystyle \mathrm{div}v*+(-\lambda )fv*}\right).$$

For such a corresponding optimal $u$ please see Figure 32.

For the solution $u$ of the primal problem obtained through the generalized method of lines, please see Figure 33.

We may observe the solutions $u$ and $u$ are qualitatively similar, as expected.

Here we present the software developed to perform such numerical results.

***********************

clear all

global m8 d L A3 A B yo u v e1 dv1 dv2 dv3 v5 v6 v3 v4 v1 v2 K5 e5 L1 L2 L3

m8=100;

d=1/m8;

e1=0.00001;

e5=0.001;

K5=10000.0;

A3=0.1;

A=3.0;

B=5.0;

for i=1:m8

uo(i,1)=5;

yo(i,1)=10.0;

end;

L=1/2;

for k=1:50

k

i=1;

m12=$+*A*uo(i,)*d/A-*A*B/A*d;$

m50(i)=1/m12;

z(i)=$m50\left(i\right)*(yo(i,)*d/A+*A*uo(i,)*d/A);$

for i=2:m8-1

m12=$+*A*uo(i,)*d/A-*A*B/A*d-m50(i-);$

m50(i)=1/m12;

z(i)=$m50\left(i\right)*(yo(i,)*d/A+*A*uo(i,)*d/A+z(i-));$

end;

w(m8,1)=0;

for i=1:m8-1

w(m8-i,1)=m50(m8-i)*w(m8-i+1)+z(m8-i);

end;

uo=w;

uo(m8/2,1)

end;

for i=1:4*m8

xo(i,1)=3.0;

end;

for i=1:1

x1(i,1)=1/2;

end;

for k1=1:10

k1

k=1;

b12=1.0;

while $(b12>10-)$ && $(k<50)$

k

k=k+1;

X=fminunc(’funFeb30LG’,xo);

b12=max(abs(xo-X))

xo=X;

end;

X1=fminunc(’funFeb31LG’,x1);

x1=X1;

end;

u(m8,1)=0;

for i=1:m8

x(i,1)=i*d;

end;

plot(x,u);

*******************************

With the auxiliary function "funFeb30LG", where

*******************************

function S=funFeb30LG(x)

global m8 d L A3 A B yo u v e1 dv2 dv1 dv3 v3 v4 v5 v6 v1 v2 K5 e5 L1 L2 L3

for i=1:m8

v1(i,1)=x(i,1);

v2(i,1)=x(m8+i,1);

v3(i,1)=x(2*m8+i,1);

v4(i,1)=x(3*m8+i,1);

end; for i=1:m8-1

dv1(i,1)=(v1(i+1,1)-v1(i,1))/d;

dv2(i,1)=(v2(i+1,1)-v2(i,1))/d;

end;

S=0;

for i=1:m8-1

S=S+$(yo(i,)*L+*yo(i,)*L*dv(i,)+dv(i,)+*B*v(i,))/(*v(i,));$

S=S+$(yo(i,)*(-L)+*yo(i,)*(-L)*dv(i,)+dv(i,)+*B*v(i,))/(*v(i,));$

S=S+$v(i,)/sqrt(L+e)//A+v(i,)/sqrt((-L)+e)//A;$

S=S+$v(i,)//sqrt(L+e)/A+v(i,)//sqrt((-L)+e)/A;$

end;

for i=1:m8-1

u(i,1)=$L*(yo(i,)*L+dv(i,))/\left(v(i,)\right)/;$

u(i,1)=u(i,1)+$(-L)*((-L)*yo(i,)+dv(i,))//\left(v(i,)\right);$

end;

*******************

Finally, we present the auxiliary function "funFeb31LG"

*********************************************

function S1=funFeb31LG(x)

global m8 d L L1 L2 L3 A3 A B yo u v e1 dv2 dv1 dv3 v5 v6 v3 v4 v1 v2 K5 e5

L=(sin(x(1,1))+1)/2;

for i=1:m8-1

dv1(i,1)=(v1(i+1,1)-v1(i,1))/d;

dv2(i,1)=(v2(i+1,1)-v2(i,1))/d;

end;

S=0;

for i=1:m8-1

S=S+$(yo(i,)*L+*yo(i,)*L*dv(i,)+dv(i,)+*B*v(i,))/(*v(i,));$

S=S+$(yo(i,)*(-L)+*yo(i,)*(-L)*dv(i,)+dv(i,)+*B*v(i,))/(*v(i,));$

S=S+$v(i,)/sqrt(L+e)//A+v(i,)/sqrt((-L)+e)//A;$

S=S+$v(i,)//sqrt(L+e)/A+v(i,)//sqrt((-L)+e)/A;$

end;

S1=-S;

**************************************

Remark 23.

Observe that the functional $J$ is convex in $A$ however, the restrictions $v*>$ and $v*>$ in Ω may cause a difference between the solution obtained through $J$ and the solution got through the primal formulation J, a so-called duality gap.

Anyway, through such a relaxation process, utilizing the dual functional $J$ we may still obtain a good qualitative approximation of the global optimal point for the primal formulation J.

Indeed, such a global solution obtained through the dual functional $J$ may be an excellent initial solution for obtaining a more accurate one through the standard Newton Method, for example.

42. A global existence result for a model in non-linear elasticity

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega =S$.

Define a functional $J:V\to \mathbb{R}$ by

$$J\left(u\right)={\displaystyle 12}\int \Omega Hijkl\gamma ij\left(u\right)\gamma kl\left(u\right)dx-\langle ui,fi\rangle L,$$

where

$$\gamma ij\left(u\right)={\displaystyle ui,j+uj,i}+{\displaystyle 12}um,ium,j,$$

$$V=\{u\in W,(\Omega ;\mathbb{R}):u=v^\mathrm{on}S\subset \partial \Omega \}.$$

We also denote $Y=Y*=L(\Omega ;\mathbb{R}),$ so that $f=(f,f,f)\in Y.$

Here $\left\{Hijkl\right\}$ is a fourth order constant, positive definite and symmetric tensor.

With such assumptions and statements in mind, we may prove the following theorem.

Theorem 11.

Assume $\left\{Hijkl\right\}$ is such that

$$\underset{\parallel u\parallel V\to \infty }{lim}J\left(u\right)=+\infty .$$

Under such hypothesis, there exists $u\in V$ such that

$$J\left(u\right)=\underset{u\in V}{min}J\left(u\right).$$

Proof. 

From the hypotheses, there exists $\alpha \in \mathbb{R}$ such that

$$\alpha =\underset{u\in V}{inf}J\left(u\right).$$

Let $\left\{un\right\}\subset V$ be a sequence such that

$$\alpha \le J\left(un\right)<\alpha +{\displaystyle n},\forall n\in \mathbb{N}.$$

Suppose, to obtain contradiction, there exists a subsequence $\left\{nk\right\}\subset \mathbb{N},$ such that

$$\parallel unk\parallel V\to \infty .$$

From the hypotheses, we have

$$J\left(unk\right)\to +\infty ,\mathrm{as}k\to \infty .$$

This contradicts

$$\underset{k\to \infty }{lim}J\left(unk\right)=\alpha \in \mathbb{R}.$$

From such results we may infer that there exists $K>$ such that

$$\parallel un\parallel V\le K,\forall n\in \mathbb{N}.$$

Consequently, from this, the Sobolev Embedding and Rellich Kondrashov theorems, there exists $u\in V\cap L\infty (\Omega ;\mathbb{R})$ for which, up to a not relabelled subsequence, we have

$$un\rightharpoonup u,\mathrm{weakly}\mathrm{in}W,(\Omega ;\mathbb{R}),$$

$$un\to u,\mathrm{strongly}\mathrm{in}L\left(\Omega \right),$$

$$un\to u,\mathrm{strongly}\mathrm{in}L\infty (\Omega ;\mathbb{R}).$$

Let $\phi \in Cc\infty \left(\Omega \right).$

Thus,

$$\begin{array}{ccc}& & \left|〈{\displaystyle \partial \left(un\right)i\partial xj}-{\displaystyle \partial \left(u\right)i\partial xj},\phi 〉L\right|\hfill \\ & =& \left|〈\left(un\right)i-\left(u\right)i,{\displaystyle \partial \phi \partial xj}〉L\right|\hfill \\ & \le & \parallel \left(un\right)i-\left(u\right)i\parallel \infty ∥{\displaystyle \partial \phi \partial xj}∥\hfill \\ & \to & ,\mathrm{as}n\to \infty .\hfill \end{array}$$

(271)

Since $\phi \in Cc\infty \left(\Omega \right)$ is arbitrary and $Cc\infty \left(\Omega \right)$ is dense in $L\left(\Omega \right)$ we may infer that

$${\displaystyle \partial \left(un\right)i\partial xj}\rightharpoonup {\displaystyle \partial \left(u\right)i\partial xj},\mathrm{weakly}\mathrm{in}L\left(\Omega \right),$$

$\forall i,j\in \{,,\}.$

Define $W=L\left(\Omega \right)$ with the norm

$$\parallel v\parallel W=sup\{\langle v,\phi \rangle L,\phi \in Cc\infty \left(\Omega \right),\parallel \phi \parallel ,\le \}.$$

We may easily verify that

$${\displaystyle \partial \left(un\right)i\partial xj}\to {\displaystyle \partial \left(u\right)i\partial xj},\mathrm{strongly}\mathrm{in}W,$$

$\forall i,j\in \{,,\}.$

Thus,

$$\left\{{\displaystyle \partial \left(un\right)i\partial xj}\right\}$$

is a Cauchy sequence in $W.$

Hence, for each $n\in \mathbb{N}$ there exists $nk\in \mathbb{N}$ such that $m,l\ge nk$, then

$$∥{\displaystyle \partial \left(um\right)i\partial xj}-{\displaystyle \partial \left(ul\right)i\partial xj}∥W<{\displaystyle k}.$$

where $n$ may be taken as an increasing subsequence in $\mathbb{N}.$

In particular, we have got

$$∥{\displaystyle \partial \left(unk+\right)i\partial xj}-{\displaystyle \partial \left(unk\right)i\partial xj}∥W<{\displaystyle k}.$$

Define now

$$gl=\left|{\displaystyle \partial \left(un\right)i\partial xj}\right|+\sum k=l-\left|{\displaystyle \partial \left(unk+\right)i\partial xj}-{\displaystyle \partial \left(unk\right)i\partial xj}\right|,$$

and

$$g=\left|{\displaystyle \partial \left(un\right)i\partial xj}\right|+\sum k=\infty \left|{\displaystyle \partial \left(unk+\right)i\partial xj}-{\displaystyle \partial \left(unk\right)i\partial xj}\right|.$$

Observe that

$$\begin{array}{ccc}\hfill \parallel g\parallel W& \le & ∥{\displaystyle \partial \left(un\right)i\partial xj}∥W+\sum k=\infty ∥{\displaystyle \partial \left(unk+\right)i\partial xj}-{\displaystyle \partial \left(unk\right)i\partial xj}∥W\hfill \\ & \le & ∥{\displaystyle \partial \left(un\right)i\partial xj}∥W+\sum k=\infty {\displaystyle k}\hfill \\ & <& +\infty .\hfill \end{array}$$

(272)

From such results we may infer that $g\left(x\right)\in \mathbb{R},\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega .$

Moreover, since an absolutely convergent series is also convergent, we may infer that

$${\displaystyle \partial \left(unl\right)i\partial xj}={\displaystyle \partial \left(un\right)i\partial xj}+\sum k=l-\left({\displaystyle \partial \left(unk+\right)i\partial xj}-{\displaystyle \partial \left(unk\right)i\partial xj}\right)\to hij,\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,$$

for some $hij\in L\left(\Omega \right),\forall i,j\in \{,,\}.$

From such results, we have

$${\displaystyle \partial \left(unl\right)i\partial xj}\to hij,\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega$$

and

$${\displaystyle \partial \left(unl\right)i\partial xj}\rightharpoonup {\displaystyle \partial \left(u\right)i\partial xj},\mathrm{weakly}\mathrm{in}L\left(\Omega \right),$$

so that

$${\displaystyle \partial \left(u\right)i\partial xj}=hij,\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega .$$

Consequently, we have got

$${\displaystyle \partial \left(unl\right)i\partial xj}\to {\displaystyle \partial \left(u\right)i\partial xj},\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega .$$

Now fix $i,j,m\in \{,,\}.$

Observe that from the Cauchy-Schwarz inequality, we have

$$\begin{array}{ccc}& & \int \Omega \left({\displaystyle \partial \left(unl\right)m\partial xj}{\displaystyle \partial \left(unl\right)m\partial xj}\right)dx\hfill \\ & \le & ∥{\displaystyle \partial \left(unl\right)m\partial xi}∥42∥{\displaystyle \partial \left(unl\right)m\partial xj}∥42\hfill \\ & \le & K,\forall l\in \mathbb{N}\hfill \end{array}$$

(273)

for some appropriate real constant $K>$.

Therefore, up to a not relabeled subsequence there exists $v\in L\left(\Omega \right)$ such that

$${\displaystyle \partial \left(unl\right)m\partial xi}{\displaystyle \partial \left(unl\right)m\partial xj}\rightharpoonup v,\mathrm{weakly}\mathrm{in}L\left(\Omega \right),$$

Since

$${\displaystyle \partial \left(unl\right)m\partial xi}{\displaystyle \partial \left(unl\right)m\partial xj}\to {\displaystyle \partial \left(u\right)m\partial xi}{\displaystyle \partial \left(u\right)m\partial xj},\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,$$

we obtain

$$v={\displaystyle \partial \left(u\right)m\partial xi}{\displaystyle \partial \left(u\right)m\partial xj},\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,$$

so that

$${\displaystyle \partial \left(unl\right)m\partial xi}{\displaystyle \partial \left(unl\right)m\partial xj}\rightharpoonup {\displaystyle \partial \left(u\right)m\partial xi}{\displaystyle \partial \left(u\right)m\partial xj},\mathrm{weakly}\mathrm{in}L\left(\Omega \right),$$

$\forall i,j,m\in \{,,\}.$

Therefore, from such results we may infer that

$$\gamma ij\left(unl\right)\rightharpoonup \gamma ij\left(u\right),\mathrm{weakly}\mathrm{in}L\left(\Omega \right),\forall i,j\in \{,,\}.$$

Moreover, since J is convex in $\left\{\gamma ij\right\}$ we finally obtain

$$\alpha =\underset{l\to \infty }{lim\; inf}J\left(unl\right)\ge J\left(u\right),$$

so that

$$J\left(u\right)=\underset{u\in V}{min}J\left(u\right).$$

The proof is complete.

□

43. A note on a general relaxation procedure for the vectorial case in the calculus of variation

Let $\Omega \subset \mathbb{R}n$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$ Consider a continuous and bounded below functional $F:V\to \mathbb{R}$ where

$$V=\{u\in W,(\Omega ;\mathbb{R}N):u=u\mathrm{on}\partial \Omega \}.$$

Define $H:V\to \mathbb{R}$ by

$$H\left(u\right)=inf\{\lambda F\left(v\right)+(-\lambda )F\left(w\right);\le \lambda \le ,v,w\in V,\lambda v+(-\lambda )w=u\}.$$

Observe that as it has been shown in a previous section, we have

$$F**\left(u\right)\le H\left(u\right)\le F\left(u\right),\forall u\in V.$$

Moreover, also as indicated in a previous section, we may obtain

$$H\left(u\right)=\underset{(\varphi ,\lambda )\in V\times [,]}{inf}\left\{\lambda F(u-(-\lambda )\varphi )+(-\lambda )F(u+\lambda \varphi )\right\},$$

where $V=W,(\Omega ;\mathbb{R}N).$

Reasoning inductively, having $Hk:V\to \mathbb{R}$, define $Hk+:V\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill Hk+\left(u\right)& =& inf\{\lambda k+Hk\left(vk+\right)+(-\lambda k+)Hk\left(wk+\right);\hfill \\ & & \le \lambda k+\le ,vk+,wk+\in V,\lambda k+vk++(-\lambda k+)wk+=u\}.\hfill \end{array}$$

(274)

Thus

$$Hk+\left(u\right)=\underset{(\varphi k+,\lambda k+)\in V\times [,]}{inf}\left\{\lambda k+Hk(u-(-\lambda k+)\varphi k+)+(-\lambda k+)Hk(u+\lambda k+\varphi k+)\right\}.$$

Observe that

$$F**\left(u\right)\le Hk+\left(u\right)\le Hk\left(u\right)\le F\left(u\right),\forall k\in \mathbb{N}.$$

Define $H:V\to \mathbb{R}$ by

$$H\left(u\right)=\underset{k\to +\infty }{lim}Hk\left(u\right)=\underset{k\in \mathbb{N}}{inf}Hk\left(u\right),\forall u\in V.$$

Suppose, to obtain contradiction, that $H$ is not convex.

Hence, there exists $u^\in V$ such that

$$\left(H\right)(u^)<H(u^),$$

where

$$\left(H\right)\left(u\right)=inf\{\lambda H\left(v\right)+(-\lambda )H\left(w\right);\le \lambda \le ,v,w\in V,\lambda v+(-\lambda )w=u\}.$$

This contradicts

$$H\left(u\right)=\underset{k\to +\infty }{lim}Hk\left(u\right)=\underset{k\in \mathbb{N}}{inf}Hk\left(u\right),\forall u\in V.$$

Therefore $H$ is convex on V so that from this and

$$F**\left(u\right)\le H\left(u\right)\le F\left(u\right),\forall u\in V$$

we may infer that

$$H\left(u\right)=F**\left(u\right),\forall u\in V.$$

44. A note on another general relaxation procedure for the vectorial case in the calculus of variation

Let $\Omega \subset \mathbb{R}n$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$ Consider a continuous and bounded below functional $F:V\to \mathbb{R}$ where

$$V=\{u\in W,(\Omega ;\mathbb{R}N):u=u\mathrm{on}\partial \Omega \}.$$

Fix $k\in \mathbb{N}.$

Define $\left(H\right)k:V\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill \left(H\right)k\left(u\right)& =& inf\left\{\sum j=k\lambda jF\left(vj\right):\le \lambda j\le \mathrm{and}vj\in V,\forall j\in \{,\dots ,k\},\right.\hfill \\ & & \left.\sum j=k\lambda j=\mathrm{and}\sum j=k\lambda jvj=u\right\}.\hfill \end{array}$$

(275)

Observe that

$$F**\left(u\right)\le \left(H\right)k+\left(u\right)\le \left(H\right)k\left(u\right)\le F\left(u\right),\forall u\in V.$$

Define $H:V\to \mathbb{R}$ by

$$H\left(u\right)=\underset{k\to \infty }{lim}\left(H\right)k\left(u\right)=\underset{k\in \mathbb{N}}{inf}\left\{\left(H\right)k\left(u\right)\right\},\forall u\in V.$$

Reasoning inductively, having $Hm:V\to \mathbb{R}$, we may obtain $\left(Hm\right)k:V\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill \left(Hm\right)k\left(u\right)& =& inf\left\{\sum j=k\lambda jHm\left(vj\right):\le \lambda j\le \mathrm{and}vj\in V,\forall j\in \{,\dots ,k\},\right.\hfill \\ & & \left.\sum j=k\lambda j=\mathrm{and}\sum j=k\lambda jvj=u\right\}.\hfill \end{array}$$

(276)

Observe that

$$F**\left(u\right)\le \left(Hm\right)k+\left(u\right)\le \left(Hm\right)k\left(u\right)\le F\left(u\right),\forall u\in V.$$

Now we define

$$Hm+\left(u\right)=\underset{k\to \infty }{lim}\left(Hm\right)k\left(u\right)=\underset{k\in \mathbb{N}}{inf}\left\{\left(Hm\right)k\left(u\right)\right\},\forall u\in V,$$

$\forall m\in \mathbb{N}.$

Therefore, we have obtained a sequence $\{Hm:V\to \mathbb{R}\}$ such that

$$F**\left(u\right)\le Hm+\left(u\right)\le Hm\left(u\right)\le F\left(u\right),\forall u\in V.$$

Thus, we may define $H:V\to \mathbb{R}$ by

$$H\left(u\right)=\underset{m\to \infty }{lim}Hm\left(u\right)=\underset{m\in \mathbb{N}}{inf}\left\{Hm\left(u\right)\right\},\forall u\in V.$$

Suppose, to obtain contradiction, that $H:V\to \mathbb{R}$ is not convex on $V.$

Hence, there exists $u^\in V$ such that

$$\left(H\right)(u^)<H(u^),$$

where

$$\left(H\right)\left(u\right)=inf\{\lambda H\left(v\right)+(-\lambda )H\left(w\right):\le \lambda \le ,v,w\in V,\lambda v+(-\lambda )w=u\},$$

$\forall u\in V.$

This contradicts

$$H\left(u\right)=\underset{m\to \infty }{lim}Hm\left(u\right)=\underset{m\in \mathbb{N}}{inf}\left\{Hm\left(u\right)\right\},\forall u\in V.$$

Therefore, $H$ is convex on V so that from this and

$$F**\left(u\right)\le H\left(u\right)\le F\left(u\right),\forall u\in V,$$

we may infer that

$$H\left(u\right)=F**\left(u\right),\forall u\in V.$$

45. A proximal relaxed general approach also suitable for the vectorial case in the calculus of variations

Let $\Omega =[,]\subset \mathbb{R}$ and consider a proximal relaxed functional $J:V\times V\times [,]\times Y*\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J(u,\varphi ,\lambda ,z*)& =& {\displaystyle \lambda }\int \Omega ((u\prime -(-\lambda )\varphi \prime )-)dx\hfill \\ & & {\displaystyle (-\lambda )}\int \Omega ((u\prime +\lambda \varphi \prime )-)dx\hfill \\ & & +{\displaystyle 12}\int \Omega (u-f)dx+{\displaystyle K}\int \Omega (u-f)dx\hfill \\ & & -\int \Omega z*(u-f)dx+{\displaystyle K}\int \Omega (z*)dx,\hfill \end{array}$$

(277)

where

$$V=\{u\in W,\left(\Omega \right):u\left(\right)=\mathrm{and}u\left(\right)=/\},$$

$V=W,\left(\Omega \right),\mathrm{and}Y=Y*=L\left(\Omega \right).$

In order to obtain a critical point of such a proximal relaxed primal formulation, we propose the following algorithm:

Set $n=$, $\epsilon =10-$ and $zn*\equiv .$

Calculate $(un,\varphi n,\lambda n)\in V\times V\times [,]$ such that

$$J(un,\varphi n,\lambda n,zn*)=\underset{(u,\varphi ,\lambda )\in V\times V\times [,]}{inf}J(u,\varphi ,\lambda ,zn*).$$

Calculate $zn+*\in Y*$ such that

$$J(un,\varphi n,\lambda n,zn+*)=\underset{z*\in Y*}{inf}J(un,\varphi n,\lambda n,z*),$$

so that indeed,

$$zn+*=K(un-f).$$

If $\parallel zn+*-zn*\parallel \infty <\epsilon ,$ then stop. Otherwise set $n:=n+$ and go to item 2.

We have obtained numerical results for $K=100$ and

$$f\left(x\right)=sin\left(\pi x\right)/.$$

For the optimal solution $u\left(x\right)$ obtained, please see Figure 34.

At this point we present the software in MAT-LAB we have developed to obtain such numerical results.

*****************************************

clear all

global m8 d u v yo e1 K z

m8=100;

d=1/m8;

e1=0.0005;

K=100.0;

for i=1:m8

yo(i,1)=sin(pi*i*d)/2;

z(i,1)=0;

end;

for i=1:2*m8+1

xo(i,1)=0.3;

x1(i,1)=0.3;

end;

k1=1;

b14=1.0;

while $(b14>10-)$ && $(k<11)$

k1

k1=k1+1;

k=1;

b12=1.0;

while $(b12>10-)$ && $(k<16)$

k

k=k+1;

X=fminunc(’funMarch24PhaseT’,xo);

b12=max(abs(X-xo))

xo=X;

u(m8/2,1)

end;

b14=max(abs(x1-xo));

z=K*(u-yo);

x1=xo;

u(m8/2,1)

end;

for i=1:m8

x(i,1)=i*d;

end;

plot(x,u)

*********************************

Here the auxiliary function "funMarch24PhaseT"

*****************************

function S=funMarch24PhaseT(x)

global m8 d u v L yo e1 K z

for i=1:m8

u(i,1)=x(i,1);

v(i,1)=x(i+m8,1);

end;

L=(sin(x(2*m8+1,1))+1)/2;

u(m8,1)=1/2;

v(m8,1)=0.0;

du(1,1)=u(1,1)/d;

dv(1,1)=v(1,1)/d;

for i=2:m8

du(i,1)=(u(i,1)-u(i-1,1))/d;

dv(i,1)=(v(i,1)-v(i-1,1))/d;

end;

d2u(1,1)=$(-*u(,)+u(,))/d;$

for i=2:m8-1

d2u(i,1)=$(u(i+,)-*u(i,)+u(i-,))/d;$

end;

S=0;

for i=1:m8

S=S+$L*((du(i,)-(-L)*dv(i,))-)/;$

S=S+$(-L)*((du(i,)+L*dv(i,))-)/;$

S=S+$(u(i,)-yo(i,))/;$

S=S+$K*(u(i,)-yo(i,))/-z(i,)*(u(i,)-yo(i,));$

end;

for i=1:m8-1

S=S+$e*du(i,);$

end;

***************************

46. Another proximal relaxed general approach also suitable for the vectorial case in the calculus of variations

Let $\Omega =[,]\subset \mathbb{R}$ and consider a proximal relaxed functional $J:V\times \left[V\right]\times B\times Y*\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J(u,\varphi ,\lambda ,z*)& =& {\displaystyle \lambda }\int \Omega ((u\prime +\lambda \varphi \prime +\lambda \varphi \prime +\lambda \varphi \prime -\varphi \prime )-)dx\hfill \\ & & +{\displaystyle \lambda }\int \Omega ((u\prime +\lambda \varphi \prime +\lambda \varphi \prime +\lambda \varphi \prime -\varphi \prime )-)dx\hfill \\ & & +{\displaystyle \lambda }\int \Omega ((u\prime +\lambda \varphi \prime +\lambda \varphi \prime +\lambda \varphi \prime -\varphi \prime )-)dx\hfill \\ & & +{\displaystyle \lambda }\int \Omega ((u\prime +\lambda \varphi \prime +\lambda \varphi \prime +\lambda \varphi \prime )-)dx\hfill \\ & & +{\displaystyle 12}\int \Omega (u-f)dx+{\displaystyle K}\int \Omega (u-f)dx\hfill \\ & & -\int \Omega z*(u-f)dx+{\displaystyle K}\int \Omega (z*)dx,\hfill \end{array}$$

(278)

where

$$V=\{u\in W,\left(\Omega \right):u\left(\right)=\mathrm{and}u\left(\right)=/\},$$

$V=W,\left(\Omega \right),Y=Y*=L\left(\Omega \right),$$f\in L\left(\Omega \right)$ and

$$B=\left\{\lambda =(\lambda ,\dots ,\lambda )\in \mathbb{R}:\lambda j\ge ,\forall j\in \{,\dots ,\}\mathrm{and}\sum j=\lambda j=\right\}.$$

In order to obtain a critical point of such a proximal relaxed primal formulation, we propose the following algorithm:

Set $n=$, $\epsilon =10-$ and $zn*\equiv .$

Calculate $(un,\varphi n,\lambda n)\in V\times \left[V\right]\times B$ such that

$$J(un,\varphi n,\lambda n,zn*)=\underset{(u,\varphi ,\lambda )\in V\times \left[V\right]\times B}{inf}J(u,\varphi ,\lambda ,zn*).$$

Calculate $zn+*\in Y*$ such that

$$J(un,\varphi n,\lambda n,zn+*)=\underset{z*\in Y*}{inf}J(un,\varphi n,\lambda n,z*),$$

so that indeed,

$$zn+*=K(un-f).$$

If $\parallel zn+*-zn*\parallel \infty <\epsilon ,$ then stop. Otherwise set $n:=n+$ and go to item 2.

We have obtained numerical results for $K=100$ and

$$f\left(x\right)=..$$

For the optimal solution $u\left(x\right)$ obtained, please see Figure 35.

At this point we present the software in MAT-LAB we have developed to obtain such numerical results.

************************

clear all

global m8 d u v yo e1 K z

m8=100;

d=1/m8;

e1=0.0007;

K=100.0;

for i=1:m8

yo(i,1)=0.0*sin(pi*i*d)/2;

z(i,1)=0;

end;

for i=1:4*m8+3

xo(i,1)=0.3;

x1(i,1)=0.3;

end;

k1=1;

b14=1.0;

while $(b14>10-)$ && $(k<11)$

k1

k1=k1+1;

k=1;

b12=1.0;

while $(b12>10-)$ && $(k<16)$

k

k=k+1;

X=fminunc(’funMarch24PhaseTC’,xo);

b12=max(abs(X-xo))

xo=X;

u(m8/2,1)

end;

b14=max(abs(x1-xo));

z=K*(u-yo);

x1=xo;

u(m8/2,1)

end;

for i=1:m8

x(i,1)=i*d;

end;

plot(x,u)

*****************************************

With the auxiliary function "funMarch24PhaseTC"

************************

function S=funMarch24PhaseTC(x)

global m8 d u v L yo e1 K z

for i=1:m8

u(i,1)=x(i,1);

v(i,1)=x(i+m8,1);

v1(i,1)=x(i+2*m8,1);

v2(i,1)=x(i+3*m8,1);

end;

L1=(sin(x(4*m8+1,1))+1)/2;

L2=min((sin(x(4*m8+2,1))+1)/2,1-L1);

L3=min((sin(x(4*m8+3,1))+1)/2,1-L1-L2);

L4=1-L1-L2-L3;

u(m8,1)=1/2;

v(m8,1)=0.0;

v1(m8,1)=0.0;

v2(m8,1)=0.0;

du(1,1)=u(1,1)/d;

dv(1,1)=v(1,1)/d;

dv1(1,1)=v1(1,1)/d;

dv2(1,1)=v2(1,1)/d;

for i=2:m8

du(i,1)=(u(i,1)-u(i-1,1))/d;

dv(i,1)=(v(i,1)-v(i-1,1))/d;

dv1(i,1)=(v1(i,1)-v1(i-1,1))/d;

dv2(i,1)=(v2(i,1)-v2(i-1,1))/d;

end;

d2u(1,1)=$(-*u(,)+u(,))/d;$

for i=2:m8-1

d2u(i,1)=$(u(i+,)-*u(i,)+u(i-,))/d;$

end;

S=0;

for i=1:m8

S=S+$L*((du(i,)+L*dv(i,)+L*dv(i,)+L*dv(i,)-dv(i,))-)/;$

S=S+$L*((du(i,)+L*dv(i,)+L*dv(i,)+L*dv(i,)-dv(i,))-)/;$

S=S+$L*((du(i,)+L*dv(i,)+L*dv(i,)+L*dv(i,)-dv(i,))-)/;$

S=S+$L*((du(i,)+L*dv(i,)+L*dv(i,)+L*dv(i,))-)/;$

S=S+$(u(i,)-yo(i,))/;$

S=S+$K*(u(i,)-yo(i,))/-z(i,)*(u(i,)-yo(i,));$

end;

for i=1:m8-1

S=S+$e*du(i,);$

end;

******************************

47. A dual variational formulation for a non-convex primal one

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular boundary denoted by $\partial \Omega$.

Consider the functional $J:V\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J\left(u\right)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx\hfill \\ & & +{\displaystyle \alpha }\int \Omega (u-\beta )dx-\langle u,f\rangle L.\hfill \end{array}$$

(279)

Here $V=W,\left(\Omega \right),$$\alpha >$, $\beta >$, $\gamma >$, and $f\in L\left(\Omega \right)\equiv Y=Y*.$

Denoting $Y=Y*=L(\Omega ;\mathbb{R}),$ define $F:Y\to \mathbb{R}$, $F:V\times Y\to \mathbb{R}$ and $F:Y\to \mathbb{R}$ by

$$F(\nabla u)={\displaystyle \gamma }\int \Omega \nabla u·\nabla udx,$$

$$F(u,v)={\displaystyle \alpha }\int \Omega (u-\beta )dx+{\displaystyle K}\int \Omega udx-\langle u,f\rangle L,$$

and

$$F\left(u\right)={\displaystyle K}\int \Omega udx.$$

Define also, $F:Y*\to \mathbb{R}$, $F˜:Y*\times Y*\times Y*\to \mathbb{R}$ and $F:Y*\to \mathbb{R}$, by

$$\begin{array}{ccc}\hfill F*(v*)& =& \underset{v\in Y}{sup}\{\langle v,v*\rangle L-F\left(v\right)\}\hfill \\ & =& {\displaystyle \gamma }\int \Omega |v*|dx,\hfill \end{array}$$

(280)

$$\begin{array}{ccc}\hfill F˜*(v*,v*,z*)& =& \underset{(u,v)\in V\times Y}{sup}\{-\langle \nabla u,v*\rangle L+\langle u,z*\rangle L\hfill \\ & & +\langle v,v*\rangle L-F(u,v)\}\hfill \\ & =& {\displaystyle 12}\int \Omega {\displaystyle (\mathrm{div}v*+z*+f)v*+K}dx+{\displaystyle \alpha }\int \Omega (v*)dx\hfill \\ & & +\beta \int \Omega v*dx,\hfill \end{array}$$

(281)

if $v*\in B*,$ where

$$B*=\{v*\in Y*:\parallel v*\parallel \infty \le K/\}.$$

Moreover,

$$\begin{array}{ccc}\hfill F*(z*)& =& \underset{u\in V}{sup}\{\langle u,z*\rangle L-F\left(u\right)\}\hfill \\ & =& {\displaystyle K}\int \Omega (z*)dx.\hfill \end{array}$$

(282)

At this point we define $J*:Y*\times B*\times Y*\to \mathbb{R}$ by

$$J*(v*,v*,z*)=-F*(v*)-F˜*(v*,v*,z*)+F*(z*).$$

Assume $(v^*,v^*,z^*)\in Y*\times B*\times Y*$ is such that

$$\delta J*(v^*,v^*,z^*)=\mathbf{0}.$$

Observe that

$$\begin{array}{ccc}\hfill J*(v^*,v^*,z^*)& =& -F*(v*)-F˜*(v*,v*,z*)+F*(z*)\hfill \\ & \le & -\langle \nabla u,v^*\rangle L+F(\nabla u)\hfill \\ & & +\langle \nabla u,v^*\rangle L+\langle u,v^*\rangle L+{\displaystyle K}\int \Omega udx\hfill \\ & & -{\displaystyle \alpha }\int \Omega v^*dx-\beta \int \Omega v^*dx\hfill \\ & & -\langle u,f\rangle L-\langle u,z^*\rangle L+{\displaystyle K}\int \Omega (z^*)dx\hfill \\ & \le & F(\nabla u)+\underset{v*\in Y*}{sup}\left\{\langle u,v*\rangle L-{\displaystyle \alpha }\int \Omega v*dx-\beta \int \Omega v*dx\right\}\hfill \\ & & -\langle u,f\rangle L+{\displaystyle K}\int \Omega udx-\langle u,z^*\rangle L+{\displaystyle K}\int \Omega (z^*)dx\hfill \\ & =& F(\nabla u)+{\displaystyle \alpha }\int \Omega (u-\beta )dx-\langle u,f\rangle L\hfill \\ & & +{\displaystyle K}\int \Omega \left(u-{\displaystyle z^*K}\right)dx\hfill \\ & =& J\left(u\right)+{\displaystyle K}\int \Omega \left(u-{\displaystyle z^*K}\right)dx,\forall u\in V.\hfill \end{array}$$

(283)

Define now $u\in V$ by

$$u={\displaystyle z^*K}.$$

From this and (283) we have

$$J*(v^*,v^*,z^*)\le \underset{u\in V}{inf}\left\{J\left(u\right)+{\displaystyle K}\int \Omega \left(u-u\right)dx\right\}.$$

Furthermore, from the variation of $J$ in $v$ we obtain

$$-{\displaystyle v^*\gamma }+\nabla \left({\displaystyle \mathrm{div}v^*+z^*+fv^*+K}\right)=\mathbf{0},$$

so that

$$v^*=\gamma \nabla \left({\displaystyle \mathrm{div}v^*+z^*+fv^*+K}\right).$$

From the variation of $J$ in $z$, we get

$${\displaystyle z^*K}-\left({\displaystyle \mathrm{div}v^*+z^*+fv^*+K}\right)=\mathbf{0}$$

so that

$$u={\displaystyle z^*K}=\left({\displaystyle \mathrm{div}v^*+z^*+fv^*+K}\right).$$

From the variation of $J$ in $v$, we obtain

$${\displaystyle v^*\alpha }-\left({\displaystyle \mathrm{div}v^*+z^*+fv^*+K}\right)+\beta =\mathbf{0}$$

so that

$$v^*=\alpha (u02-\beta ).$$

Joining the pieces, we have also

$$v^=\gamma \nabla u,$$

$$z^=Ku,$$

so that from this and

$$\mathrm{div}v^*+z^*+f=(v^*+K)u,$$

we obtain

$$\gamma \nabla u+Ku+f=\alpha (u02-\beta )u+Ku,$$

so that

$$-\gamma \nabla u+\alpha (u02-\beta )u-f=\mathbf{0},$$

that is,

$$\delta J\left(u\right)=\mathbf{0}.$$

Finally, from the Legendre transform proprieties, we also obtain

$$F*(v^*)=\langle \nabla u,v^*\rangle L-F(\nabla u),$$

$$\begin{array}{ccc}\hfill F˜(v^,v^*,z^*)& =& -\langle \nabla u,v^*\rangle L+\langle u,z^*\rangle L\hfill \\ & & +\langle ,v^*\rangle L-F(u,)\hfill \end{array}$$

(284)

and

$$F*(z*)=\langle u,z^*\rangle L-F\left(u\right).$$

Therefore

$$\begin{array}{ccc}\hfill J*(v^*,v^*,z^*)& =& -F*(v^*)-F˜*(v^*,v^*,z^*)+F*(z^*)\hfill \\ & =& F(\nabla u)+F(u,)-F\left(u\right)\hfill \\ & =& J\left(u\right).\hfill \end{array}$$

(285)

Observe now that from $\delta J\left(u\right)=\mathbf{0}$, for $K>$ sufficiently large, we have

$$J\left(u\right)=\underset{u\in V}{inf}\left\{J\left(u\right)+{\displaystyle K}\int \Omega (u-u)dx\right\}.$$

Joining the pieces we have got

$$\begin{array}{ccc}\hfill J\left(u\right)& =& \underset{u\in V}{inf}\left\{J\left(u\right)+{\displaystyle K}\int \Omega (u-u)dx\right\}\hfill \\ & =& J*(v^*,v^*,z^*).\hfill \end{array}$$

(286)

We have obtained numerical results for the case A, where $\gamma =.$, $\alpha =.$, $\beta =.$, $f\left(x\right)=10.$ and $K=120.$

For the optimal solution $u\left(x\right)$, where

$$u\left(x\right)={\displaystyle (v*)\prime +z*+fv*+K},$$

please see figure (Figure 36).

Here we present the software in MATLAB through which we have obtained such results.

*************************************

clear all

global m8 d yo z1 K e1 dv1 dv2 v3 v4 v1 v2 A A3 B L u

m8=100;

d=1/m8;

A3=0.1;

A=3.0;

B=5.0;

K=120;

e1=0.0007;

for i=1:m8

yo(i,1)=10.0;

z1(i,1)=0.0;

end;

L=1/2;

for i=1:2*m8

xo(i,1)=3.0;

end;

for k1=1:30

k1

k=1;

b12=1.0;

while $(b12>10-)$ && $(k<15)$

k

k=k+1;

X=fminunc(’funMarch24LGA7’,xo);

b12=max(abs(X-xo))

xo=X;

u(m8/2,1)

end;

for i=1:m8-1

z1(i,1)=K*(dv1(i,1)+z1(i,1)+yo(i,1))/(2*v2(i,1)+K);

end;

end;

for i=1:m8

x(i,1)=i*d;

end;

plot(x,u);

*****************************

With the auxiliary function "funMarch24LGA7"

**********************************

function S=funMarch24LGA7(x)

global m8 d yo z1 z2 K e1 dv1 dv2 v3 v4 v1 v2 A A3 B L u

for i=1:m8

v1(i,1)=x(i,1);

v2(i,1)=x(i+m8,1);

end;

for i=1:m8-1

dv1(i,1)=(v1(i+1,1)-v1(i,1))/d;

end;

S=0;

for i=1:m8-1

S=S+$v(i,)//A+/*(dv(i,)+z(i,)+yo(i,))/(*v(i,)+K);$

S=S+$v(i,)*B+v(i,)//A;$

end;

for i=1:m8-1

u(i,1)=(dv1(i,1)+z1(i,1)+yo(i,1))/(2*v2(i,1)+K);

end;

u(m8,1)=0;

***********************************

48. A convex dual variational formulation for a relaxed non-convex primal one

Let $\Omega =[,]\subset \mathbb{R}$ and consider a functional $J:V\to \mathbb{R}$ where

$$J\left(u\right)={\displaystyle 12}\int \Omega ((u\prime )-)dx+{\displaystyle 12}\int \Omega (u-f)dx,$$

where

$$V=\{u\in W,\left(\Omega \right):u\left(\right)=\mathrm{and}u\left(\right)=/\}.$$

Denoting $V=W,\left(\Omega \right)$, we define $J:V\times V\times [,]\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J(u,\varphi ,\lambda )& =& {\displaystyle \lambda }\int \Omega ((u\prime -(-\lambda )\varphi \prime )-)dx\hfill \\ & & +{\displaystyle (-\lambda )}\int \Omega ((u\prime +\lambda \varphi \prime )-)dx+{\displaystyle 12}\int \Omega (u-f)dx.\hfill \end{array}$$

(287)

Observe that

$$\begin{array}{ccc}\hfill J(u,\varphi ,\lambda )& =& -\langle (u\prime -(-\lambda )\varphi \prime )-,v*\rangle L+{\displaystyle \lambda }\int \Omega ((u\prime -(-\lambda )\varphi \prime )-)dx\hfill \\ & & -\langle (u\prime +\lambda \varphi \prime )-,v*\rangle L+{\displaystyle (-\lambda )}\int \Omega ((u\prime +\lambda \varphi \prime )-)dx\hfill \\ & & +\langle (u\prime -(-\lambda )\varphi \prime )-,v*\rangle L-\langle u\prime -(-\lambda )\varphi \prime ,v*\rangle L\hfill \\ & & +\langle (u\prime +\lambda \varphi \prime )-,v*\rangle L-\langle u\prime +\lambda \varphi \prime ,v*\rangle L\hfill \\ & & +\langle u\prime -(-\lambda )\varphi \prime ,v*\rangle L+\langle u\prime +\lambda \varphi \prime ,v*\rangle L\hfill \\ & & +{\displaystyle 12}\int \Omega (u-f)dx.\hfill \end{array}$$

(288)

Therefore

$$\begin{array}{ccc}\hfill J(u,\varphi ,\lambda )& \ge & \underset{v\in Y}{inf}\left\{-\langle v,v*\rangle L+{\displaystyle \lambda }\int \Omega \left(v\right)dx\right\}\hfill \\ & & +\underset{v\in Y}{inf}\left\{-\langle v,v*\rangle L+{\displaystyle (-\lambda )}\int \Omega \left(v\right)dx\right\}\hfill \\ & & +\underset{v˜\in Y}{inf}\{-\langle v˜,v*\rangle L+\langle v˜32-,v*\rangle L\hfill \\ & & +\underset{v˜\in Y}{inf}\{-\langle v˜,v*\rangle L+\langle v˜42-,v*\rangle L\hfill \\ & & +\underset{(u,\varphi )\in V\times V}{inf}\left\{-\langle u-(-\lambda )\varphi ,(v*)\prime \rangle L-\langle u+\lambda \varphi ,(v*)\prime \rangle L\right.\hfill \\ & & \left.+{\displaystyle 12}\int \Omega (u-f)dx+v*\left(\right)u\left(\right)+v*\left(\right)u\left(\right)\right\}\hfill \\ & =& -{\displaystyle \lambda }\int \Omega (v*)dx-{\displaystyle (-\lambda )}\int \Omega (v*)dx\hfill \\ & & -\int \Omega v*dx-\int \Omega v*dx\hfill \\ & & -\int \Omega {\displaystyle (v*)v*}dx-\int \Omega {\displaystyle (v*)v*}dx\hfill \\ & & -{\displaystyle 12}\int \Omega ((v*)\prime +\lambda f)dx-{\displaystyle 12}\int \Omega ((v*)\prime +(-\lambda )f)dx\hfill \\ & \equiv & J*(v*,v*,v*,v*,\lambda ),\hfill \end{array}$$

(289)

$\forall (u,\varphi ,\lambda )\in V\times V\times [,],\forall (v*,v*,v*,v*)\in [Y*]\times B*,$ where

$$B*=\{(v*,v*)\in Y*\times Y*:v*>\mathrm{and}v*>,\mathrm{in}\Omega \},$$

and

$$\begin{array}{ccc}\hfill J*(v*,v*,v*,v*,\lambda )& =& -{\displaystyle \lambda }\int \Omega (v*)dx-{\displaystyle (-\lambda )}\int \Omega (v*)dx\hfill \\ & & -\int \Omega v*dx-\int \Omega v*dx\hfill \\ & & -\int \Omega {\displaystyle (v*)v*}dx-\int \Omega {\displaystyle (v*)v*}dx\hfill \\ & & -{\displaystyle 12}\int \Omega ((v*)\prime +\lambda f)dx-{\displaystyle 12}\int \Omega ((v*)\prime +(-\lambda )f)dx.\hfill \end{array}$$

(290)

From such results, we may infer that

$$\underset{(u,\varphi ,\lambda )\in V\times V\times [,]}{inf}J(u,\varphi ,\lambda )\ge \underset{\lambda \in [,]}{inf}\left\{\underset{(v*,v*,v*,v*)\in [Y*]\times B*}{sup}J*(v*,v*,v*,v*,\lambda )\right\}.$$

We have developed numerical results for the cases $f\left(x\right)=sin\left(\pi x\right)/$ and $f\left(x\right)=$

For the corresponding optimal solution $u\left(x\right)$ for the case $f\left(x\right)=sin\left(\pi x\right)/$, please see Figure 37.

For the corresponding optimal solution $u\left(x\right)$ for the case $f\left(x\right)=$, please see Figure 38.

Here we present the software in MATLAB through which we have obtained such numerical results.

**************************

clear all

global m8 d yo u L v1 v2 v3 v4 dv1 dv2 K dz1 z1 e1

m8=100;

d=1/m8;

K=1.0;

e1=0.0007;

L=1/2;

for i=1:m8

yo(i,1)=0.0*sin(pi*i*d)/2;

end;

for i=1:4*m8

xo(i,1)=0.8;

end;

x1(1,1)=1/2;

for k1=1:12

k1

k=1;

b12=1.0;

while $(b12>10-)$&& $(k<15)$

k

k=k+1;

X=fminunc(’funMarch24A18’,xo);

b12=max(abs(X-xo))

u(m8/2,1)

xo=X;

end;

X1=fminunc(’funMarch24A19’,x1);

x1=X1;

u(m8/2,1)

end;

for i=1:m8

x(i,1)=i*d;

end;

plot(x,u);

************************

With the auxiliary functions "funMarch24A18" and "funMarch24A19":

*************************

function S=funMarch24A18(x)

global m8 d yo u e1 v1 v2 v3 v4 dv1 dv2 L

for i=1:m8

v1(i,1)=x(i,1);

v2(i,1)=x(i+m8,1);

v3(i,1)=x(i+2*m8,1);

v4(i,1)=x(i+3*m8,1);

end;

for i=1:m8-1

dv1(i,1)=(v1(i+1,1)-v1(i,1))/d;

dv2(i,1)=(v2(i+1,1)-v2(i,1))/d;

end;

S=0;

for i=1:m8-1

S=S+$\left(v(i,)\right)/(*v(i,))/+v(i,)//(L+e)+v(i,)+(dv(i,)+L*yo(i,))/+v(i,)//(L+e);$

S=S+$\left(v(i,)\right)/(*v(i,))/+v(i,)//((-L)+e)+v(i,)$;

S=S+$(dv(i,)+(-L)*yo(i,))/+v(i,)//((-L)+e);$

end;

S=S-v1(m8,1)/2/d-v2(m8,1)/2/d;

for i=1:m8-1

u(i,1)=L*(dv1(i,1)+L*yo(i,1))+(1-L)*(dv2(i,1)+(1-L)*yo(i,1));

end;

u(m8,1)=1/2;

****************************

*******************************

function S1=funMarch24A19(x)

global m8 d yo e1 v1 v2 v3 v4 dv1 dv2 L u

L=(sin(x(1,1))+1)/2;

S=0;

for i=1:m8-1

S=S+$\left(v(i,)\right)/(*v(i,))/+v(i,)//(L+e)+v(i,)+(dv(i,)+L*yo(i,))/+v(i,)//(L+e);$

S=S+$\left(v(i,)\right)/(*v(i,))/+v(i,)//((-L)+e)+v(i,)$

S=S+ $(dv(i,)+(-L)*yo(i,))/+v(i,)//((-L)+e);$

end;

S=S-v1(m8,1)/2/d-v2(m8,1)/2/d;

S1=-S;

***************************

49. A dual variational formulation for the shape optimization of a beam model

Let $\Omega \subset [,]\subset \mathbb{R}$ be the horizontal axis of a straight beam with a variable thickness $H\left(x\right)$.

Consider the problem of minimizing a relaxed functional $J:V\times [,]\times B\to \mathbb{R}$, where

$$\begin{array}{ccc}\hfill J(w,\lambda ,L,L)& =& {\displaystyle \lambda E}\int \Omega {\displaystyle b12}(H\left(L\right)-(-\lambda )H\left(L\right))wxxdx\hfill \\ & & +{\displaystyle (-\lambda )E}\int \Omega {\displaystyle b12}(H\left(L\right)+\lambda H\left(L\right))wxxdx,\hfill \end{array}$$

(291)

subject to

$$\begin{array}{ccc}& & \left(\lambda E{\displaystyle b12}(H\left(L\right)-(-\lambda )H\left(L\right))wxx\right)xx\hfill \\ & & +\left((-\lambda )E{\displaystyle b12}(H\left(L\right)+\lambda H\left(L\right))wxx\right)xx-P\hfill \\ & & =,\mathrm{in}\Omega .\hfill \end{array}$$

(292)

Here

$$H\left(x\right)=L\left(x\right)h,$$

$$H\left(x\right)=L\left(x\right)h,$$

$h=.m$, $b=.15m$, $E=107\mathrm{Pa}$, $P=400N$.

Also, for a simply supported beam,

$$V=\{w\in W,\left(\Omega \right):w\left(\right)=wxx\left(\right)=w\left(\right)=wxx\left(\right)=\},$$

$$\begin{array}{ccc}\hfill B& =& \left\{(L,L):\Omega \to \mathbb{R}\mathrm{measurable}:.\le L\le ,\right.\hfill \\ & & \left.-.\le L\le .,\mathrm{in}\Omega ,\int \Omega L\left(x\right)dx=.61\mathrm{and}\int \Omega L\left(x\right)dx=\right\}.\hfill \end{array}$$

(293)

Moreover, we define $Y=Y*=L\left(\Omega \right),$ and

$$\begin{array}{ccc}\hfill A& =& \left\{(w,\lambda ,L,L)\in V\times [,]\times B:\right.\hfill \\ & & \left(\lambda E{\displaystyle b12}(H\left(L\right)-(-\lambda )H\left(L\right))wxx\right)xx\hfill \\ & & +\left((-\lambda )E{\displaystyle b12}(H\left(L\right)+\lambda H\left(L\right))wxx\right)xx-P\hfill \\ & & =,\left.\mathrm{in}\Omega \right\}.\hfill \end{array}$$

(294)

Observe that

$$\begin{array}{ccc}& & \underset{(w,\lambda ,L,L)\in A}{inf}J(w,\lambda ,L,L)\hfill \\ & =& \underset{(\lambda ,L,L)\in [,]\times B}{inf}\left\{\underset{w^\in V}{sup}\left\{\underset{w\in V}{inf}\left\{J(w,\lambda ,L,L)\right.\right.\right.\hfill \\ & & -〈w^,\left(\lambda E{\displaystyle b12}(H\left(L\right)-(-\lambda )H\left(L\right))wxx\right)xx\hfill \\ & & \left.\left.\left.+\left((-\lambda )E{\displaystyle b12}(H\left(L\right)+\lambda H\left(L\right))wxx\right)xx-P〉L\right\}\right\}\right\}\hfill \\ & =& \underset{(\lambda ,L,L)\in [,]\times B}{inf}\left\{\underset{w^\in V}{sup}\left\{\underset{w\in V}{inf}\left\{{\displaystyle \lambda E}\int \Omega {\displaystyle b12}(H\left(L\right)-(-\lambda )H\left(L\right))wxxdx\right.\right.\right.\hfill \\ & & +{\displaystyle (-\lambda )E}\int \Omega {\displaystyle b12}(H\left(L\right)+\lambda H\left(L\right))wxxdx\hfill \\ & & -〈w^,\left(\lambda E{\displaystyle b12}(H\left(L\right)-(-\lambda )H\left(L\right))wxx\right)xx\hfill \\ & & \left.\left.\left.+\left((-\lambda )E{\displaystyle b12}(H\left(L\right)+\lambda H\left(L\right))wxx\right)xx-P〉L\right\}\right\}\right\}\hfill \\ & =& \underset{(\lambda ,L,L)\in [,]\times B}{inf}\left\{\underset{w^\in V}{sup}\left\{-{\displaystyle \lambda E}\int \Omega {\displaystyle b12}(H\left(L\right)-(-\lambda )H\left(L\right))w^xxdx\right.\right.\hfill \\ & & \left.\left.-{\displaystyle (-\lambda )E}\int \Omega {\displaystyle b12}(H\left(L\right)+\lambda H\left(L\right))w^xxdx+\langle w^,P\rangle L\right\}\right\}\hfill \\ & =& \underset{(\lambda ,L,L)\in [,]\times B}{inf}\left\{\underset{(M,M)\in C*}{inf}\left\{{\displaystyle \lambda Eb/12}\int \Omega {\displaystyle \left(M\right)(H\left(L\right)-(-\lambda )H\left(L\right))}dx\right.\right.\hfill \\ & & \left.\left.+{\displaystyle (-\lambda )Eb/12}\int \Omega {\displaystyle \left(M\right)(H\left(L\right)+\lambda H\left(L\right))}dx\right\}\right\},\hfill \end{array}$$

(295)

where

$$C*=\{(M,M)\in Y*\times Y*:\left(M\right)xx+\left(M\right)xx+P=,\mathrm{in}\Omega \}.$$

We have obtained numerical results through the following algorithm. It is worth highlighting the convergence criterion in this software slightly differs from the one in the algorithm.

Set $n=$, $\epsilon =10-$ and $\left(L\right)n\equiv /$, $\left(L\right)n\equiv .$, $\lambda n=/$.

Calculate $wn\in V$ such that

$$\begin{array}{ccc}& & \left(\lambda nE{\displaystyle b12}(H\left(\left(L\right)n\right)-(-\lambda )H\left(\left(L\right)n\right))\left(wn\right)xx\right)xx\hfill \\ & & +\left((-\lambda n)E{\displaystyle b12}(H\left(\left(L\right)n\right)+\lambda H\left(\left(L\right)n\right))\left(wn\right)xx\right)xx-P\hfill \\ & & =,\mathrm{in}\Omega ,\hfill \end{array}$$

(296)

Calculate $\lambda n+\in [,]$ such that

$$J(wn,\lambda n+,\left(L\right)n,\left(L\right)n)=\underset{\lambda \in [,]}{inf}J((wn,\lambda ,\left(L\right)n,\left(L\right)n).$$

Calculate $(\left(L\right)n+,\left(L\right)n+)\in B$ such that

$$J*(\left(M\right)n,\left(M\right)n,\lambda n+,\left(L\right)n+,\left(L\right)n+)=\underset{(L,L)\in B}{inf}J*(\left(M\right)n,\left(M\right)n,\lambda n+,L,L),$$

where

$$\left(M\right)n=-\lambda n+E{\displaystyle b12}(H\left(\left(L\right)n\right)-(-\lambda n+)\left(L\right)n)\left(wn\right)xx,$$

$$\left(M\right)n=-(-\lambda n+)E{\displaystyle b12}(H\left(\left(L\right)n\right)+\lambda n+\left(L\right)n)\left(wn\right)xx,$$

and

$$\begin{array}{ccc}\hfill J*(M,M)& =& {\displaystyle \lambda Eb/12}\int \Omega {\displaystyle \left(M\right)(H\left(L\right)-(-\lambda )H\left(L\right))}dx\hfill \\ & & +{\displaystyle (-\lambda )Eb/12}\int \Omega {\displaystyle \left(M\right)(H\left(L\right)+\lambda H\left(L\right))}dx.\hfill \end{array}$$

(297)

If

$$\parallel (\left(L\right)n+,\left(L\right)n+)-(\left(L\right)n,\left(L\right)n\parallel \infty <\epsilon ,$$

then stop, otherwise $n:=n+$ and go to item 2.

We have obtained numerical results for a case A with the constant values above specified.

For the optimal solution $L\left(x\right)$, please see Figure 39.

Here we present the software in MATLAB through which we have obtained such results.

********************************

clear all

global m8 d yo u L1 L2 ho Eo B L H H1 Ho Ho1

m8=100;

d=1/m8;

P=400;

Eo=$107;$

for i=1:m8 yo(i,1)=P; end;

ho=0.20;

B=0.15;

for i=1:m8

L1(i,1)=1/2;

L2(i,1)=0.3;

uo(i,1)=0.1;

Ho(i,1)=L1(i,1)*ho;

Ho1(i,1)=0.1*L2(i,1)*ho;

end;

L=1/2;

for i=1:m8

H(i,1)=L1(i,1)*ho;

H1(i,1)=L2(i,1)*ho;

end;

for i=1:2*m8

xo(i,1)=0.3;

end;

x1(1,1)=1/2;

A=zeros(2*m8,2*m8);

for i=1:m8

A(1,i)=1.0;

A(2,i+m8)=1.0;

end;

b=zeros(2*m8,1);

b(1,1)=m8*0.61;

for i=1:m8

lb(i,1)=0.3;

lb(i+m8,1)=-0.7;

end;

for i=1:m8

ub(i,1)=1;

ub(i+m8,1)=0.7;

end;

i=1;

m12=2;

m50(i)=1/m12;

z(i)=$m50\left(i\right)*(-yo(i,)*d);$

for i=2:m8-1

m12=2-m50(i-1);

m50(i)=1/m12;

z(i)=$m50\left(i\right)*(-yo(i,)*d+z(i-));$

end;

v(m8,1)=0;

for i=1:m8-1

v(m8-i,1)=m50(m8-i)*v(m8-i+1,1)+z(m8-i);

end;

k1=1;

b14=1.0;

while $(b14>10-)$ && $(k<15)$

k1

k1=k1+1;

for i=1:m8

y1(i,1)=$v(i,)/(Eo*L*B/12*(H(i,)-(-L)*H(i,))+Eo*(-L)*B/12*(H(i,)+L*H(i,)));$

end;

i=1;

m12=2;

m60(i)=1/m12;

z1(i)=$m60\left(i\right)*(-y(i,)*d);$

for i=2:m8-1

m12=2-m60(i-1);

m60(i)=1/m12;

z1(i)=$m60\left(i\right)*(-y(i,)*d+z(i-));$

end;

u(m8,1)=0;

for i=1:m8-1

u(m8-i,1)=m60(m8-i)*u(m8-i+1)+z1(m8-i);

end;

k=1;

b12=1.0;

while $(b12>10-)$ && $(k<100)$

k

k=k+1;

X=fmincon(’funMarch2024Beam1’,xo,[],[],A,b,lb,ub);

b12=abs(max(xo-X))

xo=X;

L1(m8/2,1)

end;

Ho=H;

Ho1=H1;

X1=fminunc(’funMarch2024Beam2’,x1);

x1=X1;

b14=max(abs(u-uo))

uo=u;

end;

for i=1:m8

x(i,1)=i*d;

end;

plot(x,L1);

*********************

With the auxiliary function "funMarch2024Beam1"

****************************************

function S1=funMarch2024Beam1(x)

global m8 d yo u L1 L2 ho Eo B L Ho Ho1

for i=1:m8

L1(i,1)=x(i,1);

L2(i,1)=x(i+m8,1);

end;

d2u(1,1)=$(-*u(,)+u(,))/d;$

for i=2:m8-1

d2u(i,1)=$(u(i+,)-*u(i,)+u(i-,))/d;$

end;

for i=1:m8

H(i,1)=L1(i,1)*ho;

H1(i,1)=L2(i,1)*ho;

end;

S=0;

for i=1:m8-1

S=S+$L*(Eo*B/12*(Ho(i,)-(-L)*Ho(i,))*du(i,))/(Eo*B/12*(H(i,)-(-L)*H(i,)));$

S=S+$(-L)*(Eo*B/12*(Ho(i,)+L*Ho(i,))*du(i,))/(Eo*B/12*(H(i,)+L*H(i,)));$

end;

S1=S;

********************************

And the auxiliary function "funMarch2024Beam2"

****************************************

function S=funMarch2024Beam2(x)

global m8 d yo u L1 L2 ho Eo B L Ho Ho1

L=(sin(x(1,1))+1)/2;

d2u(1,1)=$(-*u(,)+u(,))/d;$

for i=2:m8-1

d2u(i,1)=$(u(i+,)-*u(i,)+u(i-,))/d;$

end; for i=1:m8

H(i,1)=L1(i,1)*ho;

H1(i,1)=L2(i,1)*ho;

end;

S=0;

for i=1:m8-1

S=S+$L*Eo*B/12*(H(i,)-(-L)*H(i,))*du(i,);$

S=S +$(-L)*Eo*B/12*(H(i,)+L*H(i,))*du(i,);$

end;

****************************

50. A Dual Variational Formulation for a Relaxed Primal Formulation Related to a Shape Optimization Model in Elasticity

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider the problem of minimizing a relaxed functional $J:V\times [,]\times B\to \mathbb{R}$ where

$$J(u,\lambda ,\lambda ,\lambda )={\displaystyle 12}\int \Omega Hijkl(\lambda ,\lambda \left(x\right),\lambda \left(x\right))eij\left(\mathbf{u}\right)ekl\left(\mathbf{u}\right)dx,$$

subject to

$$\left(Hijkl(\lambda ,\lambda \left(x\right),\lambda \left(x\right))ekl\left(\mathbf{u}\right)\right),j+fi=,\mathrm{in}\Omega ,\forall i\{,,\}.$$

Here for simplicity $V=W,(\Omega ;\mathbb{R}),$$Y=Y*=L(\Omega ;\mathbb{R}),$$Y=Y*=L(\Omega ;\mathbb{R}\times ),$ and $f\in L(\Omega ;\mathbb{R}).$

Also, $\mathbf{u}=(u,u,u)\in V$ denotes the field of displacements resulting from the action of f,

$$\left\{eij\left(\mathbf{u}\right)\right\}=\left\{{\displaystyle 12}(ui,j+uj,i)\right\},\forall i,j\in \{,,\},$$

and $Eb\le E(\lambda ,\lambda ,\lambda )\le Ea$, $Ea\gg Eb>$, where $\lambda \left(x\right)=$ corresponds to the presence of a stronger material with Young modulus $E$ at the point $x\in \Omega$. Moreover, $\lambda \left(x\right)=$ corresponds to the presence of a much weaker material with elasticity model $E$, simulating a void space at the point $x\in \Omega .$ On the other hand, $\lambda$ and $\lambda \left(x\right)$ are a real parameter and a function related to the relaxation process for the minimization of J in $\lambda$.

Furthermore,

$$\begin{array}{ccc}\hfill E(\lambda ,\lambda \left(x\right),\lambda \left(x\right))& =& \lambda [(\lambda \left(x\right)-(-\lambda )\lambda \left(x\right))Ea+(-(\lambda \left(x\right)-(-\lambda )\lambda \left(x\right)))Eb]\hfill \\ & & +(-\lambda )[(\lambda \left(x\right)+\lambda \lambda \left(x\right))Ea+(-(\lambda \left(x\right)+\lambda \lambda \left(x\right)))Eb],\hfill \end{array}$$

(298)

$$Hijkl(\lambda ,\lambda \left(x\right),\lambda \left(x\right))=E(\lambda ,\lambda \left(x\right),\lambda \left(x\right))\mathcal{A}ijkl,$$

where

$$\mathcal{A}ijkl=\lambda ˜\delta ij\delta kl+\mu ˜(\delta ik\delta jl+\delta il\delta jk),$$

$\forall i,j,k,l\in \{,,\}.$

Here $\left\{\delta ij\right\}$ is the Kronecker delta and $\lambda ˜>$, $\mu ˜>$ are appropriate real constants.

At this point we define

$$\begin{array}{ccc}\hfill B& =& \left\{(\lambda ,\lambda ):\Omega \to \mathbb{R}\mathrm{measurable}:\le \lambda \left(x\right)\le ,\right.\hfill \\ & & \left.-.\le \lambda \left(x\right)\le .,\mathrm{in}\Omega ,\int \Omega \lambda \left(x\right)dx=cVol\left(\Omega \right),\int \Omega \lambda \left(x\right)dx=\right\},\hfill \end{array}$$

(299)

and

$$\begin{array}{ccc}\hfill A& =& \{(u,\lambda ,\lambda ,\lambda )\in V\times [,]\times B:\hfill \\ & & \left(Hijkl(\lambda ,\lambda ,\lambda )ekl\left(\mathbf{u}\right)\right),j+fi=,\mathrm{in}\Omega ,\forall i\in \{,,\}\}.\hfill \end{array}$$

(300)

Observe that

$$\begin{array}{ccc}& & \underset{(u,\lambda ,\lambda ,\lambda )\in A}{inf}J(u,\lambda ,\lambda ,\lambda )\hfill \\ & =& \underset{(\lambda ,\lambda ,\lambda )\in [,]\times B}{inf}\left\{\underset{u^\in V}{sup}\left\{\underset{u\in V}{inf}\left\{J(u,\lambda ,\lambda ,\lambda )+\langle u^i,\left(Hijkl(\lambda ,\lambda ,\lambda )ekl\left(\mathbf{u}\right)\right)j+fi\rangle L\right\}\right\}\right\}\hfill \\ & =& \underset{(\lambda ,\lambda ,\lambda )\in [,]\times B}{inf}\left\{\underset{u^\in V}{sup}\left\{\underset{u\in V}{inf}\left\{{\displaystyle 12}\int \Omega Hijkl(\lambda ,\lambda ,\lambda )eij\left(\mathbf{u}\right)ekl\left(\mathbf{u}\right)dx\right.\right.\right.\hfill \\ & & \left.\left.\left.+\langle u^i,\left(Hijkl(\lambda ,\lambda ,\lambda )ekl\left(\mathbf{u}\right)\right)j+fi\rangle L\right\}\right\}\right\}\hfill \\ & =& \underset{(\lambda ,\lambda ,\lambda )\in [,]\times B}{inf}\left\{\underset{u^\in V}{sup}\left\{-{\displaystyle 12}\int \Omega Hijkl(\lambda ,\lambda ,\lambda )eij(\mathbf{u}^)ekl(\mathbf{u}^)dx+\langle u^i,fi\rangle L\right\}\right\}\hfill \\ & =& \underset{(\lambda ,\lambda ,\lambda )\in [,]\times B}{inf}\left\{\underset{\sigma \in C*}{inf}\left\{{\displaystyle 12}\int \Omega Hijkl(\lambda ,\lambda ,\lambda )¯\sigma ij\sigma kldx\right\}\right\},\hfill \end{array}$$

(301)

where

$$\{Hijkl(\lambda ,\lambda ,\lambda )¯\}=\left\{Hijkl(\lambda ,\lambda ,\lambda )\right\}-$$

in an appropriate tensor sense and

$$C*=\{\sigma =\left\{\sigma ij\right\}\in Y*:\sigma ij,j+fi=,\mathrm{in}\Omega ,\forall i\in \{,,\}\}.$$

We have obtained numerical results concerning the optimal shape of a two-dimensional beam though the following algorithm:

Set $n=$, $\epsilon =10-$, $\lambda n=/$, $\left(\lambda \right)n\left(x\right)\equiv /$, $\left(\lambda \right)n\left(x\right)\equiv .$

Calculate $\mathbf{u}n\in V$ such that

$$\left(Hijkl(\lambda n,\left(\lambda \right)n,\left(\lambda \right)n)ekl\left(\mathbf{u}n\right)\right),j+fi=,\mathrm{in}\Omega ,\forall i\in \{,,\}.$$

Calculate $\lambda n+\in [,]$ such that

$$J(\mathbf{u}n,\lambda n+,\left(\lambda \right)n,)=\underset{\lambda \in [,]}{inf}\left\{J(\mathbf{u}n,\lambda ,\left(\lambda \right)n,)\right\}.$$

Calculate $(\left(\lambda \right)n+,\left(\lambda \right)n+)\in B$ such that

$$-J(\mathbf{u}n,\lambda n+,\left(\lambda \right)n+,\left(\lambda \right)n+)=\underset{(\lambda ,\lambda )\in B}{inf}\{-J(\mathbf{u}n,\lambda n+,\lambda ,\lambda )\}.$$

Set $\left(\lambda \right)n+\equiv$.

If $\parallel \left(\lambda \right)n+-\left(\lambda \right)n\parallel \infty \le \epsilon ,$ then stop. Otherwise $n:=n+$ and go to item 2.

We developed numerical results for a two-dimensional beam, in a two-dimensional elasticity context for two cases, namely, case A and case B.

For the case A we consider a two-dimensional beam of dimensions $m\times .m$, clamped at $x=$ and with a vertical load of $P=-42000000\left(\right)500\mathbf{j}$ applied to the point $(x,y)=(,.25)$.

For the case B, we consider a a two-dimensional beam of dimensions $m\times .m$, simply supported at $(x,y)=(,)$ and $(x,y)=(,)$, with a vertical load $P=-42000000500\mathbf{j}$ applied to the point $(x,y)=(/,.)$.

Denoting $\mathbf{u}=(u,v)$, for both cases we define the strain tensor as

$$e\left(\mathbf{u}\right)=(ex\left(\mathbf{u}\right),ey\left(\mathbf{u}\right),exy\left(\mathbf{u}\right))T,$$

where $ex\left(\mathbf{u}\right)=ux,$$ey\left(\mathbf{u}\right)=vy,$ and

$$exy\left(\mathbf{u}\right)={\displaystyle 12}(uy+vx).$$

We also set $Ea=205109Pa$ and $Eb=300Pa$, $\nu =.33$ and $c=.6091$ for both the cases.

Moreover the stress tensor $\sigma$ is given by

$$\sigma =H\left(e\right(\mathbf{u}\left)\right),$$

where

$$H={\displaystyle E(\lambda ,\lambda \left(x\right),\lambda \left(x\right))-\nu }\left\{\begin{array}{ccc}& \nu & \\ \nu & & \\ & & {\displaystyle 12}(-\nu )\end{array}\right\}.$$

(302)

For the optimal shape obtained through $\lambda$ for the case A, please see Figure 40.

For the optimal shape obtained through $\lambda$ for the case B, please see Figure 41.

Here we present the software through which we have obtained such results, in a finite differences context for the case B.

We highlight the convergence criterion in the software is a little different from the one in the algorithm above described.

**************************

clear all

global P m8 d w Ea Eb Lo d1 z1 m9 du1 du2 dv1 dv2 c3 Lo1 L u v

m8=24;

m9=22;

c3=0.95;

d=1.0/m8;

d1=0.50/m9;

Ea=$410*106*500;$

Eb=300;

w=0.30;

P=-42000000*500;

z1=(m8-1)*(m9-1);

A3=zeros(2*z1,2*z1);

for i=1:z1

A3(1,i)=1.0;

A3(2,i+z1)=1.0;

end;

L=1/2;

b=zeros(2*z1,1);

b(1,1)=c3*z1;

for i=1:z1

uo(i,1)=0.0;

uo(i+z1,1)=-0.80;

end;

for i=1:z1

u1(i,1)=1.0;

u1(i+z1,1)=0.80;

end;

for i=1:m9-1

for j=1:m8-1

Lo(i,j)=c3;

Lo1(i,j)=0.1*c3;

end;

end;

for i=1:z1*2

x1(i,1)=c3*z1;

end;

x3(1,1)=1/2;

for i=1:4*m8*m9

xo(i,1)=0.000;

end;

xw=xo;

xv=Lo;

for k2=1:22

c3=0.98*c3;

b(1,1)=c3*z1;

k2

b14=1.0;

k3=0;

while $(b14>10-.)$ && $(k<)$

k3=k3+1;

b12=1.0;

k=0;

while $(b12>10-.)$ && $(k<120)$

k=k+1;

k2

k3

k

X=fminunc(’funbeamMarch24’,xo); xo=X;

b12=max(abs(xw-xo))

xw=X;

end;

X1=fminunc(’funbeamMarch24A1’,x3);

x3=X1;

for i=1:m9-1

for j=1:m8-1

E1=$*L*((Lo(i,j)-(-L)*Lo(i,j))*Ea-(-(Lo(i,j)-(-L)*Lo(i,j)))*Eb);$

E1=E1+$*(-L)*((Lo(i,j)+L*Lo(i,j))*Ea-(-(Lo(i,j)+L*Lo(i,j)))*Eb);$

E2=$*L*(Lo(i,j)-(-L)*Lo(i,j))*Ea*(-(-L))-(-(Lo(i,j)-(-L)*Lo(i,j)))*Eb*(-(-L));$

E2=E2+$*(-L)*((Lo(i,j)+L*Lo(i,j))*Ea*L-(-(Lo(i,j)+L*Lo(i,j)))*Eb*L);$

ex=du1(i,j);

ey=dv2(i,j);

exy=1/2*(dv1(i,j)+du2(i,j));

Sx1=$E*(ex+w*ey)/(-w);$

Sy1=$E*(w*ex+ey)/(-w);$

Sxy1=$E/(*(+w\left)\right)*exy;$

Sx2=$E*(ex+w*ey)/(-w);$

Sy2=$E*(w*ex+ey)/(-w);$

Sxy2=$E/(*(+w\left)\right)*exy;$

dc31(i,j)=-(Sx1*ex+Sy1*ey+2*Sxy1*exy);

dc32(i,j)=-(Sx2*ex+Sy2*ey+2*Sxy2*exy);

end;

end;

for i=1:m9-1

for j=1:m8-1

f(j+(i-1)*(m8-1))=dc31(i,j);

f((m9-1)*(m8-1)+j+(i-1)*(m8-1))=dc32(i,j);

end;

end;

for k1=1:1

k1

X1=linprog(f,[],[],A3,b,uo,u1,x1);

x1=X1;

end;

for i=1:z1

x1(i+z1,1)=0;

end;

for i=1:m9-1

for j=1:m8-1

Lo(i,j)=X1(j+(m8-1)*(i-1));

Lo1(i,j)=X1((m8-1)*(m9-1)+j+(m8-1)*(i-1))*0.0;

end;

end;

b14=max(max(abs(Lo-xv)))

xv=Lo;

colormap(gray); imagesc(-Lo); axis equal; axis tight; axis off;pause(1e-6)

end;

end;

*******************************

With the auxiliary function "funbeamMarch24"

*********************************

function S=funbeamMarch24(x)

global P m8 d w u v Ea Eb Lo d1 m9 du1 du2 dv1 dv2 Lo1 L

for i=1:m9

for j=1:m8

u(i,j)=x(j+(m8)*(i-1));

v(i,j)=x(m8*m9+(i-1)*m8+j);

end;

end;

u(m9-1,1)=0; v(m9-1,1)=0; u(m9-1,m8-1)=0; v(m9-1,m8-1)=0;

for i=1:m9-1

for j=1:m8-1

du1(i,j)=(u(i,j+1)-u(i,j))/d;

du2(i,j)=(u(i+1,j)-u(i,j))/d1;

dv1(i,j)=(v(i,j+1)-v(i,j))/d;

dv2(i,j)=(v(i+1,j)-v(i,j))/d1;

end;

end;

S=0;

for i=1:m9-1

for j=1:m8-1

E1=$L*((Lo(i,j)-(-L)*Lo(i,j))*Ea+(-(Lo(i,j)-(-L)*Lo(i,j)))*Eb);$

E2=$(-L)*((Lo(i,j)+L*Lo(i,j))*Ea+(-(Lo(i,j)+L*Lo(i,j)))*Eb);$

ex=du1(i,j);

ey=dv2(i,j);

exy=1/2*(dv1(i,j)+du2(i,j));

Sx=$(E+E)*(ex+w*ey)/(-w);$

Sy=$(E+E)*(w*ex+ey)/(-w);$

Sxy=$(E+E)/(*(+w\left)\right)*exy;$

S=S+1/2*(Sx*ex+Sy*ey+2*Sxy*exy);

end;

end;

S=S*d*d1-P*v(2,(m8)/3)*d*d1;

*******************************

And the auxiliary function "funbeamMarch24A1"

*********************************

function S1=funbeamMarch24A1(x)

global P m8 d w u v Ea Eb Lo d1 m9 du1 du2 dv1 dv2 L Lo1

L=(sin(x(1,1))+1)/2;

for i=1:m9-1

for j=1:m8-1

du1(i,j)=(u(i,j+1)-u(i,j))/d;

du2(i,j)=(u(i+1,j)-u(i,j))/d1;

dv1(i,j)=(v(i,j+1)-v(i,j))/d;

dv2(i,j)=(v(i+1,j)-v(i,j))/d1;

end;

end;

S=0;

for i=1:m9-1

for j=1:m8-1

E1=$L*((Lo(i,j)-(-L)*Lo(i,j))*Ea+(-(Lo(i,j)-(-L)*Lo(i,j)))*Eb);$

E2=$(-L)*((Lo(i,j)+L*Lo(i,j))*Ea+(-(Lo(i,j)+L*Lo(i,j)))*Eb);$

ex=du1(i,j);

ey=dv2(i,j);

exy=1/2*(dv1(i,j)+du2(i,j));

Sx=$(E+E)*(ex+w*ey)/(-w);$

Sy=$(E+E)*(w*ex+ey)/(-w);$

Sxy=$(E+E)/(*(+w\left)\right)*exy;$

S=S+1/2*(Sx*ex+Sy*ey+2*Sxy*exy);

end;

end;

S1=S;

*********************************

51. An Existence Result for a General Parabolic Non-Linear Equation

Let $\Omega \subset \mathbb{R}m$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider the parabolic non-linear equation

$$\left\{\begin{array}{cc}{\displaystyle \partial u\partial t}=\epsilon \nabla u+g\left(u\right)+\sum j=mgj\left(u\right){\displaystyle \partial u\partial xj}+f,\hfill & \mathrm{in}\Omega \times (,T),\hfill \\ u(x,)=u^,\mathrm{in}\Omega ,\hfill & \\ u=,\mathrm{on}\partial \Omega \times [,T].\hfill \end{array}\right.$$

(303)

Here $\epsilon >$, $f\in L([,T],W,\left(\Omega \right))\cap L\infty (\Omega \times [,T]),$ $u^\in H01\left(\Omega \right)\cap L\infty \left(\Omega \right),$ where t denotes time and $[,T]$ is a time interval.

Also $g:\mathbb{R}\to \mathbb{R}$ and $gj:\mathbb{R}\to \mathbb{R}$ are continuous functions neither necessarily linear nor convex,$\forall j\in \{,\dots ,m\}$.

We assume there exist $K33>$ and $K>$ such that

$$\parallel g\parallel \infty \le {\displaystyle K33m\left(\Omega \right)/},$$

$$\parallel gj\parallel \infty \le {\displaystyle Km},$$

$\forall j\in \{,\dots ,n\}.$

At this point, we recall that fixing $\gamma >$,

$$(Id-\gamma \nabla )-:L\left(\Omega \right)\to H01\left(\Omega \right)$$

is a bounded and linear operator, so that for each $h\in L\left(\Omega \right)$ there exists a unique $u\in H01\left(\Omega \right)$ such that

$$(Id-\gamma \nabla )u=h.$$

In such a case we denote

$$u=(Id-\gamma \nabla )-h,$$

so that

$$\parallel u\parallel ,,\Omega \le \parallel (Id-\gamma \nabla )-\parallel \parallel h\parallel ,,\Omega .$$

Moreover, fixing $N\in \mathbb{N}$ and defining

$$\Delta tN={\displaystyle TN},$$

in a partial finite differences context, discretizing in t consider the approximate equation system

$${\displaystyle un+-un\Delta tN}=\epsilon \nabla un++g\left(un\right)+\sum j=mgj\left(un\right)\left(un\right)xj+fn,\mathrm{in}\Omega ,$$

$\forall n\in \{,,\dots ,N-\}.$

From such a system, for $n=$, we obtain

$$u-u^=\epsilon \nabla \left(u\right)\Delta tN+g(u^)\Delta tN+\sum j=mgj(u^)(u^)xj\Delta tN+f\Delta tN.$$

Hence

$$u=(Id-\epsilon (\nabla )\Delta tN)-\left(u^+g(u^)\Delta tN+\sum j=mgj(u^)(u^)xj\Delta tN+f\Delta tN\right),$$

so that

$$\begin{array}{ccc}& & \parallel u\parallel ,,\Omega \hfill \\ & \le & \parallel (Id-\epsilon (\nabla )\Delta tN)-\parallel \hfill \\ & & \times \left(\parallel u^\parallel ,,\Omega +\parallel g(u^)\parallel ,,\Omega \Delta tN+\sum j=m\parallel gj(u^)(u^)xj\parallel ,,\Omega \Delta tN+\parallel f\parallel ,,\Omega \Delta tN\right).\hfill \end{array}$$

(304)

Observe that there exists $K>$ such that $\parallel f\parallel \infty ,\Omega \times [,T]\le K$ so that

$$\parallel fn\parallel ,,\Omega \le K36,\forall n\in \{,,\dots ,N-\},$$

for some appropriate $K36>.$

From such results and the hypotheses, we may infer that

$$\begin{array}{ccc}& & \parallel u\parallel ,,\Omega \le \parallel (Id-\epsilon (\nabla )\Delta tN)-\parallel (\parallel u^\parallel ,,\Omega +K33\Delta tN+K\parallel u^\parallel ,,\Omega \Delta tN+K36\Delta tN)\hfill \\ & \le & \parallel (Id-\epsilon (\nabla )\Delta tN)-\parallel (\parallel u^\parallel ,,\Omega +K\parallel u^\parallel ,,\Omega \Delta tN+K\Delta tN),\hfill \end{array}$$

(305)

where

$$K=K33+K36,$$

so that

$$\parallel u\parallel ,,\Omega \le \alpha \parallel u^\parallel ,,\Omega +\alpha ,$$

where

$$\alpha =\parallel (Id-\epsilon (\nabla )\Delta tN)-\parallel (+K\Delta tN),$$

and

$$\alpha =\parallel (Id-\epsilon (\nabla )\Delta tN)-\parallel K\Delta tN.$$

In fact, generically we may similarly obtain

$$\parallel un+\parallel ,,\Omega \le \alpha \parallel un\parallel ,,\Omega +\alpha ,$$

$\forall n\in \{,,\dots ,N-\}.$

From such a result, inductively we may obtain

$$\parallel uj\parallel ,,\Omega \le \left(\alpha \right)j\parallel u^\parallel ,,\Omega +\sum k=j-\alpha k\alpha .$$

In particular for $j=N$, we get

$$\begin{array}{ccc}& & \parallel uN\parallel ,,\Omega \hfill \\ & \le & \left(\alpha \right)N\parallel u^\parallel ,,\Omega +\sum k=N-\alpha k\alpha \hfill \\ & =& \left(\alpha \right)N\parallel u^\parallel ,,\Omega +{\displaystyle -\alpha N-\alpha }\alpha \hfill \\ & =& ∥\left(Id-\epsilon (\nabla ){\displaystyle TN}\right)-∥N\left(+K{\displaystyle TN}\right)N\parallel u^\parallel ,,\Omega \hfill \\ & & +{\displaystyle -\alpha N-\alpha }\alpha .\hfill \end{array}$$

(306)

Observe that

$$\begin{array}{ccc}& & ∥\left(Id-\epsilon (\nabla ){\displaystyle TN}\right)-∥N\left(+K{\displaystyle TN}\right)N\hfill \\ & \le & \left(+K{\displaystyle TN}\right)N\hfill \\ & \to & eKT,\mathrm{as}N\to \infty .\hfill \end{array}$$

(307)

Also,

$$\begin{array}{ccc}& & \left|{\displaystyle \alpha -\alpha }\right|\hfill \\ & =& {\displaystyle ∥\left(Id-\epsilon (\nabla )TN\right)-∥KTN\left|-\alpha \right|}\hfill \\ & \le & {\displaystyle KTN|-\alpha |}\hfill \\ & =& {\displaystyle KTN\left|-∥\left(Id-\epsilon (\nabla )TN\right)-∥\left(+KTN\right)\right|}\hfill \\ & =& {\displaystyle KNT\left|-∥\left(Id-\epsilon (\nabla )TN\right)-∥\left(+KTN\right)\right|}\hfill \\ & =& {\displaystyle K\left|NT-\left(∥\left(Id-\epsilon (\nabla )TN\right)-∥-+\right)\left(NT+K\right)\right|}\hfill \\ & =& {\displaystyle K\left|NT-\left(∥\left(Id-\epsilon (\nabla )TN\right)-∥-\right)\left(NT+K\right)-NT-K\right|}\hfill \\ & =& \hfill \\ & =& {\displaystyle K\left|-\left(∥\left(Id-\epsilon (\nabla )TN\right)-∥-\right)\left(NT\right)-K-\left(∥\left(Id-\epsilon (\nabla )TN\right)-∥-\right)K\right|}\hfill \\ & =& \hfill \\ & =& {\displaystyle K\left|-\left(∥Id+\sum j=\infty \left(\epsilon (\nabla )TN\right)j∥-\right)\left(NT\right)-K-\left(∥\left(Id-\epsilon (\nabla )TN\right)-∥-\right)K\right|}\hfill \\ & =& {\displaystyle K\left|K+\left(∥Id+\sum j=\infty \left(\epsilon (\nabla )TN\right)j∥-\right)\left(NT\right)+\left(∥\left(Id-\epsilon (\nabla )TN\right)-∥-\right)K\right|}\hfill \\ & \le & {\displaystyle K\left|K+\left(\parallel Id\parallel -∥\sum j=\infty \left(\epsilon (\nabla )TN\right)j∥-\right)\left(NT\right)+\left(∥\left(Id-\epsilon (\nabla )TN\right)-∥-\right)K\right|}\hfill \\ & \le & {\displaystyle K\left|K+\left(\parallel Id\parallel -\sum j=\infty ∥\epsilon (\nabla )TN∥j-\right)\left(NT\right)+\left(∥\left(Id-\epsilon (\nabla )TN\right)-∥-\right)K\right|}\hfill \\ & \le & {\displaystyle K\left|K-\left(\sum j=\infty ∥\epsilon (\nabla )TN∥j\right)\left(NT\right)+\left(∥\left(Id-\epsilon (\nabla )TN\right)-∥-\right)K\right|}\hfill \\ & \le & {\displaystyle K\left|K-\parallel \epsilon (\nabla )\parallel -\parallel \epsilon \nabla \parallel TN+\left(∥\left(Id-\epsilon (\nabla )TN\right)-∥-\right)K\right|}\hfill \\ & \to & {\displaystyle K|K-\parallel \epsilon (\nabla )\parallel |},\mathrm{as}N\to \infty .\hfill \end{array}$$

(308)

From such results we may infer that

$$\left|{\displaystyle (-\alpha N)\alpha -\alpha }\right|\le (+\alpha N)\left|{\displaystyle \alpha -\alpha }\right|,$$

so that

$$\underset{N\to \infty }{lim\; sup}\left|{\displaystyle (-\alpha N)\alpha -\alpha }\right|\le {\displaystyle (+eKT)K|K-\parallel \epsilon \nabla \parallel |}.$$

From these results, denoting now more generically $un\equiv unN$ joining the pieces, we have got

$$\underset{N\to \infty }{lim\; sup}\parallel uNN\parallel ,,\Omega \le eKT\parallel u^\parallel ,,\Omega +{\displaystyle (+eKT)K|K-\parallel \epsilon \nabla \parallel |}.$$

Consequently, we may infer that there exists $K>$ such that

$$\parallel ujN\parallel ,,\Omega \le K,\forall j\in \{,,\dots ,N\},\forall N\in \mathbb{N}.$$

Define now

$$uN(x,t)=unN\left(x\right)\left(n+-{\displaystyle t\Delta tN}\right)+un+N\left(x\right)\left({\displaystyle t\Delta tN}-n\right),$$

if $t\in [n\Delta tN,(n+)\Delta tN],\forall (x,t)\in \Omega \times [,T].$

Observe that

$$uN(x,t)=unN\left(x\right),\mathrm{if}t=n\Delta tN,\forall n\in \{,,\dots ,N\},$$

and

$$\begin{array}{ccc}\hfill {\displaystyle \partial uN(x,t)\partial t}& =& {\displaystyle un+N-unN\Delta tN}\hfill \\ & =& \epsilon \nabla un+N+g\left(unN\right)+\sum j=mgj\left(unN\right)\left(unN\right)xj+fn,\hfill \end{array}$$

(309)

if $t\in [n\Delta tN,(n+)\Delta tN],\forall (x,t)\in \Omega \times [,T].$

Fix $\phi \in Cc\infty \left(\Omega \right).$

Thus, fixing $t\in [n\Delta tN,(n+)\Delta tN],$ we have

$$\begin{array}{ccc}\hfill \left|〈{\displaystyle \partial uN\partial t},\phi 〉L\right|& \le & \epsilon |\langle \nabla un+N,\nabla \phi \rangle L|+|\langle g\left(unN\right),\phi \rangle L|\hfill \\ & & +\int \Omega \left|\sum j=mgj\left(unN\right)\left(unN\right)xj\phi \right|dx+\left|\langle \phi ,fn\rangle L\right|\hfill \\ & \le & \epsilon \parallel un+N\parallel ,,\Omega \parallel \phi \parallel ,,\Omega +K\parallel unN\parallel ,,\Omega \parallel \phi \parallel ,,\Omega +K\parallel \phi \parallel ,,\Omega \hfill \\ & \le & K\parallel \phi \parallel ,,\Omega ,\forall \phi \in Cc\infty \left(\Omega \right),\hfill \end{array}$$

(310)

for some appropriate $K>.$

Since $\phi \in Cc\infty \left(\Omega \right)$ is arbitrary, we may conclude that

$$∥{\displaystyle \partial uN\partial t}∥H-\left(\Omega \right)\le K,\forall N\in \mathbb{N},$$

uniformly in t on $[,T],$ for some appropriate constant $K>.$

Also, from the definition of $u$ we have that there exists $K>$ such that

$$\parallel uN\parallel ,,\Omega \le K,\forall N\in \mathbb{N}$$

also uniformly in t on $[,T].$

From such results, there exist $u\in L([,T],H01\left(\Omega \right))$ and $v\in L([,T];H-\left(\Omega \right))$ such that

$$uN\rightharpoonup u,\mathrm{weakly}\mathrm{in}L((,T);W,\left(\Omega \right)),$$

and

$${\displaystyle \partial uN\partial t}\rightharpoonup v,\mathrm{weakly}-\mathrm{star}\mathrm{in}L([,T],H-\left(\Omega \right)),$$

so that we may easily obtain

$$v={\displaystyle \partial u\partial t}$$

in a distributional sense.

At this point, we provide more details about this last result.

Fix $t\in (,T)$. Thus, there exists $n\in \{,,\dots ,N-\}$ such that $t\in [n\Delta tN,(n+)\Delta tN]$.

Let $\phi \in Cc\infty (\Omega \times (,T)).$

From this, we may infer that

$$\begin{array}{ccc}& & \int \Omega {\displaystyle \partial uN\partial t}\phi (x,t)dx\hfill \\ & =& \int \Omega {\displaystyle un+N-unN\Delta tN}\phi (x,t)dx\hfill \\ & \le & \epsilon \int \Omega |\nabla un+N·\nabla \phi |dx\hfill \\ & & +\int \Omega \left|g\left(unN\right)\phi (x,t)\right|dx+\int \Omega \left|\sum j=mgj\left(unN\right)\left(unN\right)xj\phi \right|dx\hfill \\ & & +\int \Omega \left|fn\phi \right|dx\hfill \\ & \le & (K\parallel unN\parallel ,,\Omega +K20)\parallel \phi \parallel ,,\Omega \hfill \\ & \le & K\parallel \phi \parallel ,,\Omega ,\hfill \end{array}$$

(311)

for some appropriate constants $K>,K>,K20>.$

Hence,

$$\begin{array}{ccc}& & \int T\int \Omega {\displaystyle \partial uN\partial t}\phi (x,t)dxdx\hfill \\ & \le & K\int \Omega \parallel \phi \parallel ,,\Omega dt\hfill \\ & \le & K19\parallel \phi \parallel ,,\Omega \times (,T),\hfill \end{array}$$

(312)

for some appropriate $K19>$.

Since such a $\phi \in Cc\infty (\Omega \times (,T))$ is arbitrary, we may infer that

$$∥{\displaystyle \partial uN\partial t}∥H-(\Omega \times (,T))\le K15,$$

for $N\in \mathbb{N}$, for some $K15>.$

From such a result and from the Banach-Alaoglu Theorem, there exists $v\in H-(\Omega \times (,T))$ such that, up to a not relabeled subsequence

$${\displaystyle \partial uN\partial t}\rightharpoonup v,\mathrm{weakly}-\mathrm{star}\mathrm{in}H-(\Omega \times (,T)).$$

Therefore,

$$\int T\int \Omega {\displaystyle \partial uN\partial t}\phi dxdt\to \int T\int \Omega v\phi dxdt,$$

as $N\to \infty ,$$\forall \phi \in H01(\Omega \times (,T))$.

On the other hand

$$\parallel uN\parallel ,,\Omega \times (,T)\le K16,$$

$\forall N\in \mathbb{N}$, for some $K16>.$

From this and the Kakutani Theorem, there exists $u\in L(\Omega \times (,T))$ such that, up to a not relabeled subsequence,

$$uN\rightharpoonup u,\mathrm{weakly}\mathrm{in}L(\Omega \times (,T)).$$

Now fix again $\phi \in Cc\infty (\Omega \times (,T)).$

Observe that

$$\begin{array}{ccc}\hfill \int T\int \Omega u\phi tdxdt& =& \underset{N\to \infty }{lim}\int T\int \Omega uN\phi tdxdt\hfill \\ & =& -\underset{N\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uN\partial t}\phi dxdt\hfill \\ & =& -\int T\int \Omega v\phi dxdt,\hfill \end{array}$$

(313)

Since such a $\phi \in Cc\infty (\Omega \times (,T))$ is arbitrary, we may infer that

$$v={\displaystyle \partial u\partial t}$$

in a distributional sense.

Moreover, from such results we may also obtain, again up to a subsequence,

$$\underset{N\to \infty }{lim}\int \Omega {\displaystyle \partial uN\partial t}\phi dx=\int \Omega {\displaystyle \partial u\partial t}\phi dx,$$

$\forall \phi \in H01\left(\Omega \right).$

Observe also that, as a consequence of the Rellich-Kondrashov theorem, through appropriate subsequences, we have

$$uNk\left(t\right)\to u(x,t),\mathrm{strongly}\mathrm{in}L\left(\Omega \right),\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

so that, up to subsequences,

$$uNk\left(t\right)(x,t)\to u(x,t),\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

Here we emphasise the sequence $\left\{Nk\left(t\right)\right\}\subset \mathbb{N}$ may depends on t.

Since g is continuous we have that

$$g\left(uNk\left(t\right)(x,t)\right)\to g\left(u(x,t)\right),\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

Fix $t\in (,T).$

Let $\epsilon >$. From the Egorov Theorem, there exists a closed set F such that $m(\Omega \setminus F)\le \epsilon$ and $k\in \mathbb{N}$ such that if $k>k$, then

$$|g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)|\le \epsilon ,\mathrm{for}\mathrm{almost}\mathrm{all}x\in F.$$

Let $\phi \in Cc\infty \left(\Omega \right).$ Observe now that

$$\begin{array}{ccc}& & \left|\int \Omega (g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right))\phi dx\right|\hfill \\ & \le & \int \Omega |g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |dx\hfill \\ & =& \int F|g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |dx+\int \Omega \setminus F|g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |dx\hfill \\ & \le & \int F\epsilon \parallel \phi \parallel \infty dx+\int \Omega |g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |\chi \Omega \setminus Fdx\hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+(\parallel g\left(uNk\left(t\right)\right)\parallel ,,\Omega +\parallel g\left(u\right)\parallel ,,\Omega )\parallel \phi \parallel ,,\Omega \parallel \chi \Omega \setminus F\parallel ,,\Omega \hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+K21\parallel \phi \parallel ,,\Omega m(\Omega \setminus F)/\hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+K21\parallel \phi \parallel ,,\Omega \epsilon /,\forall k>k,\hfill \end{array}$$

(314)

for some appropriate constant $K21>$ which does not depend on t.

Since such a $\epsilon >$ is arbitrary, we may infer that

$$\int \Omega g\left(uNk\left(t\right)\right)\phi dx\to \int \Omega g\left(u\right)\phi dx,\mathrm{as}k\to \infty ,$$

$\forall \phi \in Cc\infty \left(\Omega \right).$

Similarly, fixing $j\in \{,\dots ,n\},$ since $g$ is continuous we have that

$$gj\left(uNk\left(t\right)(x,t)\right)\to gj\left(u(x,t)\right),\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

Fix again $t\in (,T)$

Let $\epsilon >$ (a new value). From the Egorov Theorem, there exists a closed set $F$ such that $m(\Omega \setminus F)\le \epsilon$ and $k\in \mathbb{N}$ such that if $k>k$, then

$$|gj\left(uNk\left(t\right)(x,t)\right)-gj\left(u(x,t)\right)|\le \epsilon ,\mathrm{for}\mathrm{almost}\mathrm{all}x\in F.$$

Observe now that

$$\begin{array}{ccc}& & \int \Omega |gj\left(uNk\left(t\right)(x,t)\right)-gj\left(u(x,t)\right)|dx\hfill \\ & \le & \int F|gj\left(uNk\left(t\right)(x,t)\right)-gj\left(u(x,t)\right)|dx+\int \Omega \setminus F|gj\left(uNk\left(t\right)(x,t)\right)-gj\left(u(x,t)\right)|dx\hfill \\ & \le & \int F\epsilon dx+\int \Omega |gj\left(uNk\left(t\right)(x,t)\right)-gj\left(u(x,t)\right)|\chi \Omega \setminus Fdx\hfill \\ & \le & \epsilon m\left(\Omega \right)+K12\int \Omega \chi \Omega \setminus Fdx\hfill \\ & \le & \epsilon m\left(\Omega \right)+K12\epsilon ,\forall k>k.\hfill \end{array}$$

(315)

Since such a $\epsilon >$ is arbitrary, we may infer that

$$\int \Omega |gj\left(uNk\left(t\right)\right)-gj\left(u\right)|dx\to ,\mathrm{as}k\to \infty ,$$

$\forall j\in \{,\dots ,n\}.$

Select again $\phi \in Cc\infty \left(\Omega \right).$ Since

$$\parallel gj\left(uNk\left(t\right)\right)-gj\left(u\right)\parallel ,,\Omega \to ,\mathrm{as}k\to \infty$$

and

$$\nabla uNk\left(t\right)\rightharpoonup \nabla u,\mathrm{weakly}\mathrm{in}L(\Omega ;\mathbb{R}m),$$

we obtain,

$$\begin{array}{ccc}& & \left|\int \Omega gj\left(uNk\left(t\right)\right)\left(uNk\left(t\right)\right)xj\phi dx-\int \Omega gj\left(u\right)\left(u\right)xj\phi dx\right|\hfill \\ & \le & \left|\int \Omega gj\left(uNk\left(t\right)\right)\left(uNk\left(t\right)\right)xj\phi dx-\int \Omega gj\left(u\right)\left(uNk\left(t\right)\right)xj\phi dx\right|\hfill \\ & & +\left|\int \Omega gj\left(u\right)\left(uNk\left(t\right)\right)xj\phi dx-\int \Omega gj\left(u\right)\left(u\right)xj\phi dx\right|\hfill \\ & \le & \parallel gj\left(uNk\left(t\right)\right)-gj\left(u\right)\parallel ,,\Omega K\parallel \phi \parallel \infty \hfill \\ & & +\left|\int \Omega gj\left(u\right)\left(uNk\left(t\right)\right)xj\phi dx-\int \Omega gj\left(u\right)\left(u\right)xj\phi dx\right|\hfill \\ & \to & ,\mathrm{as}k\to \infty ,\hfill \end{array}$$

(316)

$\forall j\in \{,\dots ,n\}.$

From such results, we have

$$\begin{array}{ccc}\hfill & =& \underset{k\to \infty }{lim}\left(\int \Omega {\displaystyle \partial uNk\left(t\right)\partial t}\phi dx+\epsilon \int \Omega \nabla uNk\left(t\right)·\nabla \phi dx\right.\hfill \\ & & -\int \Omega g\left(uNk\left(t\right)\right)\phi dx-\sum j=m\int \Omega gj\left(uNk\left(t\right)\right)\left(uNk\left(t\right)\right)xj\phi dx\hfill \\ & & \left.-\int \Omega fNk\left(t\right)\phi dx\right)\hfill \\ & =& \int \Omega {\displaystyle \partial u\partial t}\phi dx+\epsilon \int \Omega \nabla u·\nabla \phi dx\hfill \\ & & -\int \Omega g\left(u\right)\phi dx-\sum j=m\int \Omega gj\left(u\right)\left(u\right)xj\phi dx\hfill \\ & & -\int \Omega f\phi dx.\hfill \end{array}$$

(317)

so that, from this and by the density of $Cc\infty \left(\Omega \right)$ in $H01\left(\Omega \right)$, we have got

$$\begin{array}{ccc}& & \int \Omega {\displaystyle \partial u\partial t}\phi dx+\epsilon \int \Omega \nabla u·\nabla \phi dx\hfill \\ & & -\int \Omega g\left(u\right)\phi dx-\sum j=m\int \Omega gj\left(u\right)\left(u\right)xj\phi dx\hfill \\ & & -\int \Omega f\phi dx=,\forall \phi \in H01\left(\Omega \right),\hfill \end{array}$$

(318)

$\mathrm{a}.\mathrm{e}.\mathrm{on}[,T].$

Observe now that

$$\partial (\Omega \times (,T))=\left(\partial \Omega \times [,T]\right)\cup \left(\partial [,T]\times \Omega ¯\right).$$

Let $\phi \in Cc\infty (\Omega \times (,T)).$

Hence

$$\underset{N\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uN\partial t}\phi dxdt=\int T\int \Omega {\displaystyle \partial u\partial t}\phi dxdt.$$

From this, since $Cc\infty (\Omega \times (,T))$ is dense $L(\Omega \times (,T))$ we may infer that

$$\underset{N\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uN\partial t}\phi dxdt=\int T\int \Omega {\displaystyle \partial u\partial t}\phi dxdt,$$

$\forall \phi \in L(\Omega \times (,T)).$

Let $\phi \in C\infty (\Omega \times [,T])$ such that

$$\phi (x,T)=,\mathrm{in}\Omega .$$

From such results, we may obtain

$$\begin{array}{ccc}& & \underset{N\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uN\partial t}\phi dxdt\hfill \\ & =& \underset{N\to \infty }{lim}\left(-\int T\int \Omega uN{\displaystyle \partial \phi \partial t}dxdt-\int \Omega uN(x,)\phi (x,)dx\right)\hfill \\ & =& -\int T\int \Omega u{\displaystyle \partial \phi \partial t}dxdt-\int \Omega u(x,)\phi (x,)dx.\hfill \end{array}$$

(319)

However, since $uN\rightharpoonup u,\mathrm{weakly}\mathrm{in}L(\Omega \times (,T))$, we obtain

$$\underset{N\to \infty }{lim}\int T\int \Omega uN{\displaystyle \partial \phi \partial t}dxdt=\int T\int \Omega u{\displaystyle \partial \phi \partial t}dxdt.$$

From these last results, we may infer that

$$\begin{array}{ccc}\hfill \int \Omega u^\phi (x,)dx& =& \underset{N\to \infty }{lim}\int \Omega uN(x,)\phi (x,)dx\hfill \\ & =& \int \Omega u(x,)\phi (x,)dx,\hfill \end{array}$$

(320)

so that

$$\int \Omega u^\left(x\right)\phi (x,)dx=\int \Omega u(x,)\phi (x,)dx,$$

$\forall \phi \in C\infty (\Omega \times [,T])$ such that $\phi (x,T)=,\mathrm{in}\Omega .$

Therefore, we may infer that $u(x,)=u^\left(x\right)$ in this specified weak sense.

Similarly, it may be proven that

$$u=,\mathrm{on}\partial \Omega \times [,T],$$

in an appropriate weak sense.

Hence, we have obtained that $u$ is a solution, in a weak sense, of the parabolic non-linear equation in question.

52. An existence result for a general non-linear parabolic equation, a simpler case

Through a discussion with my colleague Prof. Maycon Araújo, we realized the geometric series in Laplace operators to express the inverse operator

$$(Id-\epsilon \Delta tN\nabla )$$

in the previous section may be not well established from a theoretical point of view in the concerning infinite dimensional function space. We will deal in more details with such an issue in a near future research.

Anyway, the present section comprises a new version including some improvements and corrections cocerning the previous one.

We thank the colleague Maycon Araújo for his valuable comments and suggestions.

About the references, the main one is [22]. Other related results may be found in [7,8].

Let $\Omega \subset \mathbb{R}m$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider the parabolic non-linear equation

$$\left\{\begin{array}{cc}{\displaystyle \partial u\partial t}=\epsilon \nabla u+g\left(u\right)+f,\hfill & \mathrm{in}\Omega \times (,T),\hfill \\ u(x,)=u^,\mathrm{in}\Omega ,\hfill & \\ u=,\mathrm{on}\partial \Omega \times [,T].\hfill \end{array}\right.$$

(321)

Here $\epsilon >$, $f\in L,\infty (\Omega \times [,T]),$ $u^\in H01\left(\Omega \right)\cap L\infty \left(\Omega \right),$ where t denotes time and $[,T]$ is a time interval.

Also $g:\mathbb{R}\to \mathbb{R}$ is a $C$ class function neither necessarily linear nor convex.

We assume there exists $K33>$ such that

$$\parallel g\parallel ,\infty \le {\displaystyle K33m\left(\Omega \right)/}.$$

52.1. The main theoretical result

At this point, we recall that fixing $\gamma >$,

$$(Id-\gamma \nabla )-:L\left(\Omega \right)\to H01\left(\Omega \right)$$

is a bounded and linear operator, so that for each $h\in L\left(\Omega \right)$ there exists a unique $u\in H01\left(\Omega \right)$ such that

$$(Id-\gamma \nabla )u=h.$$

In such a case we denote

$$u=(Id-\gamma \nabla )-h,$$

so that

$$\parallel u\parallel ,,\Omega \le \parallel (Id-\gamma \nabla )-\parallel \parallel h\parallel ,,\Omega .$$

In our discussions, we also realized that not necessarily we have

$$\parallel (Id-\gamma \nabla )-\parallel \le .$$

Indeed, for $\Delta tN$ small, depending on the domain geometry, such a norm may be larger than 1.

In order to deal with such an issue, we propose the following new development.

Let $N\in \mathbb{N}$ and define

$$\Delta tN={\displaystyle TN}.$$

Let $f\in W,\left(\Omega \right).$

For $u\in H01\left(\Omega \right)$ such that

$$(Id-\epsilon \Delta tN\nabla )u=f,$$

there exists a real constant $K>=K\left(\Delta tN\right),$ such that

$$\parallel u\parallel ,\le K\parallel f\parallel ,.$$

Observe that

$$\begin{array}{ccc}\hfill \parallel f\parallel ,& =& \parallel u-\epsilon \Delta tN\nabla u\parallel ,\hfill \\ & =& \langle u-\epsilon \Delta tN\nabla u,u-\epsilon \Delta tN\nabla u\rangle L\hfill \\ & & +\langle \nabla (u-\epsilon \Delta tN\nabla u),\nabla (u-\epsilon \Delta tN\nabla u)\rangle L\hfill \\ & =& \langle u,u\rangle L-\epsilon \Delta tN\langle u,\nabla u\rangle L+\epsilon \left(\Delta tN\right)\langle \nabla u,\nabla u\rangle L\hfill \\ & & +\langle \nabla (u-\epsilon \Delta tN\nabla u),\nabla (u-\epsilon \Delta tN\nabla u)\rangle L\hfill \\ & =& \langle u,u\rangle L+\epsilon \Delta tN\langle \nabla u,\nabla u\rangle L+\epsilon \left(\Delta tN\right)\langle \nabla u,\nabla u\rangle L\hfill \\ & & +\langle \nabla (u-\epsilon \Delta tN\nabla u),\nabla (u-\epsilon \Delta tN\nabla u)\rangle L\hfill \\ & \ge & \parallel u\parallel ,\hfill \\ & & +\langle \nabla (u-\epsilon \Delta tN\nabla u),\nabla (u-\epsilon \Delta tN\nabla u)\rangle L.\hfill \end{array}$$

(322)

On the other hand

$$\begin{array}{ccc}\hfill & & \parallel \nabla (u-\epsilon \Delta tN\nabla u)\parallel ,\hfill \\ & \ge & sup\{\langle \nabla (u-\epsilon \Delta tN\nabla u),\nabla \varphi \rangle L:\varphi \in C\infty \left(\Omega \right),\parallel \nabla \varphi \parallel ,\le ,\nabla \varphi ·\mathbf{n}=,\mathrm{on}\partial \Omega \}\hfill \end{array}$$

(323)

Let $\varphi \in W,$ be such that $\parallel \nabla \varphi \parallel ,\ne$ and

$$\nabla \varphi =\parallel \nabla \varphi \parallel ,{\displaystyle \nabla u\parallel \nabla u\parallel ,},$$

and

$$\nabla \varphi ·\mathbf{n}=,\mathrm{on}\partial \Omega .$$

From this and (323), we obtain

$$\begin{array}{ccc}& & \parallel \nabla (u-\epsilon \Delta tN\nabla u)\parallel ,\hfill \\ & \ge & sup\{\langle \nabla (u-\epsilon \Delta tN\langle \nabla u),\nabla \varphi \rangle L:\varphi \in C\infty \left(\Omega \right),\parallel \nabla \varphi \parallel ,\le ,\nabla \varphi ·\mathbf{n}=,\mathrm{on}\partial \Omega \}\hfill \\ & \ge & \langle \nabla (u-\epsilon \Delta tN\langle \nabla u),\nabla \varphi \rangle L{\displaystyle \parallel \nabla \varphi \parallel ,}\hfill \\ & =& \langle (u-\epsilon \Delta tN\langle \nabla u),-\nabla \varphi \rangle L{\displaystyle \parallel \nabla \varphi \parallel ,}\hfill \\ & =& \langle (u-\epsilon \Delta tN\langle \nabla u),-\nabla u\rangle L{\displaystyle \parallel \nabla u\parallel ,}\hfill \\ & =& \langle \nabla u,\nabla u\rangle L{\displaystyle \parallel \nabla u\parallel ,}+\epsilon \Delta tN\langle \nabla u,\nabla u\rangle L{\displaystyle \parallel \nabla u\parallel ,}\hfill \\ & =& \parallel \nabla u\parallel ,+\epsilon \Delta tN\langle \nabla u,\nabla u\rangle L{\displaystyle \parallel \nabla u\parallel ,}\hfill \\ & \ge & \parallel \nabla u\parallel ,.\hfill \end{array}$$

(324)

From such results, we may obtain

$$\begin{array}{ccc}\hfill \parallel f\parallel ,& \ge & \parallel u\parallel ,+\parallel \nabla u\parallel ,\hfill \\ & =& \parallel u\parallel ,,\hfill \end{array}$$

(325)

so that

$$\parallel u\parallel ,\le \parallel f\parallel ,.$$

Remark 24.

There is a simpler path to prove this last result.

Define

$$B=(Id-\epsilon \Delta tN\nabla )-:L\left(\Omega \right)\to H01\left(\Omega \right)\subset L\left(\Omega \right).$$

Let $\epsilon >$. Denoting

$$H^01=\{h\in W,\left(\Omega \right):\nabla h·\mathbf{n}=,\mathrm{on}\partial \Omega \},$$

observe that by density there exists $f^\in H^01$ such that

$$\parallel f-f^\parallel ,<\epsilon .$$

Thus

$$\parallel u-u^\parallel ,\le K\epsilon .$$

Observe also that $B=B*\le Id$ in $L\left(\Omega \right)$ and $-\nabla$ is positive in $H^01\left(\Omega \right)$ and $H01\left(\Omega \right)$.

Thus

$$B*(-\nabla )B\le -\nabla ,\mathrm{in}H^01.$$

Therefore,

$$\begin{array}{ccc}\hfill \langle \nabla u,\nabla u\rangle L& \le & \langle \nabla u^,\nabla u^\rangle L+\left(K\epsilon \right)\hfill \\ & =& \langle \nabla [(Id-\epsilon \Delta tN\nabla )-f^],\nabla [(Id-\epsilon \Delta tN\nabla )-f^]\rangle L+\left(K\epsilon \right)\hfill \\ & =& \langle \nabla [Bf^],\nabla [Bf^]\rangle L+\left(K\epsilon \right)\hfill \\ & =& \langle (-\nabla )[Bf^],[Bf^]\rangle L+\left(K\epsilon \right)\hfill \\ & =& \langle B*(-\nabla )[Bf^],f^\rangle L+\left(K\epsilon \right)\hfill \\ & \le & \langle (-\nabla )f^,f^\rangle L+\left(K\epsilon \right)\hfill \\ & \le & \langle \nabla f^,\nabla f^\rangle L+\left(K\epsilon \right)\hfill \\ & \le & \langle \nabla f,\nabla f\rangle L+(K72+)\epsilon 12.\hfill \end{array}$$

(326)

Thus,

$$\begin{array}{ccc}\hfill \parallel u\parallel ,+\parallel \nabla u\parallel ,& \le & \parallel Bf\parallel ,+\parallel \nabla f\parallel ,+(K72+)\epsilon 12\hfill \\ & \le & \parallel f\parallel ,+\parallel \nabla f\parallel ,+(K72+)\epsilon 12.\hfill \end{array}$$

(327)

Since such an $\epsilon >$ is arbitrary, we may infer that

$$\parallel u\parallel ,+\parallel \nabla u\parallel ,\le \parallel f\parallel ,+\parallel \nabla f\parallel ,,$$

so that

$$\parallel u\parallel ,\le \parallel f\parallel ,.$$

Now we define the norm

$$\parallel (Id-\epsilon \Delta tN\nabla )-\parallel *$$

by

$$\parallel (Id-\epsilon \Delta tN\nabla )-\parallel *=sup\{\parallel (Id-\epsilon \Delta tN\nabla )-f\parallel ,:f\in W,\left(\Omega \right)\mathrm{and}\parallel f\parallel ,\le \}.$$

From the last results

$$\parallel u\parallel ,=\parallel (Id-\epsilon \Delta tN\nabla )-f\parallel ,\le \parallel f\parallel ,,\forall f\in W,\left(\Omega \right),$$

so that

$$\parallel (Id-\epsilon \Delta tN\nabla )-\parallel *\le .$$

Now, fixing again $N\in \mathbb{N}$ and defining again

$$\Delta tN={\displaystyle TN},$$

in a partial finite differences context, discretizing in t consider the approximate equation system

$${\displaystyle un+-un\Delta tN}=\epsilon \nabla un++g\left(un\right)+fn,\mathrm{in}\Omega ,$$

$\forall n\in \{,,\dots ,N-\}.$

From such a system, for $n=$, we obtain

$$u-u^=\epsilon \nabla \left(u\right)\Delta tN+g(u^)\Delta tN+f\Delta tN.$$

Hence

$$u=(Id-\epsilon (\nabla )\Delta tN)-\left(u^+g(u^)\Delta tN+f\Delta tN\right),$$

so that

$$\begin{array}{ccc}& & \parallel u\parallel ,,\Omega \hfill \\ & \le & \parallel (Id-\epsilon (\nabla )\Delta tN)-\parallel *\hfill \\ & & \times \left(\parallel u^\parallel ,,\Omega +\parallel g(u^)\parallel ,,\Omega \Delta tN+\parallel f\parallel ,,\Omega \Delta tN\right).\hfill \end{array}$$

(328)

Observe that there exists $K>$ such that $\parallel f\parallel ,\infty ,\Omega \times [,T]\le K$ so that

$$\parallel fn\parallel ,,\Omega \le K36,\forall n\in \{,,\dots ,N-\},$$

for some appropriate $K36>.$

From such results and the hypotheses, we may infer that

$$\begin{array}{ccc}& & \parallel u\parallel ,,\Omega \le \parallel (Id-\epsilon (\nabla )\Delta tN)-\parallel *(\parallel u^\parallel ,,\Omega +K33\Delta tN+K33\parallel u^\parallel ,,\Omega \Delta tN+K36\Delta tN)\hfill \\ & \le & \parallel (Id-\epsilon (\nabla )\Delta tN)-\parallel *(\parallel u^\parallel ,,\Omega +K\parallel u^\parallel ,,\Omega \Delta tN+K\Delta tN),\hfill \end{array}$$

(329)

where

$$K=K33+K36,$$

and $K=K33,$ so that

$$\parallel u\parallel ,,\Omega \le \alpha \parallel u^\parallel ,,\Omega +\alpha ,$$

where

$$\alpha =\parallel (Id-\epsilon (\nabla )\Delta tN)-\parallel *(+K\Delta tN),$$

and

$$\alpha =\parallel (Id-\epsilon (\nabla )\Delta tN)-\parallel *K\Delta tN.$$

In fact, generically we may similarly obtain

$$\parallel un+\parallel ,,\Omega \le \alpha \parallel un\parallel ,,\Omega +\alpha ,$$

$\forall n\in \{,,\dots ,N-\}.$

From such a result, inductively we may obtain

$$\parallel uj\parallel ,,\Omega \le \left(\alpha \right)j\parallel u^\parallel ,,\Omega +\sum k=j-\alpha k\alpha .$$

In particular for $j=N$, we get

$$\begin{array}{ccc}& & \parallel uN\parallel ,,\Omega \hfill \\ & \le & \left(\alpha \right)N\parallel u^\parallel ,,\Omega +\sum k=N-\alpha k\alpha \hfill \\ & =& \left(\alpha \right)N\parallel u^\parallel ,,\Omega +{\displaystyle -\alpha N-\alpha }\alpha \hfill \\ & =& ∥\left(Id-\epsilon (\nabla ){\displaystyle TN}\right)-∥*N\left(+K{\displaystyle TN}\right)N\parallel u^\parallel ,,\Omega \hfill \\ & & +{\displaystyle -\alpha N-\alpha }\alpha .\hfill \end{array}$$

(330)

Observe that

$$\alpha =\parallel (Id-\epsilon (\nabla )\Delta tN)-\parallel *(+K\Delta tN).$$

Define now

$$\gamma \left(\Delta tN\right)={\displaystyle \Delta tN|-\alpha |}.$$

Observe that

$$\begin{array}{ccc}\hfill |-\alpha |& =& |-\parallel (Id-\epsilon (\nabla )\Delta tN)-\parallel *(+K\Delta tN)|\hfill \\ & =& |-\parallel (Id-\epsilon (\nabla )\Delta tN)-\parallel *-K\parallel (Id-\epsilon (\nabla )\Delta tN)-\parallel *\Delta tN|,\hfill \end{array}$$

(331)

so that

$$\begin{array}{ccc}\hfill \gamma \left(\Delta tN\right)& =& {\displaystyle \Delta tN|-\alpha |}\hfill \\ & =& {\displaystyle \Delta tN\left|\right|-\parallel (Id-\epsilon (\nabla )\Delta tN)-\parallel *-K\parallel (Id-\epsilon (\nabla )\Delta tN)-\parallel *\Delta tN|}\hfill \\ & =& {\displaystyle \Delta tN\left|-\parallel (Id-\epsilon (\nabla )\Delta tN)-\parallel *\Delta tN-K\parallel (Id-\epsilon (\nabla )\Delta tN)-\parallel *\right|\Delta tN}\hfill \\ & =& {\displaystyle \left|-\parallel (Id-\epsilon (\nabla )\Delta tN)-\parallel *\Delta tN-K\parallel (Id-\epsilon (\nabla )\Delta tN)-\parallel *\right|},\hfill \end{array}$$

(332)

Define

$$L=\underset{N\to \infty }{lim\; sup}\left|{\displaystyle -\parallel (Id-\epsilon (\nabla )\Delta tN)-\parallel *\Delta tN}\right|.$$

We have two possibilities:

Either

$$L=+\infty ,$$

or

$$L=K\in \mathbb{R}$$

for some

$$K\ge .$$

Observe that, up to a not relabeled subsequence, $\left\{N\right\}=\left\{Nk\right\},$ we have

$$L=\underset{N\to \infty }{lim}\left|{\displaystyle -\parallel (Id-\epsilon (\nabla )\Delta tN)-\parallel *\Delta tN}\right|.$$

If $L=+\infty$, then we may obtain

$$\underset{N\to \infty }{lim}\gamma \left(\Delta N\right)=.$$

On the other hand, if $L=K$, redefining a larger $K>$ if necessary, such that $K>K,$ we may obtain, again up to a not relabeled subsequence,

$$\underset{N\to \infty }{lim}\gamma \left(\Delta N\right)\le {\displaystyle |K-K|}\in \mathbb{R}.$$

Therefore, in any case, there exists a real constant $C>$ such that

$$\underset{N\to \infty }{lim}\gamma \left(\Delta N\right)\le C.$$

Consequently, from such results we may infer that, up to a not relabeled subsequence,

$$\underset{N\to \infty }{lim}\left|{\displaystyle (-\alpha N)\alpha -\alpha }\right|\le (+eKT)KC.$$

From these results, denoting now more generically $un\equiv unN$ joining the pieces, up to a not relabeled subsequence, we have got

$$\underset{N\to \infty }{lim\; sup}\parallel uNN\parallel ,,\Omega \le eKT\parallel u^\parallel ,,\Omega +(+eKT)KC.$$

Thus, for a not relabeled subsequence $\left\{N\right\}=\left\{Nk\right\},$ we may infer that there exists $K>$ such that

$$\parallel ujN\parallel ,,\Omega \le K,\forall j\in \{,,\dots ,N\},\forall N\in \mathbb{N}.$$

Define now

$$uN(x,t)=unN\left(x\right)\left(n+-{\displaystyle t\Delta tN}\right)+un+N\left(x\right)\left({\displaystyle t\Delta tN}-n\right),$$

if $t\in [n\Delta tN,(n+)\Delta tN],\forall (x,t)\in \Omega \times [,T].$

Observe that

$$uN(x,t)=unN\left(x\right),\mathrm{if}t=n\Delta tN,\forall n\in \{,,\dots ,N\},$$

and

$$\begin{array}{ccc}\hfill {\displaystyle \partial uN(x,t)\partial t}& =& {\displaystyle un+N-unN\Delta tN}\hfill \\ & =& \epsilon \nabla un+N+g\left(unN\right)+fn,\hfill \end{array}$$

(333)

if $t\in [n\Delta tN,(n+)\Delta tN],\forall (x,t)\in \Omega \times [,T].$

Fix $\phi \in Cc\infty \left(\Omega \right).$

Thus, fixing $t\in [n\Delta tN,(n+)\Delta tN],$ we have

$$\begin{array}{ccc}\hfill \left|〈{\displaystyle \partial uN\partial t},\phi 〉L\right|& \le & \epsilon |\langle \nabla un+N,\nabla \phi \rangle L|+|\langle g\left(unN\right),\phi \rangle L|+|\langle \phi ,fn\rangle L|\hfill \\ & \le & \epsilon \parallel un+N\parallel ,,\Omega \parallel \phi \parallel ,,\Omega +K\parallel unN\parallel ,,\Omega \parallel \phi \parallel ,,\Omega +K\parallel \phi \parallel ,,\Omega \hfill \\ & \le & K\parallel \phi \parallel ,,\Omega ,\forall \phi \in Cc\infty \left(\Omega \right),\hfill \end{array}$$

(334)

for some appropriate $K>.$

Since $\phi \in Cc\infty \left(\Omega \right)$ is arbitrary, we may conclude that

$$∥{\displaystyle \partial uN\partial t}∥H-\left(\Omega \right)\le K,\forall N\in \mathbb{N},$$

uniformly in t on $[,T],$ for some appropriate constant $K>.$

Also, from the definition of $u$ we have that there exists $K>$ such that

$$\parallel uN\parallel ,,\Omega \le K,\forall N\in \mathbb{N}$$

also uniformly in t on $[,T].$

From such results, there exist $u\in L([,T],H01\left(\Omega \right))$ and $v\in L([,T];H-\left(\Omega \right))$ such that

$$uN\rightharpoonup u,\mathrm{weakly}\mathrm{in}L((,T);W,\left(\Omega \right)),$$

and

$${\displaystyle \partial uN\partial t}\rightharpoonup v,\mathrm{weakly}-\mathrm{star}\mathrm{in}L([,T],H-\left(\Omega \right)),$$

so that we may easily obtain

$$v={\displaystyle \partial u\partial t}$$

in a distributional sense.

At this point, we provide more details about this last result.

Fix $t\in (,T)$. Thus, there exists $n\in \{,,\dots ,N-\}$ such that $t\in [n\Delta tN,(n+)\Delta tN]$.

Let $\phi \in Cc\infty (\Omega \times (,T)).$

From this, we may infer that

$$\begin{array}{ccc}& & \int \Omega {\displaystyle \partial uN\partial t}\phi (x,t)dx\hfill \\ & =& \int \Omega {\displaystyle un+N-unN\Delta tN}\phi (x,t)dx\hfill \\ & \le & \epsilon \int \Omega |\nabla un+N·\nabla \phi |dx+\int \Omega \left|g\left(unN\right)\phi (x,t)\right|dx\hfill \\ & & +\int \Omega \left|fn\phi \right|dx\hfill \\ & \le & (K\parallel unN\parallel ,,\Omega +K20)\parallel \phi \parallel ,,\Omega \hfill \\ & \le & K29\parallel \phi \parallel ,,\Omega ,\hfill \end{array}$$

(335)

for some appropriate constants $K>,K29>,K20>.$

Hence,

$$\begin{array}{ccc}& & \int T\int \Omega {\displaystyle \partial uN\partial t}\phi (x,t)dxdx\hfill \\ & \le & K29\int \Omega \parallel \phi \parallel ,,\Omega dt\hfill \\ & \le & K19\parallel \phi \parallel ,,\Omega \times (,T),\hfill \end{array}$$

(336)

for some appropriate $K19>$.

Since such a $\phi \in Cc\infty (\Omega \times (,T))$ is arbitrary, we may infer that

$$∥{\displaystyle \partial uN\partial t}∥H-(\Omega \times (,T))\le K15,$$

for $N\in \mathbb{N}$, for some $K15>.$

From such a result and from the Banach-Alaoglu Theorem, there exists $v\in H-(\Omega \times (,T))$ such that, up to a not relabeled subsequence

$${\displaystyle \partial uN\partial t}\rightharpoonup v,\mathrm{weakly}-\mathrm{star}\mathrm{in}H-(\Omega \times (,T)).$$

Therefore,

$$\int T\int \Omega {\displaystyle \partial uN\partial t}\phi dxdt\to \int T\int \Omega v\phi dxdt,$$

as $N\to \infty ,$$\forall \phi \in H01(\Omega \times (,T))$.

On the other hand

$$\parallel uN\parallel ,,\Omega \times (,T)\le K16,$$

$\forall N\in \mathbb{N}$, for some $K16>.$

From this and the Kakutani Theorem, there exists $u\in L(\Omega \times (,T))$ such that, up to a not relabeled subsequence,

$$uN\rightharpoonup u,\mathrm{weakly}\mathrm{in}L(\Omega \times (,T)).$$

Now fix again $\phi \in Cc\infty (\Omega \times (,T)).$

Observe that

$$\begin{array}{ccc}\hfill \int T\int \Omega u\phi tdxdt& =& \underset{N\to \infty }{lim}\int T\int \Omega uN\phi tdxdt\hfill \\ & =& -\underset{N\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uN\partial t}\phi dxdt\hfill \\ & =& -\int T\int \Omega v\phi dxdt,\hfill \end{array}$$

(337)

Since such a $\phi \in Cc\infty (\Omega \times (,T))$ is arbitrary, we may infer that

$$v={\displaystyle \partial u\partial t}$$

in a distributional sense.

Moreover, from such results we may also obtain, again up to a subsequence,

$$\underset{N\to \infty }{lim}\int \Omega {\displaystyle \partial uN\partial t}\phi dx=\int \Omega {\displaystyle \partial u\partial t}\phi dx,$$

$\forall \phi \in H01\left(\Omega \right).$

Observe also that, as a consequence of the Rellich-Kondrashov theorem, through appropriate subsequences, we have

$$uNk\left(t\right)\to u(x,t),\mathrm{strongly}\mathrm{in}L\left(\Omega \right),\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

so that, up to subsequences,

$$uNk\left(t\right)(x,t)\to u(x,t),\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

Here we emphasise the sequence $\left\{Nk\left(t\right)\right\}\subset \mathbb{N}$ may depends on t.

Since g is continuous we have that

$$g\left(uNk\left(t\right)(x,t)\right)\to g\left(u(x,t)\right),\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

Fix $t\in (,T).$

Let $\epsilon >$. From the Egorov Theorem, there exists a closed set F such that $m(\Omega \setminus F)\le \epsilon$ and $k\in \mathbb{N}$ such that if $k>k$, then

$$|g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)|\le \epsilon ,\mathrm{for}\mathrm{almost}\mathrm{all}x\in F.$$

Let $\phi \in Cc\infty \left(\Omega \right).$ Observe now that

$$\begin{array}{ccc}& & \left|\int \Omega (g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right))\phi dx\right|\hfill \\ & \le & \int \Omega |g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |dx\hfill \\ & =& \int F|g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |dx+\int \Omega \setminus F|g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |dx\hfill \\ & \le & \int F\epsilon \parallel \phi \parallel \infty dx+\int \Omega |g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |\chi \Omega \setminus Fdx\hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+(\parallel g\left(uNk\left(t\right)\right)\parallel ,,\Omega +\parallel g\left(u\right)\parallel ,,\Omega )\parallel \phi \parallel ,,\Omega \parallel \chi \Omega \setminus F\parallel ,,\Omega \hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+K21\parallel \phi \parallel ,,\Omega m(\Omega \setminus F)/\hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+K21\parallel \phi \parallel ,,\Omega \epsilon /,\forall k>k,\hfill \end{array}$$

(338)

for some appropriate constant $K21>$ which does not depend on t.

Since such a $\epsilon >$ is arbitrary, we may infer that

$$\int \Omega g\left(uNk\left(t\right)\right)\phi dx\to \int \Omega g\left(u\right)\phi dx,\mathrm{as}k\to \infty ,$$

$\forall \phi \in Cc\infty \left(\Omega \right).$

From such results, we have

$$\begin{array}{ccc}\hfill & =& \underset{k\to \infty }{lim}\left(\int \Omega {\displaystyle \partial uNk\left(t\right)\partial t}\phi dx+\epsilon \int \Omega \nabla uNk\left(t\right)·\nabla \phi dx\right.\hfill \\ & & -\int \Omega g\left(uNk\left(t\right)\right)\phi dx\left.-\int \Omega fNk\left(t\right)\phi dx\right)\hfill \\ & =& \int \Omega {\displaystyle \partial u\partial t}\phi dx+\epsilon \int \Omega \nabla u·\nabla \phi dx\hfill \\ & & -\int \Omega g\left(u\right)\phi dx-\int \Omega f\phi dx.\hfill \end{array}$$

(339)

so that, from this and by the density of $Cc\infty \left(\Omega \right)$ in $H01\left(\Omega \right)$, we have got

$$\begin{array}{ccc}& & \int \Omega {\displaystyle \partial u\partial t}\phi dx+\epsilon \int \Omega \nabla u·\nabla \phi dx\hfill \\ & & -\int \Omega g\left(u\right)\phi dx-\int \Omega f\phi dx=,\forall \phi \in H01\left(\Omega \right),\hfill \end{array}$$

(340)

$\mathrm{a}.\mathrm{e}.\mathrm{on}[,T].$

Observe now that

$$\partial (\Omega \times (,T))=\left(\partial \Omega \times [,T]\right)\cup \left(\partial [,T]\times \Omega ¯\right).$$

Let $\phi \in Cc\infty (\Omega \times (,T)).$

Hence

$$\underset{N\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uN\partial t}\phi dxdt=\int T\int \Omega {\displaystyle \partial u\partial t}\phi dxdt.$$

From this, since $Cc\infty (\Omega \times (,T))$ is dense $L(\Omega \times (,T))$ we may infer that

$$\underset{N\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uN\partial t}\phi dxdt=\int T\int \Omega {\displaystyle \partial u\partial t}\phi dxdt,$$

$\forall \phi \in L(\Omega \times (,T)).$

Let $\phi \in C\infty (\Omega \times [,T])$ such that

$$\phi (x,T)=,\mathrm{in}\Omega .$$

From such results, we may obtain

$$\begin{array}{ccc}& & \underset{N\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uN\partial t}\phi dxdt\hfill \\ & =& \underset{N\to \infty }{lim}\left(-\int T\int \Omega uN{\displaystyle \partial \phi \partial t}dxdt-\int \Omega uN(x,)\phi (x,)dx\right)\hfill \\ & =& -\int T\int \Omega u{\displaystyle \partial \phi \partial t}dxdt-\int \Omega u(x,)\phi (x,)dx.\hfill \end{array}$$

(341)

However, since $uN\rightharpoonup u,\mathrm{weakly}\mathrm{in}L(\Omega \times (,T))$, we obtain

$$\underset{N\to \infty }{lim}\int T\int \Omega uN{\displaystyle \partial \phi \partial t}dxdt=\int T\int \Omega u{\displaystyle \partial \phi \partial t}dxdt.$$

From these last results, we may infer that

$$\begin{array}{ccc}\hfill \int \Omega u^\phi (x,)dx& =& \underset{N\to \infty }{lim}\int \Omega uN(x,)\phi (x,)dx\hfill \\ & =& \int \Omega u(x,)\phi (x,)dx,\hfill \end{array}$$

(342)

so that

$$\int \Omega u^\left(x\right)\phi (x,)dx=\int \Omega u(x,)\phi (x,)dx,$$

$\forall \phi \in C\infty (\Omega \times [,T])$ such that $\phi (x,T)=,\mathrm{in}\Omega .$

Therefore, we may infer that $u(x,)=u^\left(x\right)$ in this specified weak sense.

Similarly, it may be proven that

$$u=,\mathrm{on}\partial \Omega \times [,T],$$

in an appropriate weak sense.

Hence, we have obtained that $u$ is a solution, in a weak sense, of the parabolic non-linear equation in question.

The objective of this section is complete.

53. An existence result for a general hyperbolic non-linear equation

Let $\Omega \subset \mathbb{R}m$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider the hyperbolic non-linear equation

$$\left\{\begin{array}{cc}{\displaystyle \partial u\partial t}=\epsilon \nabla u+g\left(u\right)+f,\hfill & \mathrm{in}\Omega \times (,T),\hfill \\ u(x,)=u^,\mathrm{in}\Omega ,\hfill & \\ u(x,T)=uf,\mathrm{in}\Omega ,\hfill \\ u=,\mathrm{on}\partial \Omega \times [,T].\hfill \end{array}\right.$$

(343)

Here $\epsilon >$, $f\in L([,T],W,\left(\Omega \right))\cap L\infty (\Omega \times [,T]),$ $u^,uf\in H01\left(\Omega \right)\cap L\infty \left(\Omega \right),$ where t denotes time and $[,T]$ is a time interval.

Also $g:\mathbb{R}\to \mathbb{R}$ is a continuous function neither necessarily linear nor convex.

We assume there exists $K33>$ such that

$$\parallel g\parallel \infty \le {\displaystyle K33m\left(\Omega \right)/},$$

Fixing $N\in \mathbb{N}$ and defining

$$\Delta tN={\displaystyle TN},$$

in a partial finite differences context, discretizing in t consider the approximate equation system

$${\displaystyle un+-un+un-\Delta tN}=\epsilon \nabla un+g\left(un\right)+fn,\mathrm{in}\Omega ,$$

$\forall n\in \{,\dots ,N-\}.$

From such a system, for $n=$, we obtain

$$u-u+u^=\epsilon \nabla \left(u\right)\Delta tN+g\left(u\right)\Delta tN+f\Delta tN.$$

Hence

$$(Idu+\epsilon \nabla \Delta tN)u=\left(u+u^-g\left(u\right)\Delta tN-f\Delta tN\right),$$

so that

$$\begin{array}{ccc}& & \parallel u\parallel ,,\Omega \hfill \\ & \le & \parallel (Id+\epsilon (\nabla )\Delta tN)-\parallel \hfill \\ & & \times \left(\parallel u\parallel ,,\Omega +\parallel u^\parallel ,,\Omega +\parallel g\left(u\right)\parallel ,,\Omega \Delta tN+\parallel f\parallel ,,\Omega \Delta tN\right).\hfill \end{array}$$

(344)

Observe that there exists $K>$ such that $\parallel f\parallel \infty ,\Omega \times [,T]\le K$ so that

$$\parallel fn\parallel ,,\Omega \le K,\forall n\in \{,,\dots ,N-\},$$

for some appropriate $K>.$

From such results and the hypotheses, we may infer that

$$\begin{array}{ccc}& & \parallel u\parallel ,,\Omega \le \parallel (Id+\epsilon (\nabla )\Delta tN)-\parallel (\parallel u\parallel ,,\Omega +K33\Delta tN+\parallel u^\parallel ,,\Omega +K\Delta tN)\hfill \\ & \le & \parallel (Id+\epsilon (\nabla )\Delta tN)-\parallel (\parallel u\parallel ,,\Omega +\parallel u^\parallel ,,\Omega +K85\Delta tN),\hfill \end{array}$$

(345)

where $K85=K33+K.$

On the other hand, through a symbolic auxiliary notation, we have

$$\begin{array}{ccc}\hfill (Id+\epsilon (\nabla )\Delta tN)-& =& {\displaystyle (Id+\epsilon (\nabla )\Delta tN)}\hfill \\ & =& {\displaystyle (Id+\epsilon (\nabla )\Delta tN/)}\hfill \\ & =& {\displaystyle 12}\left(Id-{\displaystyle \epsilon (\nabla )\Delta tN/Id+\epsilon (\nabla )\Delta tN/}\right),\hfill \end{array}$$

(346)

so that

$$\parallel (Id+\epsilon (\nabla )\Delta tN)-\parallel \le {\displaystyle 12}+∥{\displaystyle \epsilon (\nabla )\Delta tN/(Id+\epsilon (\nabla )\Delta tN/)}∥.$$

Now denote

$$\theta N=∥{\displaystyle \epsilon (\nabla )(Id+\epsilon (\nabla )\Delta tN/)}∥.$$

Thus,

$$\parallel (Id+\epsilon (\nabla )\Delta tN)-\parallel \le {\displaystyle 12}+{\displaystyle \theta N\Delta tN},$$

so that

$$\parallel u\parallel ,,\Omega \le \left({\displaystyle 12}+{\displaystyle \theta N\Delta tN}\right)(\parallel u\parallel ,,\Omega +\parallel u^\parallel ,,\Omega +K85\Delta tN).$$

Consequently, from such results, we may infer that

$$\left({\displaystyle 12}+{\displaystyle \theta N\Delta tN}\right)-\parallel u\parallel ,,\Omega \le (\parallel u\parallel ,,\Omega +\parallel u^\parallel ,,\Omega +K85\Delta tN),$$

so that

$$\left(-{\displaystyle \theta N\Delta tN/(+\theta N\Delta tN/)}\right)\parallel u\parallel ,,\Omega \le (\parallel u\parallel ,,\Omega +\parallel u^\parallel ,,\Omega +K85\Delta tN).$$

Therefore,

$$\left(-{\displaystyle \theta N\Delta tN(+\theta N\Delta tN/)}\right)\parallel u\parallel ,,\Omega \le (\parallel u\parallel ,,\Omega +\parallel u^\parallel ,,\Omega +K85\Delta tN).$$

Let $\epsilon \in \mathbb{R}$ be such that

$$<\epsilon \ll max\{\epsilon ,\}.$$

Define $\alpha ^=\epsilon \parallel \nabla \parallel$ and observe that

$${\displaystyle \theta N(+\theta N\Delta tN/)}\to \alpha ^,\mathrm{as}N\to \infty .$$

Hence, there exists $N\in \mathbb{N}$ such that if $N>N,$ then

$$\left|{\displaystyle \theta N(+\theta N\Delta tN/)}-\alpha ^\right|<\epsilon .$$

From these results, if $N>N$, we have

$$\left(-{\displaystyle \theta N\Delta tN(+\theta N\Delta tN/)}\right)\ge (-(\alpha ^+\epsilon )\Delta tN)>.$$

Therefore, defining $\alpha =\alpha ^+\epsilon ,$ we have got,

$$\left(-\alpha \Delta tN\right)\parallel u\parallel ,,\Omega \le (\parallel u\parallel ,,\Omega +\parallel u^\parallel ,,\Omega +K85\Delta tN).$$

so that

$$\parallel u\parallel ,,\Omega \le \alpha \parallel u\parallel ,,\Omega +\beta \parallel u^\parallel ,,\Omega +\gamma ,$$

where

$$\alpha =(-\alpha \Delta tN)-,$$

$$\beta =\alpha$$

and $\gamma =\alpha K85\Delta tN.$

Reasoning inductively, for $n\ge$ having

$$\parallel un-\parallel ,,\Omega \le \alpha n-\parallel un\parallel ,,\Omega +\beta n-\parallel u^\parallel ,,\Omega +\gamma n-,$$

we are going to obtain $\alpha n,\beta n$ and $\gamma$.

Similarly as above, we may obtain

$$\begin{array}{ccc}& & (-\alpha \Delta tN)\parallel un\parallel ,,\Omega \hfill \\ & \le & \parallel un+\parallel ,,\Omega +\parallel un-\parallel ,,\Omega +K85\Delta tn,\hfill \\ & \le & \parallel un+\parallel ,,\Omega +\alpha n-\parallel un\parallel ,,\Omega +\beta n-\parallel u^\parallel ,,\Omega +\gamma n-+K85\Delta tn.\hfill \end{array}$$

(347)

Thus,

$$\begin{array}{ccc}& & (-\alpha \Delta tN-\alpha n-)\parallel un\parallel ,,\Omega \hfill \\ & \le & \parallel un+\parallel ,,\Omega +\beta n-\parallel u^\parallel ,,\Omega +\gamma n-+K85\Delta tN.\hfill \end{array}$$

(348)

Consequently,

$$\parallel un\parallel ,,\Omega \le \alpha n\parallel un+\parallel ,,\Omega +\beta n\parallel u^\parallel ,,\Omega +\gamma n,$$

where

$$\alpha n={\displaystyle -\alpha \Delta tN-\alpha n-},$$

$$\beta n=\alpha n\beta n-,$$

and

$$\gamma n=\alpha n(\gamma n-+K85\Delta tN).$$

We recall that $\alpha =\epsilon \parallel \nabla \parallel +\epsilon .$ Here we assume $T\ge$ and

$$\alpha T\le {\displaystyle 12}.$$

Consider the sequence $\left\{bn\right\}\subset \mathbb{R}$ such that

$$b=/,$$

and

$$bn+={\displaystyle -bn},\forall n\in \mathbb{N}.$$

We may easily obtain by induction that

$$bn={\displaystyle nn+}.$$

Define

$$an=bn-={\displaystyle n-n},\forall n\ge .$$

Observe that

$$\begin{array}{ccc}\hfill an+{\displaystyle \alpha TN}& \le & {\displaystyle N-N}+{\displaystyle \alpha TN}\hfill \\ & \le & {\displaystyle N-N}+{\displaystyle N}\hfill \\ & =& ,\forall n\in \{,\dots ,N-\}.\hfill \end{array}$$

(349)

Observe that

$$\begin{array}{ccc}\hfill \alpha & \le & {\displaystyle -\alpha TN}\hfill \\ & =& {\displaystyle 12}+\left({\displaystyle -\alpha TN}-{\displaystyle 12}\right)\hfill \\ & \le & {\displaystyle 12}+{\displaystyle \alpha TN}\hfill \\ & \le & a+{\displaystyle \alpha TN}+{\displaystyle \alpha TN}.\hfill \end{array}$$

(350)

At this point we shall prove by induction that

$$\alpha n\le an+{\displaystyle \alpha TN}+n{\displaystyle \alpha TN},\forall n\in \{,\dots ,N-\}.$$

For $n=$ we have already proved it above.

Suppose now that for $n\ge$, we have

$$\alpha n\le an+{\displaystyle \alpha TN}+n{\displaystyle \alpha TN}.$$

Observe that

$$\begin{array}{ccc}\hfill \alpha n+& =& {\displaystyle -\alpha TN-\alpha n-}\hfill \\ & =& {\displaystyle -an}+\left({\displaystyle -\alpha TN-\alpha n-}-{\displaystyle -an}\right)\hfill \\ & =& an++\left({\displaystyle -\alpha TN-\alpha n}-{\displaystyle -an}\right)\hfill \\ & \le & an++\left(-an+\alpha n+\alpha {\displaystyle TN}\right)\hfill \\ & \le & an++{\displaystyle \alpha TN}+n{\displaystyle \alpha TN}+\alpha {\displaystyle TN}\hfill \\ & =& an++{\displaystyle \alpha TN}+(n+){\displaystyle \alpha TN}.\hfill \end{array}$$

(351)

The induction is complete, indeed we have proven that

$$\alpha n\le an+{\displaystyle \alpha TN}+n{\displaystyle \alpha TN},\forall n\in \{,\dots ,N-\}.$$

Thus, we have obtained

$$\begin{array}{ccc}\hfill \alpha n& \le & an+{\displaystyle \alpha TN}+n{\displaystyle \alpha TN}\hfill \\ & \le & an+{\displaystyle \alpha TN}+{\displaystyle \alpha TN}\hfill \\ & \le & an+{\displaystyle \alpha TN}\hfill \\ & \le & ,\forall n\in \{,\dots N-\}.\hfill \end{array}$$

(352)

Summarizing,

$$\le \alpha n\le ,\forall n\in \{,\dots ,N-\}.$$

Now denoting more generically $\alpha nN=\alpha n$ we may infer that

$$\le \alpha nN\le ,\forall n\in \{,\dots ,N-\},\forall N>N.$$

From such results we may also obtain that there exist $K15>$ and $K16>$ such that

$$|\beta nN|\le K15,$$

and

$$|\gamma nN|\le K16,$$

$\forall n\in \{,\dots N-\},\forall N>N.$

We recall that

$$uNN=uf,$$

so that since

$$\parallel uN-N\parallel ,,\Omega \le \alpha N-N\parallel uN\parallel ,,\Omega +\beta N-N\parallel u^\parallel ,,\Omega +\gamma N-N,$$

from this and these last results we may infer that

$$\parallel unN\parallel ,,\Omega \le K18,$$

$\forall n\in \{,\dots ,N-\},\forall N>N,$ for some appropriate real constant $K18>.$

Define now

$$WN(x,t)={\displaystyle un+N-unN+un-N\Delta tN},$$

if $(x,t)\in \Omega \times (n\Delta tN,(n+)\Delta tN],\forall (x,t)\in \Omega \times [,T],$ and

$$uN(x,t)=u^\left(x\right)+\left(uN\right)\prime \left(x\right)t+\int t\int \tau WN(x,\tau )d\tau d\tau ,$$

where $\left(uN\right)\prime \left(x\right)$ is such that

$$uN(x,T)=uf\left(x\right).$$

Here we highlight that

$$\begin{array}{ccc}\hfill {\displaystyle \partial uN(x,t)\partial t}& =& WN(x,t)\hfill \\ & =& {\displaystyle un+N-unN+un-N\Delta tN},\hfill \end{array}$$

(353)

if $(x,t)\in \Omega \times (n\Delta tN,(n+)\Delta tN],\forall (x,t)\in \Omega \times [,T].$

Observe that

$$\begin{array}{ccc}\hfill {\displaystyle \partial uN(x,t)\partial t}& =& {\displaystyle un+N-unN+un-N\Delta tN}\hfill \\ & =& \epsilon \nabla unN+g\left(unN\right)+fn,\hfill \end{array}$$

(354)

if $t\in (n\Delta tN,(n+)\Delta tN],\forall (x,t)\in \Omega \times [,T].$

Fix $\phi \in Cc\infty \left(\Omega \right).$

Thus, fixing $t\in (n\Delta tN,(n+)\Delta tN],$ we have

$$\begin{array}{ccc}\hfill \left|〈{\displaystyle \partial uN\partial t},\phi 〉L\right|& \le & \epsilon |\langle \nabla unN,\nabla \phi \rangle L|+|\langle g\left(unN\right),\phi \rangle L|\hfill \\ & & +|\langle \phi ,fn\rangle L|\hfill \\ & \le & \epsilon \parallel unN\parallel ,,\Omega \parallel \phi \parallel ,,\Omega +K19\parallel unN\parallel ,,\Omega \parallel \phi \parallel ,,\Omega \hfill \\ & & +K25\parallel \phi \parallel ,,\Omega \hfill \\ & \le & K26\parallel \phi \parallel ,,\Omega ,\forall \phi \in Cc\infty \left(\Omega \right),\hfill \end{array}$$

(355)

for some appropriate $K26>.$

Since $\phi \in Cc\infty \left(\Omega \right)$ is arbitrary, we may conclude that

$$∥{\displaystyle \partial uN\partial t}∥H-\left(\Omega \right)\le K,\forall N>N,$$

uniformly in t on $[,T],$ for some appropriate constant $K>.$

Also, from the definition of $u$ we have that there exists $K>$ such that

$$\parallel uN\parallel ,,\Omega \le K,\forall N>N$$

also uniformly in t on $[,T].$

From such results, there exist $u\in L([,T],H01\left(\Omega \right))$ and $v\in L([,T];H-\left(\Omega \right))$ such that

$$uN\rightharpoonup u,\mathrm{weakly}\mathrm{in}L((,T);W,\left(\Omega \right)),$$

and

$${\displaystyle \partial uN\partial t}\rightharpoonup v,\mathrm{weakly}-\mathrm{star}\mathrm{in}L([,T],H-\left(\Omega \right)),$$

so that we may easily obtain

$$v={\displaystyle \partial u\partial t}$$

in a distributional sense.

At this point, we provide more details about this last result.

Fix $t\in (,T)$. Thus, there exists $n\in \{,,\dots ,N-\}$ such that $t\in (n\Delta tN,(n+)\Delta tN]$.

Let $\phi \in Cc\infty (\Omega \times (,T)).$

From this, we may infer that

$$\begin{array}{ccc}& & \int \Omega {\displaystyle \partial uN\partial t}\phi (x,t)dx\hfill \\ & =& \int \Omega \left({\displaystyle un+N-unN+un-N\Delta tN}\right)\phi (x,t)dx\hfill \\ & \le & \epsilon \int \Omega |\nabla unN·\nabla \phi |dx\hfill \\ & & +\int \Omega \left|g\left(unN\right)\phi (x,t)\right|dx\hfill \\ & & +\int \Omega \left|fn\phi \right|dx\hfill \\ & \le & (K(\parallel unN\parallel ,,\Omega +K20)\parallel \phi \parallel ,,\Omega \hfill \\ & \le & K\parallel \phi \parallel ,,\Omega ,\hfill \end{array}$$

(356)

for some appropriate constants $K>,K>,K20>.$

Hence,

$$\begin{array}{ccc}& & \int T\int \Omega {\displaystyle \partial uN\partial t}\phi (x,t)dxdx\hfill \\ & \le & K\int \Omega \parallel \phi \parallel ,,\Omega dt\hfill \\ & \le & K19\parallel \phi \parallel ,,\Omega \times (,T),\hfill \end{array}$$

(357)

for some appropriate $K19>$.

Since such a $\phi \in Cc\infty (\Omega \times (,T))$ is arbitrary, we may infer that

$$∥{\displaystyle \partial uN\partial t}∥H-(\Omega \times (,T))\le K15,$$

for $N>N$, for some $K15>.$

From such a result and from the Banach-Alaoglu Theorem, there exists $v\in H-(\Omega \times (,T))$ such that, up to a not relabeled subsequence

$${\displaystyle \partial uN\partial t}\rightharpoonup v,\mathrm{weakly}-\mathrm{star}\mathrm{in}H-(\Omega \times (,T)).$$

Therefore,

$$\int T\int \Omega {\displaystyle \partial uN\partial t}\phi dxdt\to \int T\int \Omega v\phi dxdt,$$

as $N\to \infty ,$$\forall \phi \in H01(\Omega \times (,T))$.

On the other hand

$$\parallel uN\parallel ,,\Omega \times (,T)\le K16,$$

$\forall N>N$, for some $K16>.$

From this and the Kakutani Theorem, there exists $u\in L(\Omega \times (,T))$ such that, up to a not relabeled subsequence,

$$uN\rightharpoonup u,\mathrm{weakly}\mathrm{in}L(\Omega \times (,T)).$$

Now fix again $\phi \in Cc\infty (\Omega \times (,T)).$

Observe that

$$\begin{array}{ccc}\hfill \int T\int \Omega u\phi ttdxdt& =& \underset{N\to \infty }{lim}\int T\int \Omega uN\phi ttdxdt\hfill \\ & =& \underset{N\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uN\partial t}\phi dxdt\hfill \\ & =& \int T\int \Omega v\phi dxdt,\hfill \end{array}$$

(358)

Since such a $\phi \in Cc\infty (\Omega \times (,T))$ is arbitrary, we may infer that

$$v={\displaystyle \partial u\partial t}$$

in a distributional sense.

Moreover, from such results we may also obtain, again up to a subsequence,

$$\underset{N\to \infty }{lim}\int \Omega {\displaystyle \partial uN\partial t}\phi dx=\int \Omega {\displaystyle \partial u\partial t}\phi dx,$$

$\forall \phi \in H01\left(\Omega \right).$

Observe also that, as a consequence of the Rellich-Kondrashov theorem, through appropriate subsequences, we have

$$uNk\left(t\right)\to u(x,t),\mathrm{strongly}\mathrm{in}L\left(\Omega \right),\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

so that, up to subsequences,

$$uNk\left(t\right)(x,t)\to u(x,t),\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

Here we emphasise the sequence $\left\{Nk\left(t\right)\right\}\subset \mathbb{N}$ may depends on t.

Since g is continuous we have that

$$g\left(uNk\left(t\right)(x,t)\right)\to g\left(u(x,t)\right),\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

Fix $t\in (,T).$

Let $\epsilon >$. From the Egorov Theorem, there exists a closed set F such that $m(\Omega \setminus F)\le \epsilon$ and $k\in \mathbb{N}$ such that if $k>k$, then

$$|g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)|\le \epsilon ,\mathrm{for}\mathrm{almost}\mathrm{all}x\in F.$$

Let $\phi \in Cc\infty \left(\Omega \right).$ Observe now that

$$\begin{array}{ccc}& & \left|\int \Omega (g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right))\phi dx\right|\hfill \\ & \le & \int \Omega |g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |dx\hfill \\ & =& \int F|g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |dx+\int \Omega \setminus F|g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |dx\hfill \\ & \le & \int F\epsilon \parallel \phi \parallel \infty dx+\int \Omega |g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |\chi \Omega \setminus Fdx\hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+(\parallel g\left(uNk\left(t\right)\right)\parallel ,,\Omega +\parallel g\left(u\right)\parallel ,,\Omega )\parallel \phi \parallel ,,\Omega \parallel \chi \Omega \setminus F\parallel ,,\Omega \hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+K21\parallel \phi \parallel ,,\Omega m(\Omega \setminus F)/\hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+K21\parallel \phi \parallel ,,\Omega \epsilon /,\forall k>k,\hfill \end{array}$$

(359)

for some appropriate constant $K21>$ which does not depend on t.

Since such a $\epsilon >$ is arbitrary, we may infer that

$$\int \Omega g\left(uNk\left(t\right)\right)\phi dx\to \int \Omega g\left(u\right)\phi dx,\mathrm{as}k\to \infty ,$$

$\forall \phi \in Cc\infty \left(\Omega \right).$ From such results, we have

$$\begin{array}{ccc}\hfill & =& \underset{k\to \infty }{lim}\left(\int \Omega {\displaystyle \partial uNk\left(t\right)\partial t}\phi dx+\epsilon \int \Omega \nabla uNk\left(t\right)·\nabla \phi dx\right.\hfill \\ & & -\int \Omega g\left(uNk\left(t\right)\right)\phi dx\hfill \\ & & \left.-\int \Omega fNk\left(t\right)\phi dx\right)\hfill \\ & =& \int \Omega {\displaystyle \partial u\partial t}\phi dx+\epsilon \int \Omega \nabla u·\nabla \phi dx\hfill \\ & & -\int \Omega g\left(u\right)\phi dx\hfill \\ & & -\int \Omega f\phi dx.\hfill \end{array}$$

(360)

so that, from this and by the density of $Cc\infty \left(\Omega \right)$ in $H01\left(\Omega \right)$, we have got

$$\begin{array}{ccc}& & \int \Omega {\displaystyle \partial u\partial t}\phi dx+\epsilon \int \Omega \nabla u·\nabla \phi dx\hfill \\ & & -\int \Omega g\left(u\right)\phi dx\hfill \\ & & -\int \Omega f\phi dx=,\forall \phi \in H01\left(\Omega \right),\hfill \end{array}$$

(361)

$\mathrm{a}.\mathrm{e}.\mathrm{on}[,T].$

Hence, we have obtained that $u$ is a solution, in a weak sense, of the hyperbolic non-linear equation in question.

54. A numerical procedure combining the Euler method and the hyper-finite differences approach

Let $\Omega =[,]\subset \mathbb{R}$ and consider the equation

$$\left\{\begin{array}{cc}\epsilon u\prime \prime \left(x\right)-Au\left(x\right)+Bu\left(x\right)+=,\hfill & \mathrm{in}\Omega ,\hfill \\ u\left(\right)=,u\left(\right)=.\hfill \end{array}\right.$$

(362)

Here $A>$, $B>$ and $u\in W,\left(\Omega \right).$

We may represent such an equation, as a first order system

$$\left\{\begin{array}{cc}v\prime -Au/\epsilon +Bu)/\epsilon +/\epsilon =,\hfill & \mathrm{in}\Omega ,\hfill \\ u\prime =v,\mathrm{in}\Omega ,\hfill & \\ u\left(\right)=,u\left(\right)=.\hfill \end{array}\right.$$

(363)

Consider now such a system with generical unknown boundary conditions $u$ and $v$, that is,

$$\left\{\begin{array}{cc}v\prime -Au/\epsilon +Bu/\epsilon +/\epsilon =,\hfill & \mathrm{in}\Omega ,\hfill \\ u\prime =v,\mathrm{in}\Omega ,\hfill & \\ u\left(\right)=u^,v\left(\right)=v^.\hfill \end{array}\right.$$

(364)

Defining $d=/m$, where $m$ is total number of nodes, in finite differences we have

$$\left\{\begin{array}{cc}{\displaystyle vn-vn-d}-Aun-/\epsilon +Bun-/\epsilon +/\epsilon =,\hfill & \\ {\displaystyle un-un-d}=vn-\hfill & \\ u=u^,v=v^.\hfill \end{array}\right.$$

(365)

This is simply the explicit Euler method. We may symbolically obtain $\left\{un\right\}$ and $\left\{vn\right\}$ as functions of $u$ and $v$ (by using the MATHEMATICA or MAPLE software and by truncating the concerning polynomial solutions), through the iterations

$$\left\{\begin{array}{cc}vn=vn-+Aun-{\displaystyle d\epsilon }-Bun-{\displaystyle d\epsilon }-{\displaystyle d\epsilon },\hfill & \\ un=un-+vn-d\hfill & \\ u=u^,v=v^.\hfill \end{array}\right.$$

(366)

However, it is well known the error in this process could be big. In order to minimize such an error, we use the hyper-finite differences approach for the one-dimensional analogous of the generalized method of lines. More specifically, we will subdivide the interval $[.]$ into $N$ sub-interval of same measure, and redefine a not relabeled d as

$$d={\displaystyle mN}.$$

Hence, on each sub-interval $\left[{\displaystyle k-N},{\displaystyle kN}\right]$, using the MATHEMATICA or MAPLE software we may obtain an approximate solution

$$\{ui,k,vi,k\}$$

as functions of the initial conditions

$$\{u,k,v,k\}$$

where $i\in \{,\dots ,m\}$, $\forall k\in \{\dots ,N\}.$

In order to obtain such a solution,

$$\{ui,k,vi,k\}$$

we use following interactions

$$\left\{\begin{array}{cc}vn,k=vn-,k+Aun-,k{\displaystyle d\epsilon }-Bun-,k{\displaystyle d\epsilon }-{\displaystyle d\epsilon },\hfill & \\ un,k=un-,k+vn-,kd\hfill & \\ u,k=u^,k,v,k=v^,k.\hfill \end{array}\right.$$

(367)

Observe that for obtaining an approximate solution for the original equation in question, we must calculate $\{u^,k,v^,k\}$ though the solution of the system:

For the boundary conditions:

$$u,=,um,N=.$$

For the solution and its derivative continuity on the nodes related to the $N$ sub-intervals,

$$um,k=u,k+,vm,k=v,k+,\forall k\in \{,\dots N\}.$$

Having obtained $\{u^,k,v^,k\},\forall k\in \{,\dots ,N\}$ we may obtain

$$\{un,k,vn,k\}\forall n\in \{,\dots ,m\},\forall k\in \{,\dots ,N\}.$$

Here we present the software in Mathematica through which we have obtained the numerical results, for the case $\epsilon =.01,$$A=B=$ and $N=16$ subintervals.

**********************

m8 = 100;

N1 = 16;

d = 1.0/m8/N1;

e1 = 0.01;

A = 1.0;

B = 1.0;

For[k = 1, k < N1 + 1, k++,

Print[k];

u[0, k] = uo[k];

v[0, k] = vo[k];

For[i = 1, i < m8 + 1, i++,

z1 = $(v[i-,k]+A*d/e*u[i-,k]-B*u[i-,k]*d/e-.*d/e);$

z2 = u[i - 1, k] + v[i - 1, k]*d;

z1 = Series[z1, { uo[k], 0, 8}, { vo[k], 0, 8 }];

z2 = Series[z2, { uo[k], 0, 8 }, {vo[k], 0, 8 }];

z1 = Normal[z1];

z2 = Normal[z2];

v[i, k] = Expand[z1];

u[i, k] = Expand[z2]]];

S = $u[,]+u[m,N];$

For[k = 1, k < N1, k++,

S = S + $(u[m,k]-u[,k+])$;

S = S + $(v[m,k]-v[,k+])$];

sol = FindMinimum[

S, {uo[1], uo[2], uo[3], uo[4], uo[5], uo[6], uo[7], uo[8], uo[9],

uo[10], uo[11], uo[12], uo[13], uo[14], uo[15], uo[16], vo[1],

vo[2], vo[3], vo[4], vo[5], vo[6], vo[7], vo[8], vo[9], vo[10],

vo[11], vo[12], vo[13], vo[14], vo[15], vo[16]}]

Clear[U];

For [k = 1, k < N1 + 1, k++,

w[k] = uo[k] $/.$ sol[[2, k]]]

For[i = 1, i < N1 + 1, i++,

U[i - 1] = w[i]]

U[N1] = u[m8,N1];

For[i = 0, i < N1 + 1, i++,

Print["uo[", i + 1, "]=", U[i]]]

uo[1]=1.14453*$10$, in fact $u\left(\right)=$

uo[2]=0.817448

uo[3]=1.17018

uo[4]=1.28552

uo[5]=1.32107

uo[6]=1.33205

uo[7]=1.33546

uo[8]=1.3365

uo[9]=1.33677

uo[10]=1.33667

uo[11]=1.33596

uo[12]=1.33331

uo[13]=1.32382

uo[14]=1.2902

uo[15]=1.175

uo[16]=0.820243

uo[17]=0, in fact $u\left(\right)=.$

*******************************

Remark 25.

Observe that along the domain the solution is approximately $.33$ which is close to $.3247$, which is an approximate solution of equation $u-u-=.$ This is expected since $\epsilon =.01$ is a relatively small value.

55. A proximal numerical procedure combined with the Euler method

Let $\Omega =[,]\subset \mathbb{R}$ and consider the Ginzburg-Landau type equation

$$\left\{\begin{array}{cc}\epsilon u\prime \prime \left(x\right)-Au\left(x\right)+Bu\left(x\right)+=,\hfill & \mathrm{in}\Omega ,\hfill \\ u\left(\right)=,u\left(\right)=.\hfill \end{array}\right.$$

(368)

Here $A>$, $B>$ and $u\in W,\left(\Omega \right).$

We may represent such an equation, as a first order system

$$\left\{\begin{array}{cc}v\prime -Au/\epsilon +Bu/\epsilon +/\epsilon =,\hfill & \mathrm{in}\Omega ,\hfill \\ u\prime =v,\mathrm{in}\Omega ,\hfill & \\ u\left(\right)=,u\left(\right)=.\hfill \end{array}\right.$$

(369)

Consider now such a system with generical unknown boundary conditions $u$ and $v$, that is,

$$\left\{\begin{array}{cc}v\prime -Au/\epsilon +Bu/\epsilon +/\epsilon =,\hfill & \mathrm{in}\Omega ,\hfill \\ u\prime =v,\mathrm{in}\Omega ,\hfill & \\ u\left(\right)=u^,v\left(\right)=v^.\hfill \end{array}\right.$$

(370)

Defining $d=/m$, where $m$ is total number of nodes, in finite differences we have

$$\left\{\begin{array}{cc}{\displaystyle vn-vn-d}-Aun-/\epsilon +Bun-/\epsilon +/\epsilon =,\hfill & \\ {\displaystyle un-un-d}=vn-\hfill & \\ u=u^,v=v^.\hfill \end{array}\right.$$

(371)

This is simply the explicit Euler method. Setting $u=$, we may symbolically obtain $\left\{un\right\}$ and $\left\{vn\right\}$ as functions of $v=v^$ (by using the MATHEMATICA or MAPLE software and by truncating the concerning polynomial solutions), through the following iterations, which already include a proximal formulation about an initial fixed solution $\left\{\left(U\right)n\right\}.$

$$\left\{\begin{array}{cc}vn=vn-+Aun-{\displaystyle d\epsilon }-Bun-{\displaystyle d\epsilon }-{\displaystyle d\epsilon },\hfill & \\ un=un-+vn-d-{\displaystyle K\epsilon }(un-\left(U\right)n)d\hfill & \\ u=,v=v^.\hfill \end{array}\right.$$

(372)

$\forall n\in \{,\dots ,m\}.$

Indeed, in such a case we have

$$\left\{\begin{array}{cc}vn=vn-+Aun-{\displaystyle d\epsilon }-Bun-{\displaystyle d\epsilon }-{\displaystyle d\epsilon },\hfill & \\ un=\left(un-+vn-d+{\displaystyle Kd\epsilon }\left(U\right)n\right)/\left(+{\displaystyle Kd\epsilon }\right)\hfill & \\ u=,v=v^.\hfill \end{array}\right.$$

(373)

$\forall n\in \{,\dots ,m\}.$

We emphasize such a procedure may make the error in the explicit Euler method very small, in fact proportional to ${\displaystyle \epsilon K}.$

Thus, having obtained $un=un\left(v\right)$, we may obtain $v$ through the boundary condition $u\left(\right)=$, that is, through a solution of equation $um\left(v\right)=$.

With such an $v$ calculated, we may obtain explicitly $un=un\left(v\right)$, $\forall n\in \{,\dots m\}.$ The next step is to replace $\left\{\left(U\right)n\right\}$ by $\left\{un\right\}$ and then to repeat the process until an appropriate convergence criterion is satisfied.

We have obtained numerical results for $\epsilon =.01$, $A=B=$, $m=100$ and $K=10$.

Here we present the software through which we have obtained such results.

We highlight in this software we have fixed a total number of 800 iterations.

*************************

m8 = 100;

Clear[z1, z2, u, v, vo];

d = 1.0/m8;

e1 = 0.01;

A = 1.0;

B = 1.0;

K = 10.0;

For[i = 0, i < m8 + 1, i++,

uo[i] = 0.01];

For[k = 1, k < 800, k++, $(\mathrm{here}\mathrm{we}\mathrm{have}\mathrm{fixed}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{iterations})$

Print[k];

Clear[vo];

u[0] = 0.0;

v[0] = vo;

For[i = 1, i < m8 + 1, i++,

z1 = (v[i - 1] + A*d/e1*$u[i-]$ - B*u[i - 1]*d/e1 - 1.0*d/e1);

z2 = (u[i - 1] + v[i - 1]*d + K*uo[i]*d/e1)/(K*d/e1 + 1.0);

z1 = Series[z1, {vo, 0, 9}];

z2 = Series[z2, {vo, 0, 9}];

z1 = Normal[z1];

z2 = Normal[z2];

v[i] = Expand[z1];

u[i] = Expand[z2]];

S = $\left(u\left[m\right]\right);$

sol = FindMinimum[S, vo];

w = vo $/.$ sol[[2, 1]];

vo = w;

For[i = 0, i < m8 + 1, i++,

uo[i] = u[i]];

Print[u[m8/2]]];

For[i = 0, i < m8/10 + 1, i++,

Print["u[", 10*i, "]=", u[10*i]]]

u[0]=0.

u[10]=1.09119

u[20]=1.29955

u[30]=1.32239

u[40]=1.32427

u[50]=1.3245

u[60]=1.32386

u[70]=1.31754

u[80]=1.27924

u[90]=1.04636

u[100]=$.31252*10-18$

****************************

Remark 26.

Observe that along the domain the solution is close to $.3247$, which is an approximate solution of equation $u-u-=.$ This is expected since $\epsilon =.01$ is a relatively small value.

56. A Proximal Numerical Procedure Combined with the Euler Method for Solving Partial Differential Equations

Let $\Omega =[,]\times [,]\subset \mathbb{R}$ and consider the Ginzburg-Landau type equation

$$\left\{\begin{array}{cc}\epsilon \nabla u-Au+Bu+f=,\hfill & \mathrm{in}\Omega ,\hfill \\ u=,\mathrm{on}\partial \Omega .\hfill \end{array}\right.$$

(374)

Here $A>$, $B>$, $f\in L\left(\Omega \right)$ and $u\in W,\left(\Omega \right).$

We may represent such an equation, as a partially first order system

$$\left\{\begin{array}{cc}vx+uyy-Au/\epsilon +Bu/\epsilon +f/\epsilon =,\hfill & \mathrm{in}\Omega ,\hfill \\ ux=v,\mathrm{in}\Omega ,\hfill & \\ u=,\mathrm{on}\partial \Omega \hfill \end{array}\right.$$

(375)

Defining $d=/m$, $d=/m$ and denoting

$$m=\left[\begin{array}{cccccc}-& & & & \dots & \\ & -& & & \dots & \\ & & -& & \dots & \\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ & & \dots & & -& \\ & & \dots & \dots & & -\end{array}\right],$$

(376)

where $m$ is total number of nodes in x, and $m$ is the number of nodes in y, in a finite differences context, we may have

$$\left\{\begin{array}{cc}{\displaystyle vn-vn-d}+{\displaystyle md12}un--Aun-/\epsilon +Bun-/\epsilon +fn/\epsilon =,\hfill & \\ {\displaystyle un-un-d}=vn-\hfill & \\ u=,v=v^.\hfill \end{array}\right.$$

(377)

This is simply an adaptation of the explicit Euler method. Observe that we may obtain $\left\{un\right\}$ and $\left\{vn\right\}$ as functions of $v=v^$ through the following iterations, which already include a proximal formulation about an initial fixed solution $\left\{\left(U\right)n\right\}.$

$$\left\{\begin{array}{cc}vn=vn--{\displaystyle md12}un-d+Aun-{\displaystyle d\epsilon }-Bun-{\displaystyle d\epsilon }-{\displaystyle fnd\epsilon },\hfill & \\ un=un-+vn-d-{\displaystyle K\epsilon }(un-\left(U\right)n)d\hfill & \\ u=,v=v^.\hfill \end{array}\right.$$

(378)

$\forall n\in \{,\dots ,m\}.$

Indeed, in such a case we have, through a concerning linearization,

$$\left\{\begin{array}{cc}vn=vn--{\displaystyle md12}un-+A\left(u\right)n-un-{\displaystyle d\epsilon }-A\left(u\right)n-{\displaystyle d\epsilon }-Bun-{\displaystyle d\epsilon }-{\displaystyle fnd\epsilon },\hfill & \\ un=\left(un-+vn-d+{\displaystyle Kd\epsilon }\left(U\right)n\right)/\left(+{\displaystyle Kd\epsilon }\right)\hfill & \\ u=,v=v^.\hfill \end{array}\right.$$

(379)

$\forall n\in \{,\dots ,m\}.$

We emphasize such a procedure may make the error in the explicit Euler method very small, in fact proportional to ${\displaystyle \epsilon K}.$

Observe now that in particular for $n=$, we have

$$\begin{array}{ccc}\hfill v& =& v-f{\displaystyle d\epsilon }\hfill \\ & \equiv & \left(M\right)v+\left(y\right),\hfill \end{array}$$

(380)

where

$$\left(M\right)=Id\mathrm{identity}\mathrm{matrix}(m-)\times (m-),$$

and

$$\left(y\right)=-f{\displaystyle d\epsilon }.$$

Also,

$$\begin{array}{ccc}\hfill u& =& \left(vd+K\left(u\right){\displaystyle d\epsilon }\right)/\left(+K{\displaystyle d\epsilon }\right)\hfill \\ & \equiv & \left(M\right)v+\left(y\right),\hfill \end{array}$$

(381)

where

$$\left(M\right)={\displaystyle Idd\left(+Kd\epsilon \right)},$$

and

$$\left(y\right)=\left({\displaystyle K\left(u\right)d\epsilon }\right)/\left(+K{\displaystyle d\epsilon }\right).$$

Reasoning inductively, having

$$vn-=\left(M\right)n-v+\left(y\right)n-,$$

and

$$un-=\left(M\right)n-v+\left(y\right)n-,$$

and replacing such relations into the concerning system (379), we obtain

$$\begin{array}{ccc}\hfill vn& =& \left(M\right)n-+\left(y\right)n--{\displaystyle md12}(\left(M\right)n-+\left(y\right)n-)d\hfill \\ & & +A\left(u\right)n-(\left(M\right)n-+\left(y\right)n-){\displaystyle d\epsilon }-A\left(u\right)n-{\displaystyle d\epsilon }\hfill \\ & & -B(\left(M\right)n-+\left(y\right)n-){\displaystyle d\epsilon }-fn{\displaystyle d\epsilon }\hfill \\ & =& \left(M\right)n+\left(y\right)n,\hfill \end{array}$$

(382)

where

$$\left(M\right)n=\left(M\right)n--{\displaystyle md12}\left(\left(M\right)n-\right)d+A\left(u\right)n-\left(\left(M\right)n-\right){\displaystyle d\epsilon }-B\left(\left(M\right)n-\right){\displaystyle d\epsilon },$$

and

$$\begin{array}{ccc}\hfill \left(y\right)n& =& \left(y\right)n--{\displaystyle md12}\left(\left(y\right)n-\right)d+A\left(u\right)n-\left(\left(y\right)n-\right){\displaystyle d\epsilon }\hfill \\ & & -A\left(u\right)n-{\displaystyle d\epsilon }-B\left(\left(y\right)n-\right){\displaystyle d\epsilon }-fn{\displaystyle d\epsilon }.\hfill \end{array}$$

(383)

Moreover,

$$\begin{array}{ccc}\hfill un& =& \left(\left(M\right)n-v+\left(y\right)n-+\left(M\right)n-vd+\left(y\right)n-d+K\left(u\right)n-{\displaystyle d\epsilon }\right)/\left(+K{\displaystyle d\epsilon }\right)\hfill \\ & =& \left(M\right)nv+\left(y\right)n,\hfill \end{array}$$

(384)

where

$$\left(M\right)n={\displaystyle \left(M\right)n-+\left(M\right)n-d\left(+Kd\epsilon \right)},$$

and

$$\left(y\right)n=\left(\left(y\right)n-+\left(y\right)n-d+K\left(u\right)n-{\displaystyle d\epsilon }\right)/\left(+K{\displaystyle d\epsilon }\right).$$

Summarizing, we have obtained

$$vn=\left(M\right)nv+\left(y\right)n,$$

and

$$un=\left(M\right)nv+\left(y\right)n,$$

$\forall n\in \{,\dots ,m\}.$

Consequently, from this and the boundary condition $um=,$ we may have

$$um==\left(M\right)mv+\left(y\right)m$$

so that

$$v=-\left[\left(M\right)m\right]-\left(y\right)m.$$

From such results we have obtained $\left\{un\right\}$ and $\left\{vn\right\}$, $\forall n\in \{,\dots ,m\}.$

The next step is to replace $\left\{\left(u\right)n\right\}$ by $\left\{un\right\}$ and then to repeat the process until an appropriate convergence criterion is satisfied.

We have obtained numerical results for $\epsilon =.01$, $A=B=$, $f\equiv ,\mathrm{in}\Omega ,$$m=100$ and $K=100$.

For the solution $u=u(x,y)$ obtained, please see Figure 42.

Here we present the software in MAT-LAB through which we have obtained such results.

******************************

clear all

m8=100;

m9=100;

d=1/m8;

d1=1/m9;

e1=0.01;

A=1;

B=1;

K=100;

f=ones(m9-1,1);

for i=1:m8

uo(:,i)=1.4*ones(m9-1,1);

Yo(:,i)=f;

end;

m2=zeros(m9-1,m9-1);

for i=2:m9-2

m2(i,i)=-2.0;

m2(i,i+1)=1.0;

m2(i,i-1)=1.0;

end;

m2(1,1)=-2.0;

m2(1,2)=1.0;

m2(m9-1,m9-1)=-2.0;

m2(m9-1,m9-2)=1.0;

Id=eye(m9-1);

b12=1.0;

k=1;

while $(b12>10-10)$ &&$(k<9010)$

k

k=k+1;

M1(:,:,1)=Id;

y1(:,1)=-Yo(:,1)*d/e1;

M2(:,:,1)=Id*d/(K*d/e1+1);

y2(:,1)=K*uo(:,1)*(d/e1)/(K*d/e1+1);

for i=2:m8

M1(:,:,i)=M1(:,:,i-1)-$m/d12$*d*M2(:,:,i-1)+3*A*diag(uo(:,i-1).*uo(:,i-1))*M2(:,:,i-1)*d/e1;

M1(:,:,i)=M1(:,:,i)-B*M2(:,:,i-1)*d/e1;

y1(:,i)=y1(:,i-1)-$m/d12$*d*y2(:,i-1)+3*A*(uo(:,i-1).*uo(:,i-1)).*y2(:,i-1)*d/e1;

y1(:,i)=y1(:,i)-2*A*(uo(:,i-1).*uo(:,i-1).*uo(:,i-1))*d/e1-B*y2(:,i-1)*d/e1-Yo(:,i-1)*d/e1;

M2(:,:,i)=(M2(:,:,i-1)+d*M1(:,:,i-1))/(K*d/e1+1);

y2(:,i)=(y2(:,i-1)+d*y1(:,i-1)+K*uo(:,i)*d/e1)/(K*d/e1+1);

end;

vo(:,1)=-inv(M2(:,:,m8))*y2(:,m8);

for i=1:m8

u(:,i)=M2(:,:,i)*vo(:,1)+y2(:,i);

end;

u(m9/2,m8/2)

b12=max(max(abs(u-uo)));

uo=u;

end;

for i=1:m8

x1(i,1)=i*d;

end;

for j=1:m9-1

y3(j,1)=j*d1;

end;

mesh(x1,y3,u)

*******************************

Remark 27.

Observe that along the domain the solution is close to $.3247$, which is an approximate solution of equation $u-u-=.$ This is expected since $\epsilon =.01$ is a relatively small value.

57. A Proximal Numerical Procedure Combined with the Euler Method for First Order Systems Applied to a Flight Mechanics Model

Let $\Omega =[,tf]$ be a time interval.

Consider the first order system of ordinary differential equations given by

$$\left\{\begin{array}{cc}{\displaystyle dujdt}=fj\left(\left\{ul\right\}\right),\mathrm{on}[,tf],\forall j\in \{,\dots ,\},\hfill & \\ u\left(\right)=,u\left(\right)=.12,u\left(\right)=,u\left(tf\right)=11000.\hfill \end{array}\right.$$

(385)

Here $fj:Dj\subset \mathbb{R}\to \mathbb{R}$ is a smooth function on its domain $D$, $\forall j\in \{,\dots ,\}.$

In finite differences, such a system stands for

$$\left\{\begin{array}{cc}{\displaystyle \left(uj\right)n-\left(uj\right)n-d}=fj\left(\left\{un-\right\}\right),\forall j\in \{,\dots ,\},\hfill & \\ \left(u\right)=,\left(u\right)=.12,\left(u\right)=,\left(u\right)m=11000.\hfill \end{array}\right.$$

(386)

$\forall n\in \{,\dots ,m\},$ where $m$ is number of nodes and $d=tf/m.$ This is just the explicit Euler method. It is well known, at first the error in this procedure may be big.

However, such an error may be made very small by introducing a proximal formulation and related linearization about a fixed initial solution $\left\{\left(u\right)n\right\},$ in a Newton type approach context.

In such a case the approximate system stands for

$$\left\{\begin{array}{cc}{\displaystyle \left(u\right)n-\left(u\right)n-d}=f\left(\left\{\left(u\right)n-\right\}\right)\hfill & \\ +\sum k={\displaystyle \partial f\left(\left\{\left(u\right)n-\right\}\right)\partial uk}(\left(uk\right)n--\left(uk\right)n-)-K(\left(u\right)n-\left(\left(u\right)n\right),\hfill & \\ \\ {\displaystyle \left(uj\right)n-\left(uj\right)n-d}=fj\left(\left\{\left(u\right)n-\right\}\right)+\sum k={\displaystyle \partial fj\left(\left\{\left(u\right)n-\right\}\right)\partial uk}(\left(uk\right)n--\left(uk\right)n-),\hfill & \\ \forall j\in \{,,\},\hfill & \\ \\ \left(u\right)=,\left(u\right)=.12,\left(u\right)=,\left(u\right)m=11000.\hfill \end{array}\right.$$

(387)

Indeed, setting the boundary conditions

$$\left(u\right)=,\left(u\right)=.12,\left(u\right)=v,\left(u\right)=$$

we will calculate

$$\left\{\left(uj\right)n\left(v\right)\right\}$$

through the following iterations

$$\left\{\begin{array}{cc}\left(u\right)n=\left(\left(u\right)n-+f\left(\left\{\left(u\right)n-\right\}\right)d+\right.\hfill & \\ \left.\sum k={\displaystyle \partial f\left(\left\{\left(u\right)n-\right\}\right)\partial uk}(\left(uk\right)n--\left(uk\right)n-)d+Kd\left(\left(u\right)n\right)\right)/\left(+Kd\right),\hfill & \\ \\ \left(uj\right)n=\left(uj\right)n-+fj\left(\left\{\left(u\right)n-\right\}\right)d+\sum k={\displaystyle \partial fj\left(\left\{\left(u\right)n-\right\}\right)\partial uk}(\left(uk\right)n--\left(uk\right)n-)d,\hfill & \\ \forall j\in \{,,\},\hfill \\ \\ \left(u\right)=,\left(u\right)=.12,\left(u\right)=v,\left(u\right)=.\hfill \end{array}\right.$$

(388)

Observe that the boundary condition $u\left(tf\right)=11000$ corresponds to $\left(u\right)m\left(v\right)=11000$ so that, through this last equation we may obtain $v.$ Having obtained $v$, we may obtain $\left\{\left(uj\right)n\right\}=\left\{\left(uj\right)n\left(v\right)\right\},\forall n\in \{,\dots ,m\},$$\forall j\in \{,\dots ,\}.$

The next step is to replace $\left\{\left(uj\right)n\right\})$ by $\left\{\left(uj\right)n\right\}$ and then to repeat the process until an appropriate convergence criterion is satisfied.

We have obtained numerical results for a model in flight mechanics.

More specifically, we model an in-plan climbing motion of an airplane AIR BUS 320, through the variables $h,\gamma ,V,x$ where h denotes the airplane altitude, $\gamma$ is the angle between its velocity and the axis x, V is the airplane speed and x corresponds to its horizontal coordinate.

The concerning system of equations is given by

$$\left\{\begin{array}{cc}h\dot{}=Vsin\left(\gamma \right)\hfill & \\ \\ \gamma \dot{}={\displaystyle mfV}(Fsin(a+aF)+L)-{\displaystyle gV}cos\left(\gamma \right)\hfill & \\ \\ V\dot{}={\displaystyle mf}(Fcos(a+aF)-D)-gsin\left(\gamma \right)\hfill & \\ \\ x\dot{}=Vcos\left(\gamma \right),\mathrm{on}[,tf],\hfill & \\ \\ h\left(\right)=,\gamma \left(\right)=.12,x\left(\right)=,h\left(tf\right)=11000.\hfill \end{array}\right.$$

(389)

Here $tf=515s$, $F=240000N$, $mf=120000Kg$, $Sf=260m$, $a=.138$, $g=.m/s$,

$$\rho \left(h\right)=.225\left(-{\displaystyle .0065h288.15}\right).225Kg/m,$$

$aF=.0175$, $\left(CL\right)=,$$\left(CL\right)a=.,$$\left(CD\right)=.0175,$$K=,$$K=.06,$

$$CL=\left(CL\right)+\left(CL\right)aa,$$

$$CD=\left(CD\right)+KCL+KCL,$$

$$L={\displaystyle 12}\rho \left(h\right)VCLSf,$$

$$D={\displaystyle 12}\rho \left(h\right)VCDSf.$$

For numerical purposes, we define

$$u=h,u=\gamma (=b),u=V,u=x.$$

Here we present the software in MATHEMATICA through which we have obtained the numerical results.

****************************

m8 = 20000;

tf = 515.0;

d = tf/m8;

K5 = 10.0/d;

h1 = 11000.0;

Clear[h, b, V, x, u, a, c];

h = u[1];

b = u[2];

V = u[3];

x = u[4];

mf = 120000.0;

g = 9.8;

Sf = 260.0;

a = 0.138;

af = 0.0175;

CLo = 0.0;

CLa = 5.0;

CDo = 0.0175;

K1 = 0.0;

K2 = 0.06;

CL = CLo + CLa*a;

CD = CDo + K1*CL + K2*$CL$;

F = 240000.0;

r = $.225*(.-.0065*h/288.15).225$;

L = $/*r*V*CL*Sf;$

D1 = $/*r*V*CD*Sf;$

f[1] = V*Sin[b];

f[2] = 1/mf/V*(F*Sin[a + af] + L) - g/V*Cos[b];

f[3] = 1/mf*(F*Cos[a + af] - D1) - g*Sin[b];

f[4] = V*Cos[b];

For[i = 0, i < m8 + 1, i++,

uo[i, 1] = 11000*i/m8;

uo[i, 2] = 0.15;

uo[i, 3] = 120;

uo[i, 4] = 50000*i/m8];

Clear[u];

For[i = 1, i < 5, i++,

For[j = 1, j < 5, j++,

c[i, j] = D[f[i], u[j]]]];

uo[0, 1] = 0.0;

uo[0, 2] = 0.12;

uo[0, 3] = 120;

uo[0, 4] = 0.0;

For[k3 = 1, k3 < 30, k3++, (Here we have fixed a total of 30 iterations)

Print[k3];

Clear[vo, U];

U[0, 1] = 0.0;

U[0, 2] = 0.12;

U[0, 3] = vo;

U[0, 4] = 0.0;

For[i = 1, i < m8 + 1, i++,

Clear[u];

u[1] = uo[i - 1, 1];

u[2] = uo[i - 1, 2];

u[3] = uo[i - 1, 3];

u[4] = uo[i - 1, 4];

z1 = Expand[U[i - 1, 1] + K5*(uo[i, 1])*d + f[1]*d];

z2 = Expand[U[i - 1, 2] + 0.0*K5*(uo[i, 2])*d + f[2]*d];

z3 = Expand[U[i - 1, 3] + 0.0*K5*(uo[i, 3])*d + f[3]*d];

z4 = Expand[U[i - 1, 4] + 0.0*K5*(uo[i, 4])*d + f[4]*d];

For[k = 1, k < 5, k++,

z1 = z1 + c[1, k]*(U[i - 1, k] - uo[i - 1, k])*d;

z2 = z2 + c[2, k]*(U[i - 1, k] - uo[i - 1, k])*d;

z3 = z3 + c[3, k]*(U[i - 1, k] - uo[i - 1, k])*d;

z4 = z4 + c[4, k]*(U[i - 1, k] - uo[i - 1, k])*d;];

U[i, 1] = Expand[z1/(1.0 + K5*d)];

U[i, 2] = Expand[z2/(1.0 + 0.0*K5*d)];

U[i, 3] = Expand[z3/(1.0 + 0.0*K5*d)];

U[i, 4] = Expand[z4/(1.0 + 0.0*K5*d)]];

Print[U[m8, 1]];

S = (U[m8, 1] - h1);

sol = NSolve[S == 0, vo];

vo = vo $/.$ sol[[1, 1]];

Print[vo];

Print[U[m8, 2]];

Print[U[m8, 3]];

Print[U[m8, 4]];

For[i = 0, i < m8 + 1, i++,

For[k = 1, k < 5, k++,

uo[i, k] = U[i, k]]];

Print[U[m8/2, 1]]];

****************************

For[i = 1, i < 11, i++,

Print["h(", 2000*i*d, "s)=U[", 2000*i, ",1]=", U[2000*i, 1]]]

h(51.5s)=U[2000,1]=1099.37

h(103.s)=U[4000,1]=2199.41

h(154.5s)=U[6000,1]=3299.45

h(206.s)=U[8000,1]=4399.5

h(257.5s)=U[10000,1]=5499.6

h(309.s)=U[12000,1]=6599.74

h(360.5s)=U[14000,1]=7699.8

h(412.s)=U[16000,1]=8799.76

h(463.5s)=U[18000,1]=9899.89

h(515.s)=U[20000,1]=11000.

For[i = 1, i < 11, i++,

Print["gamma(", 2000*i*d, "s)=U[", 2000*i, ",2]=", U[2000*i, 2]]]

gamma(51.5s)=U[2000,2]=0.120754

gamma(103.s)=U[4000,2]=0.120085

gamma(154.5s)=U[6000,2]=0.117905

gamma(206.s)=U[8000,2]=0.116329

gamma(257.5s)=U[10000,2]=0.119054

gamma(309.s)=U[12000,2]=0.125181

gamma(360.5s)=U[14000,2]=0.122861

gamma(412.s)=U[16000,2]=0.111435

gamma(463.5s)=U[18000,2]=0.115118

gamma(515.s)=U[20000,2]=0.115257

For[i = 1, i < 11, i++,

Print["V(", 2000*i*d, "s)=U[", 2000*i, ",3]=", U[2000*i, 3]]]

V(51.5s)=U[2000,3]=107.325

V(103.s)=U[4000,3]=113.338

V(154.5s)=U[6000,3]=119.7

V(206.s)=U[8000,3]=126.381

V(257.5s)=U[10000,3]=133.568

V(309.s)=U[12000,3]=142.044

V(360.5s)=U[14000,3]=152.19

V(412.s)=U[16000,3]=162.209

V(463.5s)=U[18000,3]=172.269

V(515.s)=U[20000,3]=185.79

For[i = 1, i < 11, i++,

Print["x(", 2000*i*d, "s)=U[", 2000*i, ",4]=", U[2000*i, 4]]]

x(51.5s)=U[2000,4]=5318.63

x(103.s)=U[4000,4]=10930.8

x(154.5s)=U[6000,4]=16860.9

x(206.s)=U[8000,4]=23137.6

x(257.5s)=U[10000,4]=29795.8

x(309.s)=U[12000,4]=36872.5

x(360.5s)=U[14000,4]=44395.

x(412.s)=U[16000,4]=52396.6

x(463.5s)=U[18000,4]=60960.3

x(515.s)=U[20000,4]=70129.5

*************************

58. A review of the convergence of Newton’s method combined with a proximal approach

Firstly we highlight similar results to those presented in this section have been presented in my book entitled "Functional Analysis, Calculus of Variations and Numerical Methods for Models in Physics and Engineering", reference [8], in Chapter 25, page 488.

Let $f:\mathbb{R}n\to \mathbb{R}$ be a $C$ class function and consider the problem of finding a critical point of f, there is, to find a point $x^\in \mathbb{R}n$ such that

$$f\prime (x^)=\mathbf{0}.$$

Fix $k\in \mathbb{N}$ and let $xk\in \mathbb{R}n.$

Define $F:\mathbb{R}n\times \mathbb{R}n\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill F(x,xk)& =& f\left(xk\right)+f\prime \left(xk\right)·(x-xk)+{\displaystyle 12}\left[f\prime \prime \left(xk\right)(x-xk)\right]·(x-xk)\hfill \\ & & +{\displaystyle K}\parallel x-xk\parallel ,\hfill \end{array}$$

(390)

for some $K>$ to be specified.

Let $xk+\in \mathbb{R}n$ be such that

$$\left[{\displaystyle \partial F(x,xk)\partial x}\right]x=xk+=\mathbf{0},$$

so that

$$f\prime \left(xk\right)+f\prime \prime \left(xk\right)(xk+-xk)+K(xk+-xk)=\mathbf{0},$$

that is

$$xk+=xk-(f\prime \prime \left(xk\right)+KId)-f\prime \left(xk\right).$$

Now, assume $x\in \mathbb{R}n$ is such that

$$\parallel f\prime \prime \left(x\right)\parallel \le K^,\forall x\in Br\left(x\right)$$

for some $r>.$

Assume $K^>$ is such that

$$K-K^>.$$

Suppose also $<\alpha <$ is such that

$$f\prime \prime \left(x\right)\ge \alpha (K^+K)Id$$

and

$$\left(-{\displaystyle \alpha }\right)Id\le (f\prime \prime \left(x\right)+KId)-(f\prime \prime \left(y\right)+KId)\equiv H(x,y)\le \left(+{\displaystyle \alpha }\right)Id,$$

$\forall x,y\in Br\left(x\right).$

We recall that

$$\parallel f\prime \prime \left(x\right)\parallel \le K^,$$

so that

$$(K-K^)Id\le KId+f\prime \prime \left(x\right),$$

and therefore

$$(f\prime \prime \left(x\right)+KId)-\le {\displaystyle IdK-k^},$$

$\forall x\in Br\left(x\right).$

Suppose also

$$f\prime \left(x\right)-f\prime \left(y\right)=H(x,y)·(x-y),$$

where $H(x,y)$ is a symmetric matrix such that

$$\le {\displaystyle H(x,y)K-K^}\le \left(-{\displaystyle \alpha }\right)Id,$$

and

$$H(x,y)\ge \alpha (K+K^)Id,$$

$\forall x,y\in Br\left(x\right).$

Assume also $K>$ is such that

$$x\in Br(-\alpha )\left(x\right),$$

where

$$\alpha =\left(-{\displaystyle 34}\alpha \right).$$

Reasoning inductively, suppose

$$x,x,\dots ,xk+\in Br\left(x\right).$$

Observe that

$$xk+-xk+=-(f\prime \prime \left(xk+\right)+KId)-f\prime \left(xk+\right),$$

and

$$xk+-xk=-(f\prime \prime \left(xk\right)+KId)-f\prime \left(xk\right),$$

so that

$$(f\prime \prime \left(xk+\right)+KId)(xk+-xk+)=-f\prime \left(xk+\right),$$

and

$$(f\prime \prime \left(xk\right)+KId)(xk+-xk)=-f\prime \left(xk\right).$$

Hence,

$$\begin{array}{ccc}\hfill (f\prime \prime \left(xk+\right)+KId)(xk+-xk+)& =& (f\prime \prime \left(xk+\right)+KId)(xk+-xk)-f\prime \left(xk+\right)+f\prime \left(xk\right),\hfill \end{array}$$

so that

$$\begin{array}{ccc}\hfill (xk+-xk+)& =& (f\prime \prime \left(xk+\right)+KId)-[(f\prime \prime \left(xk+\right)+KId)(xk+-xk)-f\prime \left(xk+\right)+f\prime \left(xk\right)]\hfill \\ & =& (f\prime \prime \left(xk+\right)+KId)-[(f\prime \prime \left(xk+\right)+KId)(xk+-xk)\hfill \\ & & -H(xk+,xk)(xk+-xk)]\hfill \\ & =& (f\prime \prime \left(xk+\right)+KId)-\left[(f\prime \prime \left(xk+\right)+KId)(xk+-xk)\right]\hfill \\ & & -(f\prime \prime \left(xk+\right)+KId)-H(xk+,xk)(xk+-xk)\hfill \\ & =& [H(xk+,xk)-(f\prime \prime \left(xk+\right)+KId)-H(xk+,xk)](xk+-xk).\hfill \end{array}$$

(391)

Observe that

$$\begin{array}{ccc}\hfill H(xk+,xk)& \ge & \alpha (K^+K)Id\hfill \\ & \ge & \alpha (f\prime \prime \left(xk+\right)+KId),\hfill \end{array}$$

(392)

so that

$$(f\prime \prime \left(xk+\right)+KId)-H(xk+,xk)\ge \alpha Id.$$

Consequently, from such results we may infer that

$$\begin{array}{ccc}& & Id\left(-{\displaystyle 34}\alpha \right)\hfill \\ & =& Id\left(+{\displaystyle \alpha 14}\right)-\alpha Id\hfill \\ & \ge & H(xk+,xk)-(f\prime \prime \left(xk+\right)+KId)-H(xk+,xk)\hfill \\ & \ge & Id\left(-{\displaystyle \alpha 14}\right)-(K-K^)-IdH(xk+,xk)\hfill \\ & \ge & Id\left(-{\displaystyle \alpha 14}\right)-Id\left(-{\displaystyle \alpha 12}\right)\hfill \\ & =& {\displaystyle Id\alpha }\hfill \\ & \ge & .\hfill \end{array}$$

(393)

from such results we may infer that

$$\parallel H(xk+,xk)-(f\prime \prime \left(xk+\right)+KId)-H(xk+,xk)\parallel \le \left(-{\displaystyle \alpha }\right).$$

Defining

$$\alpha =\left(-{\displaystyle \alpha }\right)$$

we have got

$$\parallel xj+-xj+\parallel \le \alpha \parallel xj+-xj\parallel ,\forall j\in \{,\dots ,k\}.$$

Therefore

$$\begin{array}{ccc}\hfill \parallel xj+-xj+\parallel & \le & \alpha \parallel xj+-xj\parallel \hfill \\ & \le & \alpha 02\parallel xj-xj-\parallel \hfill \\ & ⋮& \hfill \\ & \le & \alpha j+\parallel x-x\parallel .\hfill \end{array}$$

(394)

Thus,

$$\begin{array}{ccc}\hfill \parallel xk+-x\parallel & =& \parallel xk+-xk++xk+-\dots -x+x-x\parallel \hfill \\ & \le & \parallel xk+-xk+\parallel +\parallel xk+-xk\parallel +\dots +\parallel x-x\parallel \hfill \\ & \le & \sum j=k+\alpha j\parallel x-x\parallel \hfill \\ & \le & \sum j=\infty \alpha j\parallel x-x\parallel \hfill \\ & =& {\displaystyle \alpha -\alpha }\parallel x-x\parallel .\hfill \end{array}$$

(395)

Therefore

$$\begin{array}{ccc}\hfill \parallel xk+-x\parallel & \le & \parallel xk+-x+x-x\parallel \hfill \\ & \le & \parallel xk+-x\parallel +\parallel x-x\parallel \hfill \\ & \le & {\displaystyle \alpha -\alpha }\parallel x-x\parallel +\parallel x-x\parallel \hfill \\ & =& {\displaystyle -\alpha }\parallel x-x\parallel \hfill \\ & \le & {\displaystyle -\alpha }(-\alpha )r\hfill \\ & =& r.\hfill \end{array}$$

(396)

Summarizing,

$$\parallel xk+-x\parallel <r,$$

so that

$$xk+\in Br\left(x\right).$$

The induction is complete, so that

$$xk\in Br\left(x\right),\forall k\in \mathbb{N}.$$

From such results we have also obtained

$$\parallel xk+-xk+\parallel \le \alpha \parallel xk+-xk\parallel ,\forall k\in \mathbb{N}.$$

Thus, from these results and the Banach fixed point theorem, there exists $x^\in Br¯\left(x\right)$ such that

$$\underset{k\to \infty }{lim}xk=x^.$$

Hence,

$$\begin{array}{ccc}\hfill \mathbf{0}& =& \underset{k\to \infty }{lim}xk+-xk\hfill \\ & =& \underset{k\to \infty }{lim}(-f\prime \prime \left(xk\right)+KId)-f\prime \left(xk\right)\hfill \\ & =& -(f\prime \prime (x^)+KId)-f\prime (x^).\hfill \end{array}$$

(397)

Since $det(f\prime \prime (x^)+KId)-\ne ,$ from this last equation we obtain

$$f\prime (x^)=\mathbf{0}.$$

The objective of this section is complete.

58.1. Applications to a Ginzburg-Landau type equation

Let $\Omega =[,]\subset \mathbb{R}$ and consider a functional $F:V\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill F\left(u\right)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle \alpha }\int \Omega udx\hfill \\ & & -{\displaystyle \beta }\int \Omega udx-\langle u,f\rangle L,\hfill \end{array}$$

(398)

where $V=H01\left(\Omega \right)$, $f\in L\left(\Omega \right)$, $\alpha >,\beta >$ and $\gamma >.$

Let $u\in H01\left(\Omega \right)$ and $\phi \in H01\left(\Omega \right).$

Observe that

$$\begin{array}{ccc}\hfill \delta F(u;\phi )& =& \gamma \int \Omega \nabla u·\nabla \phi dx\hfill \\ & & \alpha \int \Omega u\phi dx-\beta \int \Omega u\phi dx\hfill \\ & & -\langle \phi ,f\rangle L.\hfill \end{array}$$

(399)

Consider the problem of finding $u\in H01\left(\Omega \right)$ such that

$$\delta F(u;\phi )=,\forall \phi \in H01\left(\Omega \right).$$

Fixing $N\in \mathbb{N},$ consider now a mesh in finite differences for $\Omega$, where we define $d=/N$ and the related grid

$$\Omega N=\{(j/N,k/N)\forall j,k\in \{,,\dots ,N\}.$$

Denoting by $V$ the finite-dimensional space in a finite diferences context corresponding to V and considering the functional F, we assume there exist $u\in V$, the corresponding $uN\in VN$ and $r>$ such that the hypotheses indicated in the last section also for the corresponding function $FN:VN\to \mathbb{R}$ are satisfied so that, as developed in such a previous section, we may obtain a solution $uN:\Omega \to \mathbb{R}$ such that

$$FN\prime \left(uN\right)=\mathbf{0}$$

that is,

$$-\gamma \nabla NuN+\alpha uN-\beta uN-fN=\mathbf{0},$$

where $\nabla$ is the finite dimensional operator corresponding to the Laplace operator $\nabla .$

Also,

$$FN\prime \prime \left(u\right)=-\gamma \nabla N+\alpha \mathrm{diag}\left(u\right)-\beta Id,$$

so that

$$\begin{array}{ccc}\hfill FN\prime \left(u\right)-FN\prime \left(u\right)& =& -\gamma \nabla Nu+\alpha u13-\beta u-fN\hfill \\ & & -\left(-\gamma \nabla Nu+\alpha u23-\beta u-fN\right)\hfill \\ & =& -\gamma \nabla N(u-u)+\alpha (u13-u23)-\beta (u-u)\hfill \\ & =& -\gamma \nabla N(u-u)+\alpha (u˜)(u-u)-\beta (u-u)\hfill \\ & =& \left(-\gamma \nabla N+\alpha \mathrm{diag}u˜-\beta Id\right)(u-u)\hfill \\ & =& FN\prime \prime (u˜)(u-u)\hfill \end{array}$$

(400)

where $\left(u\right)j\le u˜j\le \left(u\right)j$, $\forall u,u\in Br\left(u\right).$

From such results, concerning the notation of the last section, we may infer that

$$\begin{array}{ccc}\hfill H(u,u)& =& FN\prime \prime (u˜(u,u))\hfill \\ & =& -\gamma \nabla N+\alpha \mathrm{diag}\left\{\left[(u˜)(u,u)\right]\right\}-\beta Id.\hfill \end{array}$$

(401)

Now fix $M,N\in \mathbb{N}.$

Observe that

$$-\gamma \nabla NuN+\alpha uN-\beta uN-fN=\mathbf{0},$$

and

$$-\gamma \nabla MuM+\alpha uM-\beta uM-fM=\mathbf{0}.$$

At this point, denoting $uN=\left\{uj,kN\right\},$ we define

$$u˜N(x,y)=\left\{\begin{array}{cc}uj,kN,\mathrm{if}(x,y)\in ((j-)d,jd]\times ((k-)d,kd],\hfill & \\ \forall j,k\in \{,\dots ,N\}.\hfill \end{array}\right.$$

(402)

We also denote for a not relabeled operator $\nabla$,

$$\nabla N(u˜N(x,y))=\left\{\begin{array}{c}{\displaystyle uj+,kN-uj,kN+uj-,kNd}+{\displaystyle uj,k+N-uj,kN+uj,k-Nd},\hfill \\ \mathrm{if}(x,y)\in \left(\right(j-)d,jd]\times \left(\right(k-)d,kd],\hfill & \\ \forall j,k\in \{,\dots ,N\}.\hfill \end{array}\right.$$

(403)

and

$$\nabla N(u˜N(x,y))=[\nabla u˜N](x-d,y-d),\mathrm{if}x\in (-d,]\mathrm{or}y\in (-d,].$$

Moreover, we define

$$uN(x,y)=(\nabla )-(\nabla N(u˜N(x,y))),\mathrm{in}\Omega .$$

Observe that

$$\begin{array}{ccc}\hfill -\gamma \nabla uN& =& -\gamma \nabla Nu˜N\hfill \\ & =& -\alpha (u˜N)+\beta u˜N+fN\hfill \\ & =& -\alpha \left(uN\right)+\beta uN+fN\hfill \\ & & -\alpha [(u˜N)-\left(uN\right)]+\beta (u˜N-uN),\hfill \end{array}$$

(404)

Similarly, we may obtain

$$\begin{array}{ccc}\hfill -\gamma \nabla uM& =& -\gamma \nabla Mu˜N\hfill \\ & =& -\alpha \left(uM\right)+\beta uM+fM\hfill \\ & & -\alpha [(u˜M)-\left(uM\right)]+\beta (u˜M-uM).\hfill \end{array}$$

(405)

Consequently, from such results, we have

$$\begin{array}{ccc}& & uN-uM\hfill \\ & =& (-\gamma \nabla +\alpha (u^N,M)-\beta Id)-\hfill \\ & & \times \left[fN-fM-\alpha (u^N)(uN-u˜N)+\alpha (u^M)(uM-u˜M)\right.\hfill \\ & & \left.+\beta (uN-u˜N)-\beta (uM-u˜M)\right]\hfill \end{array}$$

(406)

where $u$ is on the line connecting $u$, $u$ and $u$ is on the line connecting $u$ and $u$ and $u$ is on the line connecting $u$ and $uM.$

From these results, we obtain

$$\begin{array}{ccc}& & \parallel uN-uM\parallel ,,\Omega \hfill \\ & =& \parallel (-\gamma \nabla +\alpha (u^N,M)-\beta Id)-\parallel \hfill \\ & & \times \left[\parallel fN-fM\parallel ,,\Omega +\alpha \parallel (u^N)\parallel ,,\Omega \parallel (uN-u˜N)\parallel ,,\Omega +\alpha \parallel (u^M)\parallel ,,\Omega \parallel (uM-u˜M)\parallel ,,\Omega \right.\hfill \\ & & \left.+\beta \parallel (uN-u˜N)\parallel ,,\Omega +\beta \parallel (uM-u˜M)\parallel ,,\Omega \right]\hfill \\ & \le & \left[K\parallel fN-fM\parallel ,,\Omega +K\parallel (uN-u˜N)\parallel ,,\Omega +K\parallel (uM-u˜M)\parallel ,,\Omega \right]\hfill \end{array}$$

(407)

for some appropriate constants $K>,K>.$

Let $\epsilon >$.

Observe that there exists $N\in \mathbb{N}$ such that if $M,N>N$, then

$$\parallel fN-fM\parallel ,,\Omega <{\displaystyle \epsilon K},$$

$$\parallel (uN-u˜N)\parallel ,,\Omega \le {\displaystyle \epsilon K},$$

and

$$\parallel (uM-u˜M)\parallel ,,\Omega \le {\displaystyle \epsilon K},$$

so that,

$$\parallel uN-uM\parallel ,,\Omega <\epsilon .$$

Therefore, $\left\{uN\right\}$ is a Cauchy sequence in $H01\left(\Omega \right)$ so that there exists $u^\in H01\left(\Omega \right)$ such that

$$uN\to u^,\mathrm{strongly}\mathrm{in}H01\left(\Omega \right).$$

Let $\phi \in H01\left(\Omega \right).$

From such results and from the Sobolev Imbedding theorem, we may infer that

$$\begin{array}{ccc}\hfill & =& \underset{N\to \infty }{lim}\left(\gamma \langle \nabla uN,\nabla \phi \rangle L\right.\hfill \\ & & +\alpha \langle \left(uN\right),\phi \rangle L-\beta \langle uN,\phi \rangle L\hfill \\ & & \left.-\langle fN,\phi \rangle L\right)\hfill \\ & =& \left(\gamma \langle \nabla u^,\nabla \phi \rangle L\right.\hfill \\ & & +\alpha \langle u^03,\phi \rangle L-\beta \langle u^,\phi \rangle L\hfill \\ & & \left.-\langle f,\phi \rangle L\right).\hfill \end{array}$$

(408)

Thus,

$$\gamma \langle \nabla u^,\nabla \phi \rangle L+\alpha \langle u^03,\phi \rangle L-\beta \langle u^,\phi \rangle L-\langle f,\phi \rangle L=,$$

$\forall \phi \in H01\left(\Omega \right).$

From this result we may infer that $u$ is a weak solution of equation $F\prime (u^)=\mathbf{0}.$

59. On the convergence of the Newton’s method combined with a proximal formulation for a general parabolic equation

Let $\Omega \subset \mathbb{R}m$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider the parabolic non-linear equation

$$\left\{\begin{array}{cc}{\displaystyle \partial u\partial t}=\epsilon \nabla u+g\left(u\right)+f,\hfill & \mathrm{in}\Omega \times (,T),\hfill \\ u(x,)=u^,\mathrm{in}\Omega ,\hfill & \\ u=,\mathrm{on}\partial \Omega \times [,T].\hfill \end{array}\right.$$

(409)

Here $\epsilon >$, $f\in L([,T],W,\left(\Omega \right))\cap L\infty (\Omega \times [,T]),$ $u^\in H01\left(\Omega \right)\cap L\infty \left(\Omega \right),$ where t denotes time and $[,T]$ is a time interval.

Also $g:\mathbb{R}\to \mathbb{R}$ is a continuous function neither necessarily linear nor convex.

We assume there exists $r>$ such that

$$\parallel g\prime \left(u\right)\parallel \infty \le K,$$

and

$$\parallel g\left(u\right)\parallel \infty \le K^$$

$\forall u\in Br(u^),$ for some $K>$ and $K^>.$

Here

$$Br(u^)=\{u\in H01\left(\Omega \right):\parallel u-u^\parallel ,,\Omega <r\}.$$

Moreover, fixing $N\in \mathbb{N}$ and defining

$$\Delta tN={\displaystyle TN},$$

in a partial finite differences context, discretizing in t consider the approximate equation system

$${\displaystyle un+-un\Delta tN}=\epsilon \nabla un++g\left(un+\right)+fn,\mathrm{in}\Omega ,$$

$\forall n\in \{,,\dots ,N-\}.$

Fix $M\in \mathbb{N}$. In a finite elements context for the variable $x\in \mathbb{R}n$, denoting $hM=L/M$, for an appropriate fixed $L>$ consider a mesh with a concerning thickness $h$ and a related solution $u$ of the following system

$${\displaystyle un+M-unM\Delta tN}=\epsilon \nabla Mun+M+g\left(un+M\right)+fnM,\mathrm{in}\Omega ,$$

$\forall n\in \{,,\dots ,N-\}.$

Here $\nabla$ is the operator in a finite elements context corresponding to the Laplace operator $\nabla$.

We highlight in the next lines, as the meaning is clear, we may denote simply $\nabla M=\nabla .$

Observe that there exists a not relabeled $r>$, $K>$ and $K>$ such that

$$\parallel g\prime \left(uM\right)\parallel \le K,$$

and

$$-KId\le -\epsilon \nabla M-g\prime \left(uM\right)\le KId,$$

$\forall uM\in Br(u^M),$$\forall M\in \mathbb{N}.$

Observe also that there exists $N\in \mathbb{N}$ such that if $N>N$, then

$$<{\displaystyle K+K\Delta tNK+-K\Delta tN}<.$$

Indeed, we may find $\alpha \in \mathbb{R}$ such that

$$<{\displaystyle K+K\Delta tNK+-K\Delta tN}<\alpha <,\forall N>N.$$

Let $MN\subset \mathbb{N}$ be a sequence such that $MN<MN+,\forall N\in \mathbb{N}.$

Fix $N>N$.

For $n=$, we are going to calculate $u=uMn,N$ though the following iterations, which already include a proximal formulation and concerning linearization.

Set $u10=u^=u^MN,N,$

Having $u$, let $u$ be such that

$$\begin{array}{ccc}\hfill uk+-u^& =& \epsilon \nabla uk+\Delta tN+g\left(uk\right))\Delta tN\hfill \\ & & +g\prime \left(uk\right)(uk+-uk)\Delta tN+f\Delta tN-K(uk+-uk),\hfill \end{array}$$

(410)

Here we suppose $K\equiv KnN>$ is large enough so that

$$u11\in B{\displaystyle r(-\alpha )N}\left(u\right).$$

Reasoning inductively, suppose $u10,u11,\dots ,uk+\in B{\displaystyle rN}\left(u\right),$ and observe that

$$\begin{array}{ccc}& & uk+-u^-\epsilon \nabla uk+\Delta tN-g\left(uk\right))\Delta tN\hfill \\ & & -g\prime \left(uk\right)(uk+-uk)\Delta tN-f\Delta tN+K(uk+-uk)\hfill \\ & =& ,\hfill \end{array}$$

(411)

and

$$\begin{array}{ccc}& & uk+-u^-\epsilon \nabla uk+\Delta tN-g\left(uk+\right))\Delta tN\hfill \\ & & -g\prime \left(uk+\right)(uk+-uk+)\Delta tN-f\Delta tN+K(uk+-uk+)\hfill \\ & =& ,\hfill \end{array}$$

(412)

so that for an appropriate $u$,

$$\begin{array}{ccc}& & \left(Id-\epsilon \nabla N\Delta tN-g\prime \left(uk+\right)\Delta tN+KId\right)(uk+-uk+)\hfill \\ & =& \left((-g\prime (u˜k)+g\prime \left(uk+\right))\Delta tN+KId\right)\left(uk+-uk\right).\hfill \end{array}$$

(413)

Hence,

$$\begin{array}{ccc}& & \parallel uk+-uk+\parallel \hfill \\ & \le & ∥\left(Id-\epsilon \nabla N\Delta tN-g\prime \left(uk+\right)\Delta tN+KId\right)-\left((-g\prime (u˜k)+g\prime \left(uk+\right))\Delta tN+KId\right)∥\hfill \\ & & \times ∥uk+-uk∥\hfill \\ & \le & {\displaystyle K+K\Delta tNK+-K\Delta tN}∥uk+-uk∥\hfill \\ & \le & \alpha ∥uk+-uk∥.\hfill \end{array}$$

(414)

Thus, we have got

$$\parallel uj+-uj+\parallel \le \alpha \parallel uj+-uj\parallel ,\forall j\in \{,\dots ,k\}.$$

Therefore

$$\begin{array}{ccc}\hfill \parallel uj+-uj+\parallel & \le & \alpha \parallel uj+-uj\parallel \hfill \\ & \le & \alpha 02\parallel uj-uj-\parallel \hfill \\ & ⋮& \hfill \\ & \le & \alpha j+\parallel u11-u10\parallel .\hfill \end{array}$$

(415)

Thus,

$$\begin{array}{ccc}\hfill \parallel uk+-u11\parallel & =& \parallel uk+-uk++uk+-\dots -u12+u12-u11\parallel \hfill \\ & \le & \parallel uk+-uk+\parallel +\parallel uk+-uk\parallel +\dots +\parallel u12-u11\parallel \hfill \\ & \le & \sum j=k+\alpha j\parallel u11-u10\parallel \hfill \\ & \le & \sum j=\infty \alpha j\parallel u11-u10\parallel \hfill \\ & =& {\displaystyle \alpha -\alpha }\parallel u11-u10\parallel .\hfill \end{array}$$

(416)

Therefore

$$\begin{array}{ccc}\hfill \parallel uk+-u10\parallel & \le & \parallel uk+-u11+u11-u10\parallel \hfill \\ & \le & \parallel uk+-u11\parallel +\parallel u11-u10\parallel \hfill \\ & \le & {\displaystyle \alpha -\alpha }\parallel u11-u10\parallel +\parallel u11-u10\parallel \hfill \\ & =& {\displaystyle -\alpha }\parallel u11-u10\parallel \hfill \\ & <& {\displaystyle -\alpha }(-\alpha ){\displaystyle rN}\hfill \\ & =& {\displaystyle rN}.\hfill \end{array}$$

(417)

Summarizing,

$$\parallel uk+-u10\parallel <{\displaystyle rN},$$

so that

$$uk+\in B{\displaystyle rN}\left(u01\right).$$

The induction is complete, so that

$$uk\in B{\displaystyle rN}\left(u01\right),\forall k\in \mathbb{N}.$$

From such results we have also obtained

$$\parallel uk+-uk+\parallel \le \alpha \parallel uk+-uk\parallel ,\forall k\in \mathbb{N}.$$

Thus, from these results and the Banach fixed point theorem, there exists $u=uMN,N\in B{\displaystyle rN}\left(u01\right)$ such that

$$\underset{k\to \infty }{lim}uk=u=uMN,N.$$

$$\begin{array}{ccc}\hfill \mathbf{0}& =& \underset{k\to \infty }{lim}\left(uk+-u^\right.\hfill \\ & & -\epsilon \nabla uk+\Delta tN-g\left(uk\right))\Delta tN\hfill \\ & & \left.-g\prime \left(uk\right)(uk+-uk)\Delta tN-f\Delta tN+K(uk+-uk)\right)\hfill \\ & =& u-u^-\epsilon \nabla u\Delta tN-g\left(u\right)\Delta tN-f\Delta tN,\hfill \end{array}$$

(418)

so that

$${\displaystyle u-u^\Delta tN}=\epsilon \nabla u+g\left(u\right)+f,\mathrm{in}\Omega ,$$

Reasoning inductively again having $u\in B{\displaystyle rN}(u^)$ and $uj\in B{\displaystyle rN}\left(uj-\right),\forall j\in \{,\dots ,n\}$ similarly as we have obtained $u$ in the last lines, we may obtain

$$un+=un+MN,N\in B{\displaystyle rN}\left(un\right),$$

such that

$${\displaystyle un+-un\Delta tN}=\epsilon \nabla un++g\left(un+\right)+fn,\mathrm{in}\Omega .$$

The induction on n is also complete.

Fix $n\in \{,\dots ,N-\}$.

Observe that

$$\begin{array}{ccc}\hfill \parallel un-u^\parallel & =& \parallel un-un-+un--un-+\dots -u+u-u\parallel \hfill \\ & \le & \parallel un-un-\parallel +\dots +\parallel u-u^\parallel \hfill \\ & \le & {\displaystyle nN}r\hfill \\ & <& r\hfill \end{array}$$

(419)

Summarizing $un\in Br(u^),\forall n\in \{,,\dots ,N-\}.$

From these results, denoting now more generically $un\equiv unMN,N=unN$, we may infer that there exists $K>$ such that

$$\parallel ujN\parallel \le K,\forall j\in \{,,\dots ,N\},\forall N\in \mathbb{N}.$$

With a completely analogous reasoning, we may obtain that

$$\parallel ujN\parallel ,,\Omega \le K^,\forall j\in \{,,\dots ,N\},\forall N\in \mathbb{N},$$

for some $K^>.$

Define now

$$uN(x,t)=unN\left(x\right)\left(n+-{\displaystyle t\Delta tN}\right)+un+N\left(x\right)\left({\displaystyle t\Delta tN}-n\right),$$

if $t\in [n\Delta tN,(n+)\Delta tN],\forall (x,t)\in \Omega \times [,T].$

Observe that

$$uN(x,t)=unN\left(x\right),\mathrm{if}t=n\Delta tN,\forall n\in \{,,\dots ,N\},$$

and

$$\begin{array}{ccc}\hfill {\displaystyle \partial uN(x,t)\partial t}& =& {\displaystyle un+N-unN\Delta tN}\hfill \\ & =& \epsilon \nabla un+N+g\left(un+N\right)+fn,\hfill \end{array}$$

(420)

if $t\in [n\Delta tN,(n+)\Delta tN],\forall (x,t)\in \Omega \times [,T].$

Fix $\phi \in Cc\infty \left(\Omega \right).$

Thus, fixing $t\in [n\Delta tN,(n+)\Delta tN],$ we have

$$\begin{array}{ccc}\hfill \left|〈{\displaystyle \partial uN\partial t},\phi 〉L\right|& \le & \epsilon |\langle \nabla un+N,\nabla \phi \rangle L|+|\langle g\left(un+N\right),\phi \rangle L|\hfill \\ & & +|\langle \phi ,fn\rangle L|\hfill \\ & \le & \epsilon \parallel un+N\parallel ,,\Omega \parallel \phi \parallel ,,\Omega +K18\parallel un+N\parallel ,,\Omega \parallel \phi \parallel ,,\Omega +K\parallel \phi \parallel ,,\Omega \hfill \\ & \le & K\parallel \phi \parallel ,,\Omega ,\forall \phi \in Cc\infty \left(\Omega \right),\hfill \end{array}$$

(421)

for some appropriate $K>.$

Since $\phi \in Cc\infty \left(\Omega \right)$ is arbitrary, we may conclude that

$$∥{\displaystyle \partial uN\partial t}∥H-\left(\Omega \right)\le K,\forall N>N,$$

uniformly in t on $[,T],$ for some appropriate constant $K>.$

Also, from the definition of $u$ we have that there exists $K>$ such that

$$\parallel uN\parallel ,,\Omega \le K,\forall N\in \mathbb{N}$$

also uniformly in t on $[,T].$

From such results, there exist $u\in L([,T],H01\left(\Omega \right))$ and $v\in L([,T];H-\left(\Omega \right))$ such that

$$uN\rightharpoonup u,\mathrm{weakly}\mathrm{in}L((,T);W,\left(\Omega \right)),$$

and

$${\displaystyle \partial uN\partial t}\rightharpoonup v,\mathrm{weakly}-\mathrm{star}\mathrm{in}L([,T],H-\left(\Omega \right)),$$

so that we may easily obtain

$$v={\displaystyle \partial u\partial t}$$

in a distributional sense.

At this point, we provide more details about this last result.

Fix $t\in (,T)$. Thus, there exists $n\in \{,,\dots ,N-\}$ such that $t\in [n\Delta tN,(n+)\Delta tN]$.

Let $\phi \in Cc\infty (\Omega \times (,T)).$

From this, we may infer that

$$\begin{array}{ccc}& & \int \Omega {\displaystyle \partial uN\partial t}\phi (x,t)dx\hfill \\ & =& \int \Omega {\displaystyle un+N-unN\Delta tN}\phi (x,t)dx\hfill \\ & \le & \epsilon \int \Omega |\nabla un+N·\nabla \phi |dx\hfill \\ & & +\int \Omega |g\left(un+N\right)\phi (x,t)|dx+\int \Omega \left|fn\phi \right|dx\hfill \\ & \le & (K\parallel un+N\parallel ,,\Omega +K20)\parallel \phi \parallel ,,\Omega \hfill \\ & \le & K\parallel \phi \parallel ,,\Omega ,\hfill \end{array}$$

(422)

for some appropriate constants $K>,K>,K20>.$

Hence,

$$\begin{array}{ccc}& & \int T\int \Omega {\displaystyle \partial uN\partial t}\phi (x,t)dxdx\hfill \\ & \le & K\int \Omega \parallel \phi \parallel ,,\Omega dt\hfill \\ & \le & K19\parallel \phi \parallel ,,\Omega \times (,T),\hfill \end{array}$$

(423)

for some appropriate $K19>$.

Since such a $\phi \in Cc\infty (\Omega \times (,T))$ is arbitrary, we may infer that

$$∥{\displaystyle \partial uN\partial t}∥H-(\Omega \times (,T))\le K15,$$

for $N\in \mathbb{N}$, for some $K15>.$

From such a result and from the Banach-Alaoglu Theorem, there exists $v\in H-(\Omega \times (,T))$ such that, up to a not relabeled subsequence

$${\displaystyle \partial uN\partial t}\rightharpoonup v,\mathrm{weakly}-\mathrm{star}\mathrm{in}H-(\Omega \times (,T)).$$

Therefore,

$$\int T\int \Omega {\displaystyle \partial uN\partial t}\phi dxdt\to \int T\int \Omega v\phi dxdt,$$

as $N\to \infty ,$$\forall \phi \in H01(\Omega \times (,T))$.

On the other hand

$$\parallel uN\parallel ,,\Omega \times (,T)\le K16,$$

$\forall N\in \mathbb{N}$, for some $K16>.$

From this and the Kakutani Theorem, there exists $u\in L(\Omega \times (,T))$ such that, up to a not relabeled subsequence,

$$uN\rightharpoonup u,\mathrm{weakly}\mathrm{in}L(\Omega \times (,T)).$$

Now fix again $\phi \in Cc\infty (\Omega \times (,T)).$

Observe that

$$\begin{array}{ccc}\hfill \int T\int \Omega u\phi tdxdt& =& \underset{N\to \infty }{lim}\int T\int \Omega uN\phi tdxdt\hfill \\ & =& -\underset{N\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uN\partial t}\phi dxdt\hfill \\ & =& -\int T\int \Omega v\phi dxdt,\hfill \end{array}$$

(424)

Since such a $\phi \in Cc\infty (\Omega \times (,T))$ is arbitrary, we may infer that

$$v={\displaystyle \partial u\partial t}$$

in a distributional sense.

Moreover, from such results we may also obtain, again up to a subsequence,

$$\underset{N\to \infty }{lim}\int \Omega {\displaystyle \partial uN\partial t}\phi dx=\int \Omega {\displaystyle \partial u\partial t}\phi dx,$$

$\forall \phi \in H01\left(\Omega \right).$

Observe also that, as a consequence of the Rellich-Kondrashov theorem, through appropriate subsequences, we have

$$uNk\left(t\right)\to u(x,t),\mathrm{strongly}\mathrm{in}L\left(\Omega \right),\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

so that, up to subsequences,

$$uNk\left(t\right)(x,t)\to u(x,t),\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

Here we emphasise the sequence $\left\{Nk\left(t\right)\right\}\subset \mathbb{N}$ may depends on t.

Since g is continuous we have that

$$g\left(uNk\left(t\right)(x,t)\right)\to g\left(u(x,t)\right),\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

Fix $t\in (,T).$

Let $\epsilon >$. From the Egorov Theorem, there exists a closed set F such that $m(\Omega \setminus F)\le \epsilon$ and $k\in \mathbb{N}$ such that if $k>k$, then

$$|g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)|\le \epsilon ,\mathrm{for}\mathrm{almost}\mathrm{all}x\in F.$$

Let $\phi \in Cc\infty \left(\Omega \right).$ Observe now that

$$\begin{array}{ccc}& & \left|\int \Omega (g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right))\phi dx\right|\hfill \\ & \le & \int \Omega |g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |dx\hfill \\ & =& \int F|g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |dx+\int \Omega \setminus F|g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |dx\hfill \\ & \le & \int F\epsilon \parallel \phi \parallel \infty dx+\int \Omega |g\left(uNk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |\chi \Omega \setminus Fdx\hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+(\parallel g\left(uNk\left(t\right)\right)\parallel ,,\Omega +\parallel g\left(u\right)\parallel ,,\Omega )\parallel \phi \parallel ,,\Omega \parallel \chi \Omega \setminus F\parallel ,,\Omega \hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+K21\parallel \phi \parallel ,,\Omega m(\Omega \setminus F)/\hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+K21\parallel \phi \parallel ,,\Omega \epsilon /,\forall k>k,\hfill \end{array}$$

(425)

for some appropriate constant $K21>$ which does not depend on t.

Since such a $\epsilon >$ is arbitrary, we may infer that

$$\int \Omega g\left(uNk\left(t\right)\right)\phi dx\to \int \Omega g\left(u\right)\phi dx,\mathrm{as}k\to \infty ,$$

$\forall \phi \in Cc\infty \left(\Omega \right).$

From such results, we have

$$\begin{array}{ccc}\hfill & =& \underset{k\to \infty }{lim}\left(\int \Omega {\displaystyle \partial uNk\left(t\right)\partial t}\phi dx+\epsilon \int \Omega \nabla uNk\left(t\right)·\nabla \phi dx\right.\hfill \\ & & \left.-\int \Omega g\left(uNk\left(t\right)\right)\phi dx-\int \Omega fNk\left(t\right)\phi dx\right)\hfill \\ & =& \int \Omega {\displaystyle \partial u\partial t}\phi dx+\epsilon \int \Omega \nabla u·\nabla \phi dx\hfill \\ & & -\int \Omega g\left(u\right)\phi dx-\int \Omega f\phi dx.\hfill \end{array}$$

(426)

so that, from this and by the density of $Cc\infty \left(\Omega \right)$ in $H01\left(\Omega \right)$, we have got

$$\begin{array}{ccc}& & \int \Omega {\displaystyle \partial u\partial t}\phi dx+\epsilon \int \Omega \nabla u·\nabla \phi dx\hfill \\ & & -\int \Omega g\left(u\right)\phi dx-\int \Omega f\phi dx=,\forall \phi \in H01\left(\Omega \right),\hfill \end{array}$$

(427)

$\mathrm{a}.\mathrm{e}.\mathrm{on}[,T].$

Observe now that

$$\partial (\Omega \times (,T))=\left(\partial \Omega \times [,T]\right)\cup \left(\partial [,T]\times \Omega ¯\right).$$

Let $\phi \in Cc\infty (\Omega \times (,T)).$

Hence

$$\underset{N\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uN\partial t}\phi dxdt=\int T\int \Omega {\displaystyle \partial u\partial t}\phi dxdt.$$

From this, since $Cc\infty (\Omega \times (,T))$ is dense $L(\Omega \times (,T))$ we may infer that

$$\underset{N\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uN\partial t}\phi dxdt=\int T\int \Omega {\displaystyle \partial u\partial t}\phi dxdt,$$

$\forall \phi \in L(\Omega \times (,T)).$

Let $\phi \in C\infty (\Omega \times [,T])$ such that

$$\phi (x,T)=,\mathrm{in}\Omega .$$

From such results, we may obtain

$$\begin{array}{ccc}& & \underset{N\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uN\partial t}\phi dxdt\hfill \\ & =& \underset{N\to \infty }{lim}\left(-\int T\int \Omega uN{\displaystyle \partial \phi \partial t}dxdt-\int \Omega uN(x,)\phi (x,)dx\right)\hfill \\ & =& -\int T\int \Omega u{\displaystyle \partial \phi \partial t}dxdt-\int \Omega u(x,)\phi (x,)dx.\hfill \end{array}$$

(428)

However, since $uN\rightharpoonup u,\mathrm{weakly}\mathrm{in}L(\Omega \times (,T))$, we obtain

$$\underset{N\to \infty }{lim}\int T\int \Omega uN{\displaystyle \partial \phi \partial t}dxdt=\int T\int \Omega u{\displaystyle \partial \phi \partial t}dxdt.$$

From these last results, we may infer that

$$\begin{array}{ccc}\hfill \int \Omega u^\phi (x,)dx& =& \underset{N\to \infty }{lim}\int \Omega uN(x,)\phi (x,)dx\hfill \\ & =& \int \Omega u(x,)\phi (x,)dx,\hfill \end{array}$$

(429)

so that

$$\int \Omega u^\left(x\right)\phi (x,)dx=\int \Omega u(x,)\phi (x,)dx,$$

$\forall \phi \in C\infty (\Omega \times [,T])$ such that $\phi (x,T)=,\mathrm{in}\Omega .$

Therefore, we may infer that $u(x,)=u^\left(x\right)$ in this specified weak sense.

Similarly, it may be proven that

$$u=,\mathrm{on}\partial \Omega \times [,T],$$

in an appropriate weak sense.

Hence, we have obtained that $u$ is a solution, in a weak sense, of the parabolic non-linear equation in question.

60. More results on the convergence of Newton’s method combined with a proximal approach for a parabolic equation

Let $\Omega \subset \mathbb{R}m$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider the parabolic non-linear equation

$$\left\{\begin{array}{cc}{\displaystyle \partial u\partial t}=\epsilon \nabla u+g\left(u\right)+f,\hfill & \mathrm{in}\Omega \times (,T),\hfill \\ u(x,)=u^,\mathrm{in}\Omega ,\hfill & \\ u=,\mathrm{on}\partial \Omega \times [,T].\hfill \end{array}\right.$$

(430)

Here $\epsilon >$, $f\in L([,T],W,\left(\Omega \right))\cap L\infty (\Omega \times [,T]),$ $u^\in H01\left(\Omega \right)\cap L\infty \left(\Omega \right),$ where t denotes time and $[,T]$ is a time interval.

Also $g:\mathbb{R}\to \mathbb{R}$ is a continuous function neither necessarily linear nor convex.

We assume there exists $r>$ such that

$$\parallel g\prime \left(u\right)\parallel \infty \le K,$$

and

$$\parallel g\left(u\right)\parallel \infty \le K^$$

$\forall u\in Br(u^),$ for some $K>$ and $K^>.$

Here

$$Br(u^)=\{u\in H01\left(\Omega \right):\parallel u-u^\parallel ,,\Omega <r\}.$$

About the references, the main ones are [6,22]. Other related results may be found in [7,8].

Consider the operator

$$-\nabla :D\subset L\left(\Omega \right)\to L\left(\Omega \right),$$

where

$$D=H01\left(\Omega \right)\cap H\left(\Omega \right)$$

is dense in $L\left(\Omega \right)$ concerning the $L\left(\Omega \right)$ norm.

From the standard spectral analysis theory (please see [8], for details), denoting the spectral decomposition of $-\nabla$ by

$$\left\{E\right(\lambda ),\lambda \in [,+\infty \left)\right\},$$

we may obtain

$$-\nabla u=\int \infty \lambda dE\left(\lambda \right)u,$$

$\forall u\in D.$

Fix $M\in \mathbb{N}$ and define

$$(-\nabla )M=\int M\lambda dE\left(\lambda \right),$$

which is a bounded operator in $D.$

Also from the standard spectral theory, we have

$$\underset{M\to \infty }{lim}\parallel (-\nabla )Mu-(-\nabla )u\parallel ,=,\forall u\in D.$$

Similarly, defining the self-adjoint densely defined in $L\left(\Omega \right),$

$$Bj:H01\left(\Omega \right)\to L\left(\Omega \right),$$

by

$$Bj\left(u\right)=i{\displaystyle \partial u\partial xj},$$

Denoting by

$$\{Ej\left(\lambda \right),\lambda \in \mathbb{R}\},$$

the spectral decomposition of $B$ and fixing again $M\in \mathbb{N}$, we define

$$\left(Bj\right)M=-i\int -MM\lambda dEj\left(\lambda \right),$$

so that

$$\left(Bj\right)M\left(u\right)=-i\int -MM\lambda dEj\left(\lambda \right)u,\forall u\in H01\left(\Omega \right).$$

We may also infer that

$$\parallel \left(Bj\right)M\left(u\right)-(-i)Bj\left(u\right)\parallel ,\to ,\mathrm{as}M\to \infty ,$$

$\forall u\in H01\left(\Omega \right).$

Finally, we define

$$\nabla M=(\left(B\right)M,\dots ,\left(Bm\right)M),$$

which is a bounded linear operator from $H01\left(\Omega \right)$ into $L\left(\Omega \right).$

60.1. The main result

Fix again $M\in \mathbb{N}$ (a new value).

Moreover, fixing $N\in \mathbb{N}$ and defining

$$\Delta tN={\displaystyle TN},$$

in a partial finite differences context, discretizing in t consider the approximate equation system

$${\displaystyle un+-un\Delta tN}=\epsilon (\nabla )Mun++g\left(un+\right)+fn,\mathrm{in}\Omega ,$$

$\forall n\in \{,,\dots ,N-\}.$

Denoting $un=unM$ we may write

$${\displaystyle un+M-unM\Delta tN}=\epsilon \nabla Mun+M+g\left(un+M\right)+fn,\mathrm{in}\Omega ,$$

$\forall n\in \{,,\dots ,N-\}.$

Observe that there exists a not relabeled $r>$, $K=K\left(M\right)>$ and $K>$ such that

$$\parallel g\prime \left(uM\right)\parallel \le K,$$

and

$$-KId\le -\epsilon \nabla M-g\prime \left(uM\right)\le KId,$$

$\forall uM\in Br(u^M).$

Fix $K\gg .$

Observe also that there exists $N=N\left(M\right)\in \mathbb{N}$ such that if $N>N$, then

$$<{\displaystyle K+K\Delta tNK+-K\Delta tN}<.$$

Indeed, we may find $\alpha \in \mathbb{R}$ such that

$$<{\displaystyle K+K\Delta tNK+-K\Delta tN}<\alpha <,\forall N>N.$$

Let $N=NM\subset \mathbb{N}$ be a sequence such that $NM<NM+,\forall M\in \mathbb{N}.$

and $NM>N\left(M\right).$

For $n=$, we are going to calculate $u=uMn,NM$ though the following iterations, which already include a proximal formulation and concerning linearization.

Set $u10=u^=u^M,NM,$

Having $u$, let $u$ be such that

$$\begin{array}{ccc}\hfill uk+-u^& =& \epsilon (\nabla )Muk+\Delta tN+g\left(uk\right))\Delta tN\hfill \\ & & +g\prime \left(uk\right)(uk+-uk)\Delta tN+f\Delta tN-K(uk+-uk),\hfill \end{array}$$

(431)

Here we suppose $r>$ is large enough so that

$$u11\in B{\displaystyle r(-\alpha )NM}\left(u\right).$$

Reasoning inductively, suppose $u10,u11,\dots ,uk+\in B{\displaystyle rNM}\left(u\right),$ and observe that

$$\begin{array}{ccc}& & uk+-u^-\epsilon (\nabla )Muk+\Delta tN-g\left(uk\right))\Delta tN\hfill \\ & & -g\prime \left(uk\right)(uk+-uk)\Delta tN-f\Delta tN+K(uk+-uk)\hfill \\ & =& ,\hfill \end{array}$$

(432)

and

$$\begin{array}{ccc}& & uk+-u^-\epsilon (\nabla )Muk+\Delta tN-g\left(uk+\right))\Delta tN\hfill \\ & & -g\prime \left(uk+\right)(uk+-uk+)\Delta tN-f\Delta tN+K(uk+-uk+)\hfill \\ & =& ,\hfill \end{array}$$

(433)

so that for an appropriate $u$,

$$\begin{array}{ccc}& & \left(Id-\epsilon \nabla M\Delta tN-g\prime \left(uk+\right)\Delta tN+KId\right)(uk+-uk+)\hfill \\ & =& \left((-g\prime (u˜k)+g\prime \left(uk+\right))\Delta tN+KId\right)\left(uk+-uk\right).\hfill \end{array}$$

(434)

Hence,

$$\begin{array}{ccc}& & \parallel uk+-uk+\parallel \hfill \\ & \le & ∥\left(Id-\epsilon \nabla M\Delta tN-g\prime \left(uk+\right)\Delta tN+KId\right)-\left((-g\prime (u˜k)+g\prime \left(uk+\right))\Delta tN+KId\right)∥\hfill \\ & & \times ∥uk+-uk∥\hfill \\ & \le & {\displaystyle K+K\Delta tNK+-K\Delta tN}∥uk+-uk∥\hfill \\ & \le & \alpha ∥uk+-uk∥.\hfill \end{array}$$

(435)

Thus, we have got

$$\parallel uj+-uj+\parallel \le \alpha \parallel uj+-uj\parallel ,\forall j\in \{,\dots ,k\}.$$

Therefore

$$\begin{array}{ccc}\hfill \parallel uj+-uj+\parallel & \le & \alpha \parallel uj+-uj\parallel \hfill \\ & \le & \alpha 02\parallel uj-uj-\parallel \hfill \\ & ⋮& \hfill \\ & \le & \alpha j+\parallel u11-u10\parallel .\hfill \end{array}$$

(436)

Thus,

$$\begin{array}{ccc}\hfill \parallel uk+-u11\parallel & =& \parallel uk+-uk++uk+-\dots -u12+u12-u11\parallel \hfill \\ & \le & \parallel uk+-uk+\parallel +\parallel uk+-uk\parallel +\dots +\parallel u12-u11\parallel \hfill \\ & \le & \sum j=k+\alpha j\parallel u11-u10\parallel \hfill \\ & \le & \sum j=\infty \alpha j\parallel u11-u10\parallel \hfill \\ & =& {\displaystyle \alpha -\alpha }\parallel u11-u10\parallel .\hfill \end{array}$$

(437)

Therefore

$$\begin{array}{ccc}\hfill \parallel uk+-u10\parallel & \le & \parallel uk+-u11+u11-u10\parallel \hfill \\ & \le & \parallel uk+-u11\parallel +\parallel u11-u10\parallel \hfill \\ & \le & {\displaystyle \alpha -\alpha }\parallel u11-u10\parallel +\parallel u11-u10\parallel \hfill \\ & =& {\displaystyle -\alpha }\parallel u11-u10\parallel \hfill \\ & <& {\displaystyle -\alpha }(-\alpha ){\displaystyle rNM}\hfill \\ & =& {\displaystyle rNM}.\hfill \end{array}$$

(438)

Summarizing,

$$\parallel uk+-u10\parallel <{\displaystyle rNM},$$

so that

$$uk+\in B{\displaystyle rNM}\left(u01\right).$$

The induction is complete, so that

$$uk\in B{\displaystyle rNM}\left(u01\right),\forall k\in \mathbb{N}.$$

From such results we have also obtained

$$\parallel uk+-uk+\parallel \le \alpha \parallel uk+-uk\parallel ,\forall k\in \mathbb{N}.$$

Thus, from these results and the Banach fixed point theorem, there exists $u=uM,NM\in B{\displaystyle rNM}\left(u01\right)$ such that

$$\underset{k\to \infty }{lim}uk=u=uM,N=NM.$$

$$\begin{array}{ccc}\hfill \mathbf{0}& =& \underset{k\to \infty }{lim}\left(uk+-u^\right.\hfill \\ & & -\epsilon (\nabla )Muk+\Delta tN-g\left(uk\right))\Delta tN\hfill \\ & & \left.-g\prime \left(uk\right)(uk+-uk)\Delta tN-f\Delta tN+K(uk+-uk)\right)\hfill \\ & =& u-u^-\epsilon (\nabla )Mu\Delta tN-g\left(u\right)\Delta tN-f\Delta tN,\hfill \end{array}$$

(439)

so that

$${\displaystyle u-u^\Delta tN}=\epsilon (\nabla )Mu+g\left(u\right)+f,\mathrm{in}\Omega ,$$

Reasoning inductively again having $u\in B{\displaystyle rNM}(u^)$ and $uj\in B{\displaystyle rNM}\left(uj-\right),\forall j\in \{,\dots ,n\}$ similarly as we have obtained $u$ in the last lines, we may obtain

$$un+=un+M,NM\in B{\displaystyle rNM}\left(un\right),$$

such that

$${\displaystyle un+-un\Delta tN}=\epsilon (\nabla )Mun++g\left(un+\right)+fn,\mathrm{in}\Omega ,$$

where we recall to have denoted $N=NM.$

The induction on n is also complete.

Fix $n\in \{,\dots ,NM-\}$.

Observe that

$$\begin{array}{ccc}\hfill \parallel un-u^\parallel & =& \parallel un-un-+un--un-+\dots -u+u-u\parallel \hfill \\ & \le & \parallel un-un-\parallel +\dots +\parallel u-u^\parallel \hfill \\ & \le & {\displaystyle nNM}r\hfill \\ & <& r\hfill \end{array}$$

(440)

Summarizing $un\in Br(u^),\forall n\in \{,,\dots ,NM-\}.$

From these results, denoting now more generically $un\equiv unM,N=NM=unM$, we may infer that there exists $K>$ such that

$$\parallel ujM\parallel \le K,\forall j\in \{,,\dots ,NM\},\forall M\in \mathbb{N}.$$

With a completely analogous reasoning, we may obtain that

$$\parallel ujM\parallel ,,\Omega \le K^,\forall j\in \{,,\dots ,NM\}.$$

for some $K^>.$

Recalling again, we have denoted $N=NM,$ define now

$$uM(x,t)=unM\left(x\right)\left(n+-{\displaystyle t\Delta tN}\right)+un+M\left(x\right)\left({\displaystyle t\Delta tN}-n\right),$$

if $t\in [n\Delta tN,(n+)\Delta tN],\forall (x,t)\in \Omega \times [,T].$

Observe that

$$uM(x,t)=unM\left(x\right),\mathrm{if}t=n\Delta tN,\forall n\in \{,,\dots ,N=NM\},$$

and

$$\begin{array}{ccc}\hfill {\displaystyle \partial uM(x,t)\partial t}& =& {\displaystyle un+M-unM\Delta tN}\hfill \\ & =& \epsilon (\nabla )Mun+M+g\left(un+M\right)+fn,\hfill \end{array}$$

(441)

if $t\in [n\Delta tN,(n+)\Delta tN],\forall (x,t)\in \Omega \times [,T].$

Fix $\phi \in Cc\infty \left(\Omega \right).$

Thus, fixing $t\in [n\Delta tN,(n+)\Delta tN],$ we have

$$\begin{array}{ccc}\hfill \left|〈{\displaystyle \partial uM\partial t},\phi 〉L\right|& \le & \epsilon |\langle \nabla Mun+M,\nabla M\phi \rangle L|+|\langle g\left(un+M\right),\phi \rangle L|\hfill \\ & & +|\langle \phi ,fn\rangle L|\hfill \\ & \le & \epsilon \parallel un+M\parallel ,,\Omega \parallel \phi \parallel ,,\Omega +K18\parallel un+M\parallel ,,\Omega \parallel \phi \parallel ,,\Omega +K\parallel \phi \parallel ,,\Omega \hfill \\ & \le & K\parallel \phi \parallel ,,\Omega ,\forall \phi \in Cc\infty \left(\Omega \right),\hfill \end{array}$$

(442)

for some appropriate $K>.$

Since $\phi \in Cc\infty \left(\Omega \right)$ is arbitrary, we may conclude that

$$∥{\displaystyle \partial uM\partial t}∥H-\left(\Omega \right)\le K,\forall M>N,$$

uniformly in t on $[,T],$ for some appropriate constant $K>$ and $N\in \mathbb{N}.$

Also, from the definition of $u$ we have that there exists $K>$ such that

$$\parallel uM\parallel ,,\Omega \le K,\forall M\in \mathbb{N}$$

also uniformly in t on $[,T].$

From such results, there exist $u\in L([,T],H01\left(\Omega \right))$ and $v\in L([,T];H-\left(\Omega \right))$ such that

$$uM\rightharpoonup u,\mathrm{weakly}\mathrm{in}L((,T);W,\left(\Omega \right)),$$

and

$${\displaystyle \partial uM\partial t}\rightharpoonup v,\mathrm{weakly}-\mathrm{star}\mathrm{in}L([,T],H-\left(\Omega \right)),$$

so that we may easily obtain

$$v={\displaystyle \partial u\partial t}$$

in a distributional sense.

At this point, we provide more details about this last result.

Fix $t\in (,T)$. Thus, there exists $n\in \{,,\dots ,NM-\}$ such that $t\in [n\Delta tN,(n+)\Delta tN]$.

Let $\phi \in Cc\infty (\Omega \times (,T)).$

From this, we may infer that

$$\begin{array}{ccc}& & \int \Omega {\displaystyle \partial uM\partial t}\phi (x,t)dx\hfill \\ & =& \int \Omega {\displaystyle un+M-unM\Delta tN}\phi (x,t)dx\hfill \\ & \le & \epsilon \int \Omega |\nabla Mun+M·\nabla M\phi |dx\hfill \\ & & +\int \Omega |g\left(un+M\right)\phi (x,t)|dx+\int \Omega \left|fn\phi \right|dx\hfill \\ & \le & (K\parallel un+M\parallel ,,\Omega +K20)\parallel \phi \parallel ,,\Omega \hfill \\ & \le & K\parallel \phi \parallel ,,\Omega ,\hfill \end{array}$$

(443)

for some appropriate constants $K>,K>,K20>.$

Hence,

$$\begin{array}{ccc}& & \int T\int \Omega {\displaystyle \partial uM\partial t}\phi (x,t)dxdx\hfill \\ & \le & K\int \Omega \parallel \phi \parallel ,,\Omega dt\hfill \\ & \le & K19\parallel \phi \parallel ,,\Omega \times (,T),\hfill \end{array}$$

(444)

for some appropriate $K19>$.

Since such a $\phi \in Cc\infty (\Omega \times (,T))$ is arbitrary, we may infer that

$$∥{\displaystyle \partial uM\partial t}∥H-(\Omega \times (,T))\le K15,$$

for $N\in \mathbb{N}$, for some $K15>.$

From such a result and from the Banach-Alaoglu Theorem, there exists $v\in H-(\Omega \times (,T))$ such that, up to a not relabeled subsequence

$${\displaystyle \partial uM\partial t}\rightharpoonup v,\mathrm{weakly}-\mathrm{star}\mathrm{in}H-(\Omega \times (,T)).$$

Therefore,

$$\int T\int \Omega {\displaystyle \partial uM\partial t}\phi dxdt\to \int T\int \Omega v\phi dxdt,$$

as $N\to \infty ,$$\forall \phi \in H01(\Omega \times (,T))$.

On the other hand

$$\parallel uM\parallel ,,\Omega \times (,T)\le K16,$$

$\forall M\in \mathbb{N}$, for some $K16>.$

From this and the Kakutani Theorem, there exists $u\in L(\Omega \times (,T))$ such that, up to a not relabeled subsequence,

$$uM\rightharpoonup u,\mathrm{weakly}\mathrm{in}L(\Omega \times (,T)).$$

Now fix again $\phi \in Cc\infty (\Omega \times (,T)).$

Observe that

$$\begin{array}{ccc}\hfill \int T\int \Omega u\phi tdxdt& =& \underset{M\to \infty }{lim}\int T\int \Omega uM\phi tdxdt\hfill \\ & =& -\underset{M\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uM\partial t}\phi dxdt\hfill \\ & =& -\int T\int \Omega v\phi dxdt,\hfill \end{array}$$

(445)

Since such a $\phi \in Cc\infty (\Omega \times (,T))$ is arbitrary, we may infer that

$$v={\displaystyle \partial u\partial t}$$

in a distributional sense.

Moreover, from such results we may also obtain, again up to a subsequence,

$$\underset{M\to \infty }{lim}\int \Omega {\displaystyle \partial uM\partial t}\phi dx=\int \Omega {\displaystyle \partial u\partial t}\phi dx,$$

$\forall \phi \in H01\left(\Omega \right).$

Observe also that, as a consequence of the Rellich-Kondrashov theorem, through appropriate subsequences, we have

$$uMk\left(t\right)\to u(x,t),\mathrm{strongly}\mathrm{in}L\left(\Omega \right),\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

so that, up to subsequences,

$$uMk\left(t\right)(x,t)\to u(x,t),\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

Here we emphasise the sequence $\left\{Mk\left(t\right)\right\}\subset \mathbb{N}$ may depends on t.

Since g is continuous we have that

$$g\left(uMk\left(t\right)(x,t)\right)\to g\left(u(x,t)\right),\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

Fix $t\in (,T).$

Let $\epsilon >$. From the Egorov Theorem, there exists a closed set F such that $m(\Omega \setminus F)\le \epsilon$ and $k\in \mathbb{N}$ such that if $k>k$, then

$$|g\left(uMk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)|\le \epsilon ,\mathrm{for}\mathrm{almost}\mathrm{all}x\in F.$$

Let $\phi \in Cc\infty \left(\Omega \right).$ Observe now that

$$\begin{array}{ccc}& & \left|\int \Omega (g\left(uMk\left(t\right)(x,t)\right)-g\left(u(x,t)\right))\phi dx\right|\hfill \\ & \le & \int \Omega |g\left(uMk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |dx\hfill \\ & =& \int F|g\left(uMk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |dx+\int \Omega \setminus F|g\left(uMk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |dx\hfill \\ & \le & \int F\epsilon \parallel \phi \parallel \infty dx+\int \Omega |g\left(uMk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |\chi \Omega \setminus Fdx\hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+(\parallel g\left(uMk\left(t\right)\right)\parallel ,,\Omega +\parallel g\left(u\right)\parallel ,,\Omega )\parallel \phi \parallel ,,\Omega \parallel \chi \Omega \setminus F\parallel ,,\Omega \hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+K21\parallel \phi \parallel ,,\Omega m(\Omega \setminus F)/\hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+K21\parallel \phi \parallel ,,\Omega \epsilon /,\forall k>k,\hfill \end{array}$$

(446)

for some appropriate constant $K21>$ which does not depend on t.

Since such a $\epsilon >$ is arbitrary, we may infer that

$$\int \Omega g\left(uMk\left(t\right)\right)\phi dx\to \int \Omega g\left(u\right)\phi dx,\mathrm{as}k\to \infty ,$$

$\forall \phi \in Cc\infty \left(\Omega \right).$

From such results, we have

$$\begin{array}{ccc}\hfill & =& \underset{k\to \infty }{lim}\left(\int \Omega {\displaystyle \partial uMk\left(t\right)\partial t}\phi dx-\epsilon \int \Omega (\nabla )Mk\left(t\right)uMk\left(t\right)\phi dx\right.\hfill \\ & & \left.-\int \Omega g\left(uMk\left(t\right)\right)\phi dx-\int \Omega fNk\left(t\right)\phi dx\right)\hfill \\ & =& \int \Omega {\displaystyle \partial u\partial t}\phi dx+\epsilon \int \Omega \nabla u·\nabla \phi dx\hfill \\ & & -\int \Omega g\left(u\right)\phi dx-\int \Omega f\phi dx.\hfill \end{array}$$

(447)

so that, from this and by the density of $Cc\infty \left(\Omega \right)$ in $H01\left(\Omega \right)$, we have got

$$\begin{array}{ccc}& & \int \Omega {\displaystyle \partial u\partial t}\phi dx+\epsilon \int \Omega \nabla u·\nabla \phi dx\hfill \\ & & -\int \Omega g\left(u\right)\phi dx-\int \Omega f\phi dx=,\forall \phi \in H01\left(\Omega \right),\hfill \end{array}$$

(448)

$\mathrm{a}.\mathrm{e}.\mathrm{on}[,T].$

Observe now that

$$\partial (\Omega \times (,T))=\left(\partial \Omega \times [,T]\right)\cup \left(\partial [,T]\times \Omega ¯\right).$$

Let $\phi \in Cc\infty (\Omega \times (,T)).$

Hence

$$\underset{M\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uM\partial t}\phi dxdt=\int T\int \Omega {\displaystyle \partial u\partial t}\phi dxdt.$$

From this, since $Cc\infty (\Omega \times (,T))$ is dense $L(\Omega \times (,T))$ we may infer that

$$\underset{M\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uM\partial t}\phi dxdt=\int T\int \Omega {\displaystyle \partial u\partial t}\phi dxdt,$$

$\forall \phi \in L(\Omega \times (,T)).$

Let $\phi \in C\infty (\Omega \times [,T])$ such that

$$\phi (x,T)=,\mathrm{in}\Omega .$$

From such results, we may obtain

$$\begin{array}{ccc}& & \underset{M\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uM\partial t}\phi dxdt\hfill \\ & =& \underset{M\to \infty }{lim}\left(-\int T\int \Omega uM{\displaystyle \partial \phi \partial t}dxdt-\int \Omega uM(x,)\phi (x,)dx\right)\hfill \\ & =& -\int T\int \Omega u{\displaystyle \partial \phi \partial t}dxdt-\int \Omega u(x,)\phi (x,)dx.\hfill \end{array}$$

(449)

However, since $uM\rightharpoonup u,\mathrm{weakly}\mathrm{in}L(\Omega \times (,T))$, we obtain

$$\underset{M\to \infty }{lim}\int T\int \Omega uM{\displaystyle \partial \phi \partial t}dxdt=\int T\int \Omega u{\displaystyle \partial \phi \partial t}dxdt.$$

From these last results, we may infer that

$$\begin{array}{ccc}\hfill \int \Omega u^\phi (x,)dx& =& \underset{M\to \infty }{lim}\int \Omega uM(x,)\phi (x,)dx\hfill \\ & =& \int \Omega u(x,)\phi (x,)dx,\hfill \end{array}$$

(450)

so that

$$\int \Omega u^\left(x\right)\phi (x,)dx=\int \Omega u(x,)\phi (x,)dx,$$

$\forall \phi \in C\infty (\Omega \times [,T])$ such that $\phi (x,T)=,\mathrm{in}\Omega .$

Therefore, we may infer that $u(x,)=u^\left(x\right)$ in this specified weak sense.

Similarly, it may be proven that

$$u=,\mathrm{on}\partial \Omega \times [,T],$$

in an appropriate weak sense.

Hence, we have obtained that $u$ is a solution, in a weak sense, of the parabolic non-linear equation in question.

61. On the convergence of Newton’s method for a more general non-linear parabolic equation

Let $\Omega \subset \mathbb{R}m$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider the parabolic non-linear equation

$$\left\{\begin{array}{cc}{\displaystyle \partial u\partial t}=\epsilon \nabla u+g(u,\nabla u)+f,\hfill & \mathrm{in}\Omega \times (,T),\hfill \\ u(x,)=u^,\mathrm{in}\Omega ,\hfill & \\ u=,\mathrm{on}\partial \Omega \times [,T].\hfill \end{array}\right.$$

(451)

Here $\epsilon >$, $f\in L([,T],W,\left(\Omega \right))\cap L\infty (\Omega \times [,T]),$ $u^\in H01\left(\Omega \right)\cap L\infty \left(\Omega \right),$ where t denotes time and $[,T]$ is a time interval.

Also $g:\mathbb{R}\to \mathbb{R}$ is a $C$ class function neither necessarily linear nor convex.

We assume

$$\parallel g\parallel ,\infty \le K,$$

for an appropriate real constant $K>.$

Moreover, from now and on, we denote

$$Br(u^)=\{u\in H01\left(\Omega \right):\parallel u-u^\parallel ,,\Omega <r\}.$$

About the references, the main ones are [6,22]. Other related results may be found in [7,8].

Consider the operator

$$-\nabla :D\subset L\left(\Omega \right)\to L\left(\Omega \right),$$

where

$$D=H01\left(\Omega \right)\cap H\left(\Omega \right)$$

is dense in $L\left(\Omega \right)$ concerning the $L\left(\Omega \right)$ norm.

From the standard spectral analysis theory (please see [8], for details), denoting the spectral decomposition of $-\nabla$ by

$$\left\{E\right(\lambda ),\lambda \in [,+\infty \left)\right\},$$

we may obtain

$$-\nabla u=\int \infty \lambda dE\left(\lambda \right)u,$$

$\forall u\in D.$

Fix $M\in \mathbb{N}$ and define

$$(-\nabla )M=\int M\lambda dE\left(\lambda \right),$$

which is a bounded operator in $D.$

Also from the standard spectral theory, we have

$$\underset{M\to \infty }{lim}\parallel (-\nabla )Mu-(-\nabla )u\parallel ,=,\forall u\in D.$$

Similarly, we define the self-adjoint densely defined in $L\left(\Omega \right),$

$$Bj:H01\left(\Omega \right)\to L\left(\Omega \right),$$

by

$$Bj\left(u\right)=i{\displaystyle \partial u\partial xj},$$

Denoting by

$$\{Ej\left(\lambda \right),\lambda \in \mathbb{R}\},$$

the spectral decomposition of $B$ and fixing again $M\in \mathbb{N}$, we also define

$$\left(Bj\right)M=-i\int -MM\lambda dEj\left(\lambda \right),$$

so that

$$\left(Bj\right)M\left(u\right)=-i\int -MM\lambda dEj\left(\lambda \right)u,\forall u\in H01\left(\Omega \right).$$

We may infer that

$$\parallel \left(Bj\right)M\left(u\right)-(-i)Bj\left(u\right)\parallel ,\to ,\mathrm{as}M\to \infty ,$$

$\forall u\in H01\left(\Omega \right).$

Finally, we define

$$\nabla M=(\left(B\right)M,\dots ,\left(Bm\right)M),$$

which is a bounded linear operator from $H01\left(\Omega \right)$ into $L\left(\Omega \right).$

61.1. The main result

Fix again $M\in \mathbb{N}$ (a new value).

Moreover, fixing $N\in \mathbb{N}$ and defining

$$\Delta tN={\displaystyle TN},$$

in a partial finite differences context, discretizing in t consider the approximate equation system

$${\displaystyle un+-un\Delta tN}=\epsilon (\nabla )Mun++g(un+,(\nabla M)un+)+fn,\mathrm{in}\Omega ,$$

$\forall n\in \{,,\dots ,N-\}.$

Denoting $un=unM$ we may write

$${\displaystyle un+M-unM\Delta tN}=\epsilon \nabla Mun+M+g(un+M,(\nabla M)un+M)+fn,\mathrm{in}\Omega ,$$

$\forall n\in \{,,\dots ,N-\}.$

We assume g is also such that there exists $r>$, $K=K\left(M\right)>$ and $K>$ such that

$$\parallel gu(u,\nabla Mu)+(\nabla M*)gv(u,(\nabla M)u)\parallel \le K,$$

where we have denoted $v=\nabla Mu,$ and

$$\parallel -\epsilon \nabla M-GM\prime \left(u\right)\parallel \le K,$$

$\forall u\in Br(u^M),$ where we have denoted

$$GM\left(u\right)=g(u,(\nabla M)u),$$

and

$$GM\prime \left(u\right)=gu(u,(\nabla M)u)+(\nabla M*)gv(u,(\nabla M)u).$$

Fix $K\gg .$

Observe also that there exists $N=N\left(M\right)\in \mathbb{N}$ such that if $N>N$, then

$$<{\displaystyle K+K\Delta tNK+-K\Delta tN}<.$$

Indeed, we may find $\alpha \in \mathbb{R}$ such that

$$<{\displaystyle K+K\Delta tNK+-K\Delta tN}<\alpha <,\forall N>N.$$

Let $\{N=NM\}\subset \mathbb{N}$ be a sequence such that $NM<NM+,\forall M\in \mathbb{N}.$

and $NM>N\left(M\right).$

For $n=$, we are going to calculate $u=uMn,NM$ though the following iterations, which already include a proximal formulation and concerning linearization.

Set $u10=u^=u^M,NM,$

Having $u$, let $u$ be such that

$$\begin{array}{ccc}\hfill uk+-u^& =& \epsilon (\nabla )Muk+\Delta tN+GM\left(uk\right)\Delta tN\hfill \\ & & +GM\prime \left(uk\right)(uk+-uk)\Delta tN+f\Delta tN-K(uk+-uk),\hfill \end{array}$$

(452)

Here we suppose $r>$ is large enough so that

$$u11\in B{\displaystyle r(-\alpha )NM}\left(u\right).$$

Reasoning inductively, suppose $u10,u11,\dots ,uk+\in B{\displaystyle rNM}\left(u\right),$ and observe that

$$\begin{array}{ccc}& & uk+-u^-\epsilon (\nabla )Muk+\Delta tN-GM\left(uk\right)\Delta tN\hfill \\ & & -GM\prime \left(uk\right)(uk+-uk)\Delta tN-f\Delta tN+K(uk+-uk)\hfill \\ & =& ,\hfill \end{array}$$

(453)

and

$$\begin{array}{ccc}& & uk+-u^-\epsilon (\nabla )Muk+\Delta tN-GM\left(uk+\right)\Delta tN\hfill \\ & & -GM\prime \left(uk+\right)(uk+-uk+)\Delta tN-f\Delta tN+K(uk+-uk+)\hfill \\ & =& ,\hfill \end{array}$$

(454)

so that for an appropriate $u$,

$$\begin{array}{ccc}& & \left(Id-\epsilon \nabla M\Delta tN-GM\prime \left(uk+\right)\Delta tN+KId\right)(uk+-uk+)\hfill \\ & =& \left((-GM\prime (u˜k)+GM\prime \left(uk+\right))\Delta tN+KId\right)\left(uk+-uk\right).\hfill \end{array}$$

(455)

Hence,

$$\begin{array}{ccc}& & \parallel uk+-uk+\parallel \hfill \\ & \le & ∥\left(Id-\epsilon \nabla M\Delta tN-GM\prime \left(uk+\right)\Delta tN+KId\right)-\left((-GM\prime (u˜k)+GM\prime \left(uk+\right))\Delta tN+KId\right)∥\hfill \\ & & \times ∥uk+-uk∥\hfill \\ & \le & {\displaystyle K+K\Delta tNK+-K\Delta tN}∥uk+-uk∥\hfill \\ & \le & \alpha ∥uk+-uk∥.\hfill \end{array}$$

(456)

Thus, we have got

$$\parallel uj+-uj+\parallel \le \alpha \parallel uj+-uj\parallel ,\forall j\in \{,\dots ,k\}.$$

Therefore

$$\begin{array}{ccc}\hfill \parallel uj+-uj+\parallel & \le & \alpha \parallel uj+-uj\parallel \hfill \\ & \le & \alpha 02\parallel uj-uj-\parallel \hfill \\ & ⋮& \hfill \\ & \le & \alpha j+\parallel u11-u10\parallel .\hfill \end{array}$$

(457)

Thus,

$$\begin{array}{ccc}\hfill \parallel uk+-u11\parallel & =& \parallel uk+-uk++uk+-\dots -u12+u12-u11\parallel \hfill \\ & \le & \parallel uk+-uk+\parallel +\parallel uk+-uk\parallel +\dots +\parallel u12-u11\parallel \hfill \\ & \le & \sum j=k+\alpha j\parallel u11-u10\parallel \hfill \\ & \le & \sum j=\infty \alpha j\parallel u11-u10\parallel \hfill \\ & =& {\displaystyle \alpha -\alpha }\parallel u11-u10\parallel .\hfill \end{array}$$

(458)

Therefore

$$\begin{array}{ccc}\hfill \parallel uk+-u10\parallel & \le & \parallel uk+-u11+u11-u10\parallel \hfill \\ & \le & \parallel uk+-u11\parallel +\parallel u11-u10\parallel \hfill \\ & \le & {\displaystyle \alpha -\alpha }\parallel u11-u10\parallel +\parallel u11-u10\parallel \hfill \\ & =& {\displaystyle -\alpha }\parallel u11-u10\parallel \hfill \\ & <& {\displaystyle -\alpha }(-\alpha ){\displaystyle rNM}\hfill \\ & =& {\displaystyle rNM}.\hfill \end{array}$$

(459)

Summarizing,

$$\parallel uk+-u10\parallel <{\displaystyle rNM},$$

so that

$$uk+\in B{\displaystyle rNM}\left(u01\right).$$

The induction is complete, so that

$$uk\in B{\displaystyle rNM}\left(u01\right),\forall k\in \mathbb{N}.$$

From such results we have also obtained

$$\parallel uk+-uk+\parallel \le \alpha \parallel uk+-uk\parallel ,\forall k\in \mathbb{N}.$$

Thus, from these results and the Banach fixed point theorem, there exists $u=uM,NM\in B{\displaystyle rNM}\left(u01\right)$ such that

$$\underset{k\to \infty }{lim}uk=u=uM,N=NM.$$

$$\begin{array}{ccc}\hfill \mathbf{0}& =& \underset{k\to \infty }{lim}\left(uk+-u^\right.\hfill \\ & & -\epsilon (\nabla )Muk+\Delta tN-GM\left(uk\right)\Delta tN\hfill \\ & & \left.-GM\prime \left(uk\right)(uk+-uk)\Delta tN-f\Delta tN+K(uk+-uk)\right)\hfill \\ & =& u-u^-\epsilon (\nabla )Mu\Delta tN-GM\left(u\right)\Delta tN-f\Delta tN,\hfill \end{array}$$

(460)

so that

$${\displaystyle u-u^\Delta tN}=\epsilon (\nabla )Mu+GM\left(u\right)+f,\mathrm{in}\Omega ,$$

Reasoning inductively again having $u\in B{\displaystyle rNM}(u^)$ and $uj\in B{\displaystyle rNM}\left(uj-\right),\forall j\in \{,\dots ,n\}$ similarly as we have obtained $u$ in the last lines, we may obtain

$$un+=un+M,NM\in B{\displaystyle rNM}\left(un\right),$$

such that

$${\displaystyle un+-un\Delta tN}=\epsilon (\nabla )Mun++g(un+,(\nabla M)un+)+fn,\mathrm{in}\Omega ,$$

where we recall to have denoted $N=NM.$

The induction on n is also complete.

Fix $n\in \{,\dots ,NM-\}$.

Observe that

$$\begin{array}{ccc}\hfill \parallel un-u^\parallel & =& \parallel un-un-+un--un-+\dots -u+u-u\parallel \hfill \\ & \le & \parallel un-un-\parallel +\dots +\parallel u-u^\parallel \hfill \\ & \le & {\displaystyle nNM}r\hfill \\ & <& r\hfill \end{array}$$

(461)

Summarizing $un\in Br(u^),\forall n\in \{,,\dots ,NM-\}.$

From these results, denoting now more generically $un\equiv unM,N=NM=unM$, we may infer that there exists $K>$ such that

$$\parallel ujM\parallel \le K,\forall j\in \{,,\dots ,NM\},\forall M\in \mathbb{N}.$$

With a completely analogous reasoning, we may obtain that

$$\parallel ujM\parallel ,,\Omega \le K^,\forall j\in \{,,\dots ,NM\}.$$

for some $K^>.$

Recalling again, we have denoted $N=NM,$ define now

$$uM(x,t)=unM\left(x\right)\left(n+-{\displaystyle t\Delta tN}\right)+un+M\left(x\right)\left({\displaystyle t\Delta tN}-n\right),$$

if $t\in [n\Delta tN,(n+)\Delta tN],\forall (x,t)\in \Omega \times [,T].$

Observe that

$$uM(x,t)=unM\left(x\right),\mathrm{if}t=n\Delta tN,\forall n\in \{,,\dots ,N=NM\},$$

and

$$\begin{array}{ccc}\hfill {\displaystyle \partial uM(x,t)\partial t}& =& {\displaystyle un+M-unM\Delta tN}\hfill \\ & =& \epsilon (\nabla )Mun+M+g(un+M,(\nabla M)un+M)+fn,\hfill \end{array}$$

(462)

if $t\in [n\Delta tN,(n+)\Delta tN],\forall (x,t)\in \Omega \times [,T].$

Fix $\phi \in Cc\infty \left(\Omega \right).$

Thus, fixing $t\in [n\Delta tN,(n+)\Delta tN],$ we have

$$\begin{array}{ccc}\hfill \left|〈{\displaystyle \partial uM\partial t},\phi 〉L\right|& \le & \epsilon |\langle (\nabla M)un+M,(\nabla M)\phi \rangle L|+|\langle g(un+M,(\nabla M)un+M),\phi \rangle L|\hfill \\ & & +|\langle \phi ,fn\rangle L|\hfill \\ & \le & \epsilon \parallel un+M\parallel ,,\Omega \parallel \phi \parallel ,,\Omega +K18\parallel un+M\parallel ,,\Omega \parallel \phi \parallel ,,\Omega +K\parallel \phi \parallel ,,\Omega \hfill \\ & \le & K\parallel \phi \parallel ,,\Omega ,\forall \phi \in Cc\infty \left(\Omega \right),\hfill \end{array}$$

(463)

for some appropriate $K>.$

Since $\phi \in Cc\infty \left(\Omega \right)$ is arbitrary, we may conclude that

$$∥{\displaystyle \partial uM\partial t}∥H-\left(\Omega \right)\le K,\forall M>N,$$

uniformly in t on $[,T],$ for some appropriate constant $K>$ and $N\in \mathbb{N}.$

Also, from the definition of $u$ we have that there exists $K>$ such that

$$\parallel uM\parallel ,,\Omega \le K,\forall M\in \mathbb{N}$$

also uniformly in t on $[,T].$

From such results, there exist $u\in L([,T],H01\left(\Omega \right))$ and $v\in L([,T];H-\left(\Omega \right))$ such that

$$uM\rightharpoonup u,\mathrm{weakly}\mathrm{in}L((,T);W,\left(\Omega \right)),$$

and

$${\displaystyle \partial uM\partial t}\rightharpoonup v,\mathrm{weakly}-\mathrm{star}\mathrm{in}L([,T],H-\left(\Omega \right)),$$

so that we may easily obtain

$$v={\displaystyle \partial u\partial t}$$

in a distributional sense.

At this point, we provide more details about this last result.

Fix $t\in (,T)$. Thus, there exists $n\in \{,,\dots ,NM-\}$ such that $t\in [n\Delta tN,(n+)\Delta tN]$.

Let $\phi \in Cc\infty (\Omega \times (,T)).$

From this, we may infer that

$$\begin{array}{ccc}& & \int \Omega {\displaystyle \partial uM\partial t}\phi (x,t)dx\hfill \\ & =& \int \Omega {\displaystyle un+M-unM\Delta tN}\phi (x,t)dx\hfill \\ & \le & \epsilon \int \Omega |(\nabla M)un+M·\nabla M\phi |dx\hfill \\ & & +\int \Omega |g(un+M,(\nabla M)un+M)\phi (x,t)|dx+\int \Omega \left|fn\phi \right|dx\hfill \\ & \le & (K\parallel un+M\parallel ,,\Omega +K20)\parallel \phi \parallel ,,\Omega \hfill \\ & \le & K\parallel \phi \parallel ,,\Omega ,\hfill \end{array}$$

(464)

for some appropriate constants $K>,K>,K20>.$

Hence,

$$\begin{array}{ccc}& & \int T\int \Omega {\displaystyle \partial uM\partial t}\phi (x,t)dxdx\hfill \\ & \le & K\int \Omega \parallel \phi \parallel ,,\Omega dt\hfill \\ & \le & K19\parallel \phi \parallel ,,\Omega \times (,T),\hfill \end{array}$$

(465)

for some appropriate $K19>$.

Since such a $\phi \in Cc\infty (\Omega \times (,T))$ is arbitrary, we may infer that

$$∥{\displaystyle \partial uM\partial t}∥H-(\Omega \times (,T))\le K15,$$

for $N\in \mathbb{N}$, for some $K15>.$

From such a result and from the Banach-Alaoglu Theorem, there exists $v\in H-(\Omega \times (,T))$ such that, up to a not relabeled subsequence

$${\displaystyle \partial uM\partial t}\rightharpoonup v,\mathrm{weakly}-\mathrm{star}\mathrm{in}H-(\Omega \times (,T)).$$

Therefore,

$$\int T\int \Omega {\displaystyle \partial uM\partial t}\phi dxdt\to \int T\int \Omega v\phi dxdt,$$

as $N\to \infty ,$$\forall \phi \in H01(\Omega \times (,T))$.

On the other hand

$$\parallel uM\parallel ,,\Omega \times (,T)\le K16,$$

$\forall M\in \mathbb{N}$, for some $K16>.$

From this and the Kakutani Theorem, there exists $u\in L(\Omega \times (,T))$ such that, up to a not relabeled subsequence,

$$uM\rightharpoonup u,\mathrm{weakly}\mathrm{in}L(\Omega \times (,T)).$$

Now fix again $\phi \in Cc\infty (\Omega \times (,T)).$

Observe that

$$\begin{array}{ccc}\hfill \int T\int \Omega u\phi tdxdt& =& \underset{M\to \infty }{lim}\int T\int \Omega uM\phi tdxdt\hfill \\ & =& -\underset{M\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uM\partial t}\phi dxdt\hfill \\ & =& -\int T\int \Omega v\phi dxdt,\hfill \end{array}$$

(466)

Since such a $\phi \in Cc\infty (\Omega \times (,T))$ is arbitrary, we may infer that

$$v={\displaystyle \partial u\partial t}$$

in a distributional sense.

Moreover, from such results we may also obtain, again up to a subsequence,

$$\underset{M\to \infty }{lim}\int \Omega {\displaystyle \partial uM\partial t}\phi dx=\int \Omega {\displaystyle \partial u\partial t}\phi dx,$$

$\forall \phi \in H01\left(\Omega \right).$

Observe also that, as a consequence of the Rellich-Kondrashov theorem, through appropriate subsequences, we have

$$uMk\left(t\right)\to u(x,t),\mathrm{strongly}\mathrm{in}L\left(\Omega \right),\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

so that, up to subsequences,

$$uMk\left(t\right)(x,t)\to u(x,t),\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

Observe also that

$$(\nabla M)uM(x,t)\in L\left(\Omega \right),\forall M\in \mathbb{N},\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

Fixing $t\in (,T),$ from such a result, similarly as it has been obtained in reference [22], up to a subsequence, we may obtain a measurable functions $h$ such that

$$(\nabla M)uM(x,t)\to h\in \mathbb{R},\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega .$$

From such a result, also similarly as it has been made in reference [22], we may obtain a measurable vectorial function $h$ such that

$$(\nabla M)uM(x,t)\to h\in \mathbb{R}m,\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega .$$

Consequently, considering that

$$(\nabla M)uM(x,t)\rightharpoonup \nabla u(x,t),\mathrm{weakly}\mathrm{in}L\left(\Omega \right),$$

we obtain

$$(\nabla M)uM(x,t)\to \nabla u(x,t),\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega .$$

Here we emphasise the sequence $\{M=Mk\left(t\right)\}\subset \mathbb{N}$ may depends on t.

Since g is continuous we have that

$$g(uMk\left(t\right)(x,t),(\nabla M)uMk\left(t\right)(x,t))\to g(u(x,t),\nabla u(x,t)),\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

Fix $t\in (,T).$

Let $\epsilon >$. From the Egorov Theorem, there exists a closed set F such that $m(\Omega \setminus F)\le \epsilon$ and $k\in \mathbb{N}$ such that if $k>k$, then

$$|g(uMk\left(t\right)(x,t),(\nabla M)uMk\left(t\right)(x,t))-g(u(x,t),(\nabla M)uMk\left(t\right)(x,t))|\le \epsilon ,\mathrm{for}\mathrm{almost}\mathrm{all}x\in F.$$

Let $\phi \in Cc\infty \left(\Omega \right).$ Observe now that

$$\begin{array}{ccc}& & \left|\int \Omega (g(uMk\left(t\right)(x,t),(\nabla M)uMk\left(t\right)(x,t))-g(u(x,t),\nabla u(x,t)))\phi dx\right|\hfill \\ & \le & \int \Omega |g(uMk\left(t\right)(x,t),(\nabla M)uMk\left(t\right)(x,t))-g(u(x,t),\nabla u(x,t))\left|\right|\phi |dx\hfill \\ & =& \int F|g(uMk\left(t\right)(x,t),(\nabla M)uMk\left(t\right)(x,t))-g(u(x,t),\nabla u(x,t))\left|\right|\phi |dx\hfill \\ & & +\int \Omega \setminus F|g(uMk\left(t\right)(x,t),\nabla MuMk\left(t\right)(x,t))-g(u(x,t),\nabla u(x,t))\left|\right|\phi |dx\hfill \\ & \le & \int F\epsilon \parallel \phi \parallel \infty dx\hfill \\ & & +\int \Omega |g(uMk\left(t\right)(x,t),(\nabla M)uMk\left(t\right)(x,t))-g(u(x,t),\nabla u(x,t))\left|\right|\phi |\chi \Omega \setminus Fdx\hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+(\parallel g(uMk\left(t\right),(\nabla M)uMk\left(t\right)(x,t))\parallel ,,\Omega \hfill \\ & & +\parallel g(u,\nabla u)\parallel ,,\Omega )\parallel \phi \parallel ,,\Omega \parallel \chi \Omega \setminus F\parallel ,,\Omega \hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+K21\parallel \phi \parallel ,,\Omega m(\Omega \setminus F)/\hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+K21\parallel \phi \parallel ,,\Omega \epsilon /,\forall k>k,\hfill \end{array}$$

(467)

for some appropriate constant $K21>$ which does not depend on t.

Since such a $\epsilon >$ is arbitrary, we may infer that

$$\int \Omega g(uMk\left(t\right),(\nabla M)uMk\left(t\right)(x,t))\phi dx\to \int \Omega g(u,\nabla u)\phi dx,\mathrm{as}k\to \infty ,$$

$\forall \phi \in Cc\infty \left(\Omega \right).$

From such results, we have

$$\begin{array}{ccc}\hfill & =& \underset{k\to \infty }{lim}\left(\int \Omega {\displaystyle \partial uMk\left(t\right)\partial t}\phi dx-\epsilon \int \Omega (\nabla )Mk\left(t\right)uMk\left(t\right)\phi dx\right.\hfill \\ & & \left.-\int \Omega g(uMk\left(t\right),(\nabla M)uMk\left(t\right)(x,t))\phi dx-\int \Omega fNk\left(t\right)\phi dx\right)\hfill \\ & =& \int \Omega {\displaystyle \partial u\partial t}\phi dx+\epsilon \int \Omega \nabla u·\nabla \phi dx\hfill \\ & & -\int \Omega g(u,\nabla u)\phi dx-\int \Omega f\phi dx.\hfill \end{array}$$

(468)

so that, from this and by the density of $Cc\infty \left(\Omega \right)$ in $H01\left(\Omega \right)$, we have got

$$\begin{array}{ccc}& & \int \Omega {\displaystyle \partial u\partial t}\phi dx+\epsilon \int \Omega \nabla u·\nabla \phi dx\hfill \\ & & -\int \Omega g(u,\nabla u)\phi dx-\int \Omega f\phi dx=,\forall \phi \in H01\left(\Omega \right),\hfill \end{array}$$

(469)

$\mathrm{a}.\mathrm{e}.\mathrm{on}[,T].$

Observe now that

$$\partial (\Omega \times (,T))=\left(\partial \Omega \times [,T]\right)\cup \left(\partial [,T]\times \Omega ¯\right).$$

Let $\phi \in Cc\infty (\Omega \times (,T)).$

Hence

$$\underset{M\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uM\partial t}\phi dxdt=\int T\int \Omega {\displaystyle \partial u\partial t}\phi dxdt.$$

From this, since $Cc\infty (\Omega \times (,T))$ is dense $L(\Omega \times (,T))$ we may infer that

$$\underset{M\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uM\partial t}\phi dxdt=\int T\int \Omega {\displaystyle \partial u\partial t}\phi dxdt,$$

$\forall \phi \in L(\Omega \times (,T)).$

Let $\phi \in C\infty (\Omega \times [,T])$ such that

$$\phi (x,T)=,\mathrm{in}\Omega .$$

From such results, we may obtain

$$\begin{array}{ccc}& & \underset{M\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uM\partial t}\phi dxdt\hfill \\ & =& \underset{M\to \infty }{lim}\left(-\int T\int \Omega uM{\displaystyle \partial \phi \partial t}dxdt-\int \Omega uM(x,)\phi (x,)dx\right)\hfill \\ & =& -\int T\int \Omega u{\displaystyle \partial \phi \partial t}dxdt-\int \Omega u(x,)\phi (x,)dx.\hfill \end{array}$$

(470)

However, since $uM\rightharpoonup u,\mathrm{weakly}\mathrm{in}L(\Omega \times (,T))$, we obtain

$$\underset{M\to \infty }{lim}\int T\int \Omega uM{\displaystyle \partial \phi \partial t}dxdt=\int T\int \Omega u{\displaystyle \partial \phi \partial t}dxdt.$$

From these last results, we may infer that

$$\begin{array}{ccc}\hfill \int \Omega u^\phi (x,)dx& =& \underset{M\to \infty }{lim}\int \Omega uM(x,)\phi (x,)dx\hfill \\ & =& \int \Omega u(x,)\phi (x,)dx,\hfill \end{array}$$

(471)

so that

$$\int \Omega u^\left(x\right)\phi (x,)dx=\int \Omega u(x,)\phi (x,)dx,$$

$\forall \phi \in C\infty (\Omega \times [,T])$ such that $\phi (x,T)=,\mathrm{in}\Omega .$

Therefore, we may infer that $u(x,)=u^\left(x\right)$ in this specified weak sense.

Similarly, it may be proven that

$$u=,\mathrm{on}\partial \Omega \times [,T],$$

in an appropriate weak sense.

Hence, we have obtained that $u$ is a solution, in a weak sense, of the parabolic non-linear equation in question.

61.2. An existence result for a general parabolic non-linear equation, a new development and result for a simpler case

Let $\Omega \subset \mathbb{R}m$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider the parabolic non-linear equation

$$\left\{\begin{array}{cc}{\displaystyle \partial u\partial t}=\epsilon \nabla u+g\left(u\right)+\sum j=mgj\left(u\right){\displaystyle \partial u\partial xj}+f,\hfill & \mathrm{in}\Omega \times (,T),\hfill \\ u(x,)=u^,\mathrm{in}\Omega ,\hfill & \\ u=,\mathrm{on}\partial \Omega \times [,T].\hfill \end{array}\right.$$

(472)

Here $\epsilon >$, $f\in L([,T],W,\left(\Omega \right))\cap L\infty (\Omega \times [,T]),$ $u^\in H01\left(\Omega \right)\cap L\infty \left(\Omega \right),$ where t denotes time and $[,T]$ is a time interval.

Also $g:\mathbb{R}\to \mathbb{R}$ and $gj:\mathbb{R}\to \mathbb{R}$ are continuous functions neither necessarily linear nor convex,$\forall j\in \{,\dots ,m\}$.

We assume there exist $K33>$ and $K>$ such that

$$\parallel g\parallel \infty \le {\displaystyle K33m\left(\Omega \right)/},$$

$$\parallel gj\parallel \infty \le {\displaystyle Km},$$

$\forall j\in \{,\dots ,n\}.$

At this point, we recall that fixing $\gamma >$,

$$(Id-\gamma \nabla )-:L\left(\Omega \right)\to H01\left(\Omega \right)$$

is a bounded and linear operator, so that for each $h\in L\left(\Omega \right)$ there exists a unique $u\in H01\left(\Omega \right)$ such that

$$(Id-\gamma \nabla )u=h.$$

In such a case we denote

$$u=(Id-\gamma \nabla )-h,$$

so that

$$\parallel u\parallel ,,\Omega \le \parallel (Id-\gamma \nabla )-\parallel \parallel h\parallel ,,\Omega .$$

Moreover, fixing $N\in \mathbb{N}$, define

$$\Delta tN={\displaystyle TN}.$$

Fix $M\in \mathbb{N}.$

Let $\{N=NM\}\in \mathbb{N}$ be a subsequence such that $NM<NM+,\forall M\in \mathbb{N}$ and

$$\parallel (\nabla M)\Delta tNM\parallel <{\displaystyle M},$$

and

$$\parallel (\nabla M)\Delta tNM\parallel <{\displaystyle M}.$$

Denoting $N=NM$, in a partial finite differences context, discretizing in t and also denoting $\left\{un\right\}=\left\{unM\right\},$ consider the approximate equation system

$${\displaystyle un+-un\Delta tN}=\epsilon (\nabla )Mun++g\left(un\right)+\sum j=mgj\left(un\right)\left(Bj\right)M\left(un\right)+fn,\mathrm{in}\Omega ,$$

$\forall n\in \{,,\dots ,N-\}.$

From such a system, for $n=$, we obtain

$$u-u^=\epsilon (\nabla )M\left(u\right)\Delta tN+g(u^)\Delta tN+\sum j=mgj(u^)\left(Bj\right)Mu^\Delta tN+f\Delta tN.$$

Hence

$$u=(Id-\epsilon (\nabla )M\Delta tN)-\left(u^+g(u^)\Delta tN+\sum j=mgj(u^)\left(Bj\right)Mu^\Delta tN+f\Delta tN\right),$$

so that

$$\begin{array}{ccc}& & \parallel u\parallel ,,\Omega \hfill \\ & \le & \parallel (Id-\epsilon (\nabla )M\Delta tN)-\parallel \hfill \\ & & \times \left(\parallel u^\parallel ,,\Omega +\parallel g(u^)\parallel ,,\Omega \Delta tN+\sum j=m\parallel gj(u^)\left(Bj\right)Mu\parallel ,,\Omega \Delta tN+\parallel f\parallel ,,\Omega \Delta tN\right).\hfill \end{array}$$

(473)

Observe that there exists $K>$ such that $\parallel f\parallel \infty ,\Omega \times [,T]\le K$ so that

$$\parallel fn\parallel ,,\Omega \le K36,\forall n\in \{,,\dots ,N-\},$$

for some appropriate $K36>.$

From such results and the hypotheses, we may infer that

$$\begin{array}{ccc}& & \parallel u\parallel ,,\Omega \le \parallel (Id-\epsilon (\nabla )M\Delta tN)-\parallel (\parallel u^\parallel ,,\Omega +K33\Delta tN+K\parallel u^\parallel ,,\Omega \Delta tN+K36\Delta tN)\hfill \\ & \le & \parallel (Id-\epsilon (\nabla )M\Delta tN)-\parallel (\parallel u^\parallel ,,\Omega +K\parallel u^\parallel ,,\Omega \Delta tN+K\Delta tN),\hfill \end{array}$$

(474)

where

$$K=K33+K36,$$

so that

$$\parallel u\parallel ,,\Omega \le \alpha \parallel u^\parallel ,,\Omega +\alpha ,$$

where

$$\alpha =\parallel (Id-\epsilon (\nabla )M\Delta tN)-\parallel (+K\Delta tN),$$

and

$$\alpha =\parallel (Id-\epsilon (\nabla )M\Delta tN)-\parallel K\Delta tN.$$

In fact, generically we may similarly obtain

$$\parallel un+\parallel ,,\Omega \le \alpha \parallel un\parallel ,,\Omega +\alpha ,$$

$\forall n\in \{,,\dots ,N-\}.$

From such a result, inductively we may obtain

$$\parallel uj\parallel ,,\Omega \le \left(\alpha \right)j\parallel u^\parallel ,,\Omega +\sum k=j-\alpha k\alpha .$$

In particular for $j=NM=N$, we get

$$\begin{array}{ccc}& & \parallel uNM\parallel ,,\Omega \hfill \\ & \le & \left(\alpha \right)N\parallel u^\parallel ,,\Omega +\sum k=N-\alpha k\alpha \hfill \\ & =& \left(\alpha \right)N\parallel u^\parallel ,,\Omega +{\displaystyle -\alpha N-\alpha }\alpha \hfill \\ & =& ∥\left(Id-\epsilon (\nabla )M{\displaystyle TN}\right)-∥N\left(+K{\displaystyle TN}\right)N\parallel u^\parallel ,,\Omega \hfill \\ & & +{\displaystyle -\alpha N-\alpha }\alpha .\hfill \end{array}$$

(475)

Observe that

$$\begin{array}{ccc}& & ∥\left(Id-\epsilon (\nabla )M{\displaystyle TN}\right)-∥N\left(+K{\displaystyle TN}\right)N\hfill \\ & \le & Kb\left(+K{\displaystyle TN}\right)N\hfill \\ & \le & KbeKT,\forall M\in \mathbb{N},\hfill \end{array}$$

(476)

for some real constant $Kb>.$

Moreover, since

$$\parallel (\nabla )M\Delta tNM\parallel </M,\forall M\in \mathbb{N},$$

there exists a real constant $Ks>$ such that

$$\left|{\displaystyle -\alpha NM-\alpha }\alpha \right|<Ks,\forall M\in \mathbb{N}.$$

Consequently, we may infer that there exists $K>$ such that

$$\parallel ujM\parallel ,,\Omega \le K,\forall j\in \{,,\dots ,NM\},\forall M\in \mathbb{N}.$$

Define now

$$uM(x,t)=unM\left(x\right)\left(n+-{\displaystyle t\Delta tN}\right)+un+M\left(x\right)\left({\displaystyle t\Delta tN}-n\right),$$

if $t\in [n\Delta tN,(n+)\Delta tN],\forall (x,t)\in \Omega \times [,T].$

Observe that

$$uM(x,t)=unM\left(x\right),\mathrm{if}t=n\Delta tN,\forall n\in \{,,\dots ,NM\},$$

and

$$\begin{array}{ccc}\hfill {\displaystyle \partial uM(x,t)\partial t}& =& {\displaystyle un+M-unM\Delta tN}\hfill \\ & =& \epsilon (\nabla )Mun+M+g\left(unM\right)+\sum j=mgj\left(unM\right)\left(Bj\right)M\left(unM\right)+fn,\hfill \end{array}$$

(477)

if $t\in [n\Delta tN,(n+)\Delta tN],\forall (x,t)\in \Omega \times [,T].$

Fix $\phi \in Cc\infty \left(\Omega \right).$

Thus, fixing $t\in [n\Delta tN,(n+)\Delta tN],$ we have

$$\begin{array}{ccc}\hfill \left|〈{\displaystyle \partial uM\partial t},\phi 〉L\right|& \le & \epsilon |\langle (\nabla )Mun+M,\nabla \phi \rangle L|+|\langle g\left(unM\right),\phi \rangle L|\hfill \\ & & +\int \Omega \left|\sum j=mgj\left(unM\right)\left(Bj\right)M\left(unM\right)\phi \right|dx+\left|\langle \phi ,fn\rangle L\right|\hfill \\ & \le & \epsilon \parallel un+M\parallel ,,\Omega \parallel \phi \parallel ,,\Omega +K\parallel unM\parallel ,,\Omega \parallel \phi \parallel ,,\Omega +K\parallel \phi \parallel ,,\Omega \hfill \\ & \le & K\parallel \phi \parallel ,,\Omega ,\forall \phi \in Cc\infty \left(\Omega \right),\hfill \end{array}$$

(478)

for some appropriate $K>.$

Since $\phi \in Cc\infty \left(\Omega \right)$ is arbitrary, we may conclude that

$$∥{\displaystyle \partial uM\partial t}∥H-\left(\Omega \right)\le K,\forall M\in \mathbb{N},$$

uniformly in t on $[,T],$ for some appropriate constant $K>.$

Also, from the definition of $u$ we have that there exists $K>$ such that

$$\parallel uM\parallel ,,\Omega \le K,\forall M\in \mathbb{N}$$

also uniformly in t on $[,T].$

From such results, there exist $u\in L([,T],H01\left(\Omega \right))$ and $v\in L([,T];H-\left(\Omega \right))$ such that

$$uM\rightharpoonup u,\mathrm{weakly}\mathrm{in}L((,T);W,\left(\Omega \right)),$$

and

$${\displaystyle \partial uM\partial t}\rightharpoonup v,\mathrm{weakly}-\mathrm{star}\mathrm{in}L([,T],H-\left(\Omega \right)),$$

so that we may easily obtain

$$v={\displaystyle \partial u\partial t}$$

in a distributional sense.

At this point, we provide more details about this last result.

Fix $t\in (,T)$. Thus, there exists $n\in \{,,\dots ,N-\}$ such that $t\in [n\Delta tN,(n+)\Delta tN]$.

Let $\phi \in Cc\infty (\Omega \times (,T)).$

From this, we may infer that

$$\begin{array}{ccc}& & \int \Omega {\displaystyle \partial uM\partial t}\phi (x,t)dx\hfill \\ & =& \int \Omega {\displaystyle un+M-unM\Delta tN}\phi (x,t)dx\hfill \\ & \le & \epsilon \int \Omega |(\nabla )Mun+M·\nabla \phi |dx\hfill \\ & & +\int \Omega \left|g\left(unM\right)\phi (x,t)\right|dx+\int \Omega \left|\sum j=mgj\left(unM\right)\left(Bj\right)M\left(unM\right)\phi \right|dx\hfill \\ & & +\int \Omega \left|fn\phi \right|dx\hfill \\ & \le & (K\parallel unM\parallel ,,\Omega +K20)\parallel \phi \parallel ,,\Omega \hfill \\ & \le & K\parallel \phi \parallel ,,\Omega ,\hfill \end{array}$$

(479)

for some appropriate constants $K>,K>,K20>.$

Hence,

$$\begin{array}{ccc}& & \int T\int \Omega {\displaystyle \partial uM\partial t}\phi (x,t)dxdx\hfill \\ & \le & K\int \Omega \parallel \phi \parallel ,,\Omega dt\hfill \\ & \le & K19\parallel \phi \parallel ,,\Omega \times (,T),\hfill \end{array}$$

(480)

for some appropriate $K19>$.

Since such a $\phi \in Cc\infty (\Omega \times (,T))$ is arbitrary, we may infer that

$$∥{\displaystyle \partial uM\partial t}∥H-(\Omega \times (,T))\le K15,$$

for $M\in \mathbb{N}$, for some $K15>.$

From such a result and from the Banach-Alaoglu Theorem, there exists $v\in H-(\Omega \times (,T))$ such that, up to a not relabeled subsequence

$${\displaystyle \partial uM\partial t}\rightharpoonup v,\mathrm{weakly}-\mathrm{star}\mathrm{in}H-(\Omega \times (,T)).$$

Therefore,

$$\int T\int \Omega {\displaystyle \partial uM\partial t}\phi dxdt\to \int T\int \Omega v\phi dxdt,$$

as $N\to \infty ,$$\forall \phi \in H01(\Omega \times (,T))$.

On the other hand

$$\parallel uM\parallel ,,\Omega \times (,T)\le K16,$$

$\forall M\in \mathbb{N}$, for some $K16>.$

From this and the Kakutani Theorem, there exists $u\in L(\Omega \times (,T))$ such that, up to a not relabeled subsequence,

$$uM\rightharpoonup u,\mathrm{weakly}\mathrm{in}L(\Omega \times (,T)).$$

Now fix again $\phi \in Cc\infty (\Omega \times (,T)).$

Observe that

$$\begin{array}{ccc}\hfill \int T\int \Omega u\phi tdxdt& =& \underset{M\to \infty }{lim}\int T\int \Omega uM\phi tdxdt\hfill \\ & =& -\underset{M\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uM\partial t}\phi dxdt\hfill \\ & =& -\int T\int \Omega v\phi dxdt,\hfill \end{array}$$

(481)

Since such a $\phi \in Cc\infty (\Omega \times (,T))$ is arbitrary, we may infer that

$$v={\displaystyle \partial u\partial t}$$

in a distributional sense.

Moreover, from such results we may also obtain, again up to a subsequence,

$$\underset{M\to \infty }{lim}\int \Omega {\displaystyle \partial uM\partial t}\phi dx=\int \Omega {\displaystyle \partial u\partial t}\phi dx,$$

$\forall \phi \in H01\left(\Omega \right).$

Observe also that, as a consequence of the Rellich-Kondrashov theorem, through appropriate subsequences, we have

$$uMk\left(t\right)\to u(x,t),\mathrm{strongly}\mathrm{in}L\left(\Omega \right),\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

so that, up to subsequences,

$$uMk\left(t\right)(x,t)\to u(x,t),\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

Here we emphasise the sequence $\left\{Mk\left(t\right)\right\}\subset \mathbb{N}$ may depends on t.

Since g is continuous we have that

$$g\left(uMk\left(t\right)(x,t)\right)\to g\left(u(x,t)\right),\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

Fix $t\in (,T).$

Let $\epsilon >$. From the Egorov Theorem, there exists a closed set F such that $m(\Omega \setminus F)\le \epsilon$ and $k\in \mathbb{N}$ such that if $k>k$, then

$$|g\left(uMk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)|\le \epsilon ,\mathrm{for}\mathrm{almost}\mathrm{all}x\in F.$$

Let $\phi \in Cc\infty \left(\Omega \right).$ Observe now that

$$\begin{array}{ccc}& & \left|\int \Omega (g\left(uMk\left(t\right)(x,t)\right)-g\left(u(x,t)\right))\phi dx\right|\hfill \\ & \le & \int \Omega |g\left(uMk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |dx\hfill \\ & =& \int F|g\left(uMk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |dx+\int \Omega \setminus F|g\left(uMk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |dx\hfill \\ & \le & \int F\epsilon \parallel \phi \parallel \infty dx+\int \Omega |g\left(uMk\left(t\right)(x,t)\right)-g\left(u(x,t)\right)\left|\right|\phi |\chi \Omega \setminus Fdx\hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+(\parallel g\left(uMk\left(t\right)\right)\parallel ,,\Omega +\parallel g\left(u\right)\parallel ,,\Omega )\parallel \phi \parallel ,,\Omega \parallel \chi \Omega \setminus F\parallel ,,\Omega \hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+K21\parallel \phi \parallel ,,\Omega m(\Omega \setminus F)/\hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+K21\parallel \phi \parallel ,,\Omega \epsilon /,\forall k>k,\hfill \end{array}$$

(482)

for some appropriate constant $K21>$ which does not depend on t.

Since such a $\epsilon >$ is arbitrary, we may infer that

$$\int \Omega g\left(uMk\left(t\right)\right)\phi dx\to \int \Omega g\left(u\right)\phi dx,\mathrm{as}k\to \infty ,$$

$\forall \phi \in Cc\infty \left(\Omega \right).$

Similarly, fixing $j\in \{,\dots ,n\},$ since $g$ is continuous we have that

$$gj\left(uMk\left(t\right)(x,t)\right)\to gj\left(u(x,t)\right),\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

Fix again $t\in (,T)$

Let $\epsilon >$ (a new value). From the Egorov Theorem, there exists a closed set $F$ such that $m(\Omega \setminus F)\le \epsilon$ and $k\in \mathbb{N}$ such that if $k>k$, then

$$|gj\left(uMk\left(t\right)(x,t)\right)-gj\left(u(x,t)\right)|\le \epsilon ,\mathrm{for}\mathrm{almost}\mathrm{all}x\in F.$$

Observe now that

$$\begin{array}{ccc}& & \int \Omega |gj\left(uMk\left(t\right)(x,t)\right)-gj\left(u(x,t)\right)|dx\hfill \\ & \le & \int F|gj\left(uMk\left(t\right)(x,t)\right)-gj\left(u(x,t)\right)|dx+\int \Omega \setminus F|gj\left(uMk\left(t\right)(x,t)\right)-gj\left(u(x,t)\right)|dx\hfill \\ & \le & \int F\epsilon dx+\int \Omega |gj\left(uMk\left(t\right)(x,t)\right)-gj\left(u(x,t)\right)|\chi \Omega \setminus Fdx\hfill \\ & \le & \epsilon m\left(\Omega \right)+K12\int \Omega \chi \Omega \setminus Fdx\hfill \\ & \le & \epsilon m\left(\Omega \right)+K12\epsilon ,\forall k>k.\hfill \end{array}$$

(483)

Since such a $\epsilon >$ is arbitrary, we may infer that

$$\int \Omega |gj\left(uMk\left(t\right)\right)-gj\left(u\right)|dx\to ,\mathrm{as}k\to \infty ,$$

$\forall j\in \{,\dots ,m\}.$

Select again $\phi \in Cc\infty \left(\Omega \right).$ Since

$$\parallel gj\left(uMk\left(t\right)\right)-gj\left(u\right)\parallel ,,\Omega \to ,\mathrm{as}k\to \infty$$

and

$$(\nabla M)uMk\left(t\right)\rightharpoonup \nabla u,\mathrm{weakly}\mathrm{in}L(\Omega ;\mathbb{R}m),$$

we obtain,

$$\begin{array}{ccc}& & \left|\int \Omega gj\left(uMk\left(t\right)\right)\left(Bj\right)M\left(uMk\left(t\right)\right)\phi dx-\int \Omega gj\left(u\right)\left(u\right)xj\phi dx\right|\hfill \\ & \le & \left|\int \Omega gj\left(uMk\left(t\right)\right)\left(Bj\right)M\left(uMk\left(t\right)\right)\phi dx-\int \Omega gj\left(u\right)\left(Bj\right)M\left(uMk\left(t\right)\right)\phi dx\right|\hfill \\ & & +\left|\int \Omega gj\left(u\right)\left(Bj\right)M\left(uMk\left(t\right)\right)\phi dx-\int \Omega gj\left(u\right)\left(u\right)xj\phi dx\right|\hfill \\ & \le & \parallel gj\left(uMk\left(t\right)\right)-gj\left(u\right)\parallel ,,\Omega K\parallel \phi \parallel \infty \hfill \\ & & +\left|\int \Omega gj\left(u\right)\left(Bj\right)M\left(uMk\left(t\right)\right)\phi dx-\int \Omega gj\left(u\right)\left(u\right)xj\phi dx\right|\hfill \\ & \to & ,\mathrm{as}k\to \infty ,\hfill \end{array}$$

(484)

$\forall j\in \{,\dots ,m\}.$

From such results, we have

$$\begin{array}{ccc}\hfill & =& \underset{k\to \infty }{lim}\left(\int \Omega {\displaystyle \partial uMk\left(t\right)\partial t}\phi dx-\epsilon \int \Omega (\nabla )MuMk\left(t\right)\phi dx\right.\hfill \\ & & -\int \Omega g\left(uMk\left(t\right)\right)\phi dx-\sum j=m\int \Omega gj\left(uMk\left(t\right)\right)\left(Bj\right)M\left(uMk\left(t\right)\right)\phi dx\hfill \\ & & \left.-\int \Omega fMk\left(t\right)\phi dx\right)\hfill \\ & =& \int \Omega {\displaystyle \partial u\partial t}\phi dx+\epsilon \int \Omega \nabla u·\nabla \phi dx\hfill \\ & & -\int \Omega g\left(u\right)\phi dx-\sum j=m\int \Omega gj\left(u\right)\left(u\right)xj\phi dx\hfill \\ & & -\int \Omega f\phi dx.\hfill \end{array}$$

(485)

so that, from this and by the density of $Cc\infty \left(\Omega \right)$ in $H01\left(\Omega \right)$, we have got

$$\begin{array}{ccc}& & \int \Omega {\displaystyle \partial u\partial t}\phi dx+\epsilon \int \Omega \nabla u·\nabla \phi dx\hfill \\ & & -\int \Omega g\left(u\right)\phi dx-\sum j=m\int \Omega gj\left(u\right)\left(u\right)xj\phi dx\hfill \\ & & -\int \Omega f\phi dx=,\forall \phi \in H01\left(\Omega \right),\hfill \end{array}$$

(486)

$\mathrm{a}.\mathrm{e}.\mathrm{on}[,T].$

Observe now that

$$\partial (\Omega \times (,T))=\left(\partial \Omega \times [,T]\right)\cup \left(\partial [,T]\times \Omega ¯\right).$$

Let $\phi \in Cc\infty (\Omega \times (,T)).$

Hence

$$\underset{M\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uM\partial t}\phi dxdt=\int T\int \Omega {\displaystyle \partial u\partial t}\phi dxdt.$$

From this, since $Cc\infty (\Omega \times (,T))$ is dense $L(\Omega \times (,T))$ we may infer that

$$\underset{M\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uM\partial t}\phi dxdt=\int T\int \Omega {\displaystyle \partial u\partial t}\phi dxdt,$$

$\forall \phi \in L(\Omega \times (,T)).$

Let $\phi \in C\infty (\Omega \times [,T])$ such that

$$\phi (x,T)=,\mathrm{in}\Omega .$$

From such results, we may obtain

$$\begin{array}{ccc}& & \underset{M\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uM\partial t}\phi dxdt\hfill \\ & =& \underset{M\to \infty }{lim}\left(-\int T\int \Omega uM{\displaystyle \partial \phi \partial t}dxdt-\int \Omega uM(x,)\phi (x,)dx\right)\hfill \\ & =& -\int T\int \Omega u{\displaystyle \partial \phi \partial t}dxdt-\int \Omega u(x,)\phi (x,)dx.\hfill \end{array}$$

(487)

However, since $uN\rightharpoonup u,\mathrm{weakly}\mathrm{in}L(\Omega \times (,T))$, we obtain

$$\underset{M\to \infty }{lim}\int T\int \Omega uM{\displaystyle \partial \phi \partial t}dxdt=\int T\int \Omega u{\displaystyle \partial \phi \partial t}dxdt.$$

From these last results, we may infer that

$$\begin{array}{ccc}\hfill \int \Omega u^\phi (x,)dx& =& \underset{M\to \infty }{lim}\int \Omega uN(x,)\phi (x,)dx\hfill \\ & =& \int \Omega u(x,)\phi (x,)dx,\hfill \end{array}$$

(488)

so that

$$\int \Omega u^\left(x\right)\phi (x,)dx=\int \Omega u(x,)\phi (x,)dx,$$

$\forall \phi \in C\infty (\Omega \times [,T])$ such that $\phi (x,T)=,\mathrm{in}\Omega .$

Therefore, we may infer that $u(x,)=u^\left(x\right)$ in this specified weak sense.

Similarly, it may be proven that

$$u=,\mathrm{on}\partial \Omega \times [,T],$$

in an appropriate weak sense.

Hence, we have obtained that $u$ is a solution, in a weak sense, of the parabolic non-linear equation in question.

62. On the convergence of Newton’s method combined with a proximal approach for an eigenvalue problem

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider the eigenvalue problem of finding $u\in V$ and $\lambda \in \mathbb{R}$ such that

$$\left\{\begin{array}{cc}-\epsilon \nabla u+g\left(u\right)-\lambda u=,\hfill & \mathrm{in}\Omega ,\hfill \\ u=,\mathrm{on}\partial \Omega ,\hfill & \\ \int \Omega udx=\parallel u\parallel ,,\Omega =.\hfill \end{array}\right.$$

(489)

Here $\epsilon >$, $V=H01\left(\Omega \right),$ and $g:\mathbb{R}\to \mathbb{R}$ is a $C$ class function, such that either g is linear or such that

$$g\left(tx\right)=tg\left(x\right),\forall t>,\forall x\in \mathbb{R}$$

About the references, related and similar results have been published in [7].

Concerning the numerical example in the last section, related models in superconductivity are addressed in [14,15].

Finally, details on the Sobolev spaces involved may be found in [6].

62.1. The main theoretical result

In a finite differences or finite elements context (please, see reference [19] for details on the standard finite differences method), already including a proximal formulation, we shall look for a sequence $\left\{un\right\}\subset \mathbb{R}N$ for an appropriate $N\in \mathbb{N}$ such that

$$-\epsilon \nabla un++g\left(un+\right)-{\displaystyle un\parallel un\parallel }+K(un+-un)=\mathbf{0},$$

$\forall n\in \mathbb{N}\cup \left\{\right\}.$

Here we highlight $\nabla$ refers to an appropriate linear and bounded finite dimensional approximation of the standard Laplace operator.

Now considering a concerning linearization of g, such an equation approximately stands for

$$-\epsilon \nabla un++g\left(un\right)+g\prime \left(un\right)(un+-un)-{\displaystyle un\parallel un\parallel }+K(un+-un)=\mathbf{0},$$

$\forall n\in \mathbb{N}\cup \left\{\right\}.$

Denoting by $I$ the identity matrix, assume $u\in \mathbb{R}N$ is such that there exists $r>$ such that

$$KId\le g\prime \left(u\right)\le KId,$$

$\forall u\in Br\left(u\right),$ for some $K,K>.$

Suppose there exists a symmetric matrix $H(u,v)$ such that

$${\displaystyle u\parallel u\parallel }-{\displaystyle v\parallel v\parallel }=H(u,v)(u-v),$$

and

$$-KId\le H(u,v)\le KId,$$

$\forall u,v\in Br\left(u\right),$ for some $K>$ and also there exists a symmetric matrix $H(u,v)$ such that

$$g\left(u\right)-g\left(v\right)=H(u,v)(u-v),$$

and

$$KId\le H(u,v)\le KId,$$

$\forall u,v\in Br\left(u\right),$ for some $K,K>.$ Moreover, we assume that these last constants, $K>$ and $<\alpha <$ are such that

$$K+K>K+K,$$

$$(-\alpha )(-\epsilon \nabla +KId)+KId-KId-KId\ge \alpha KId=KId-(-\alpha )KId,$$

so that

$$(-\epsilon \nabla +KId+KId)-(-KId+KId+KId+KId)\le (-\alpha )Id.$$

Recall that

$$-\epsilon \nabla +g\prime \left(u\right)+KId\ge -\epsilon \nabla +KId+KId\ge$$

and

$$\le -K-K+K+K\le -H(u,v)+H(u,v)+g\prime \left(v\right)+KId\le -KId+KId+KId+KId,$$

$\forall u,v\in Br\left(u\right)$ so that

$$\begin{array}{ccc}& & (-\epsilon \nabla +g\prime \left(u\right)+KId)-(-H(u,v)+H(u,v)+g\prime \left(v\right)+KId)\hfill \\ & \le & (-\epsilon \nabla +KId+KId)-(-KId+KId+KId+KId)\hfill \\ & \le & (-\alpha )Id,\hfill \end{array}$$

(490)

$\forall u,v\in Br\left(u\right).$

Summarizing, defining $\alpha =-\alpha$ we have got

$$\parallel (-\epsilon \nabla +g\prime \left(u\right)+KId)-(-H(u,v)+H(u,v)+g\prime \left(v\right)+KId)\parallel \le \alpha <,$$

$\forall u,v\in Br\left(u\right).$

Suppose $K>$ and $\alpha >$ are such that $u\in Br(-\alpha )\left(u\right).$

Reasoning inductively, suppose also

$$u,u,\dots ,un+\in Br\left(u\right).$$

From the results above we have

$$-\epsilon \nabla un++g\left(un+\right)+g\prime \left(un+\right)(un+-un+)-{\displaystyle un+\parallel un+\parallel }+K(un+-un+)=\mathbf{0},$$

and

$$-\epsilon \nabla un++g\left(un\right)+g\prime \left(un\right)(un+-un)-{\displaystyle un\parallel un\parallel }+K(un+-un)=\mathbf{0},$$

so that

$$\begin{array}{ccc}& & -\epsilon (\nabla un+-\nabla un+)+g\prime \left(un+\right)(un+-un+)+K(un+-un+)\hfill \\ & =& -g\left(un+\right)+g\left(un\right)+{\displaystyle un+\parallel un+\parallel }-{\displaystyle un\parallel un\parallel }+g\prime \left(un\right)(un+-un)+K(un+-un)\hfill \\ & =& (-H(un+,un)+H(un+,un)+g\prime \left(un\right)+KId)(un+-un).\hfill \end{array}$$

(491)

Therefore

$$\begin{array}{ccc}& & un+-un+\hfill \\ & =& ((-\epsilon \nabla +g\prime \left(un+\right)+KId)-(-H(un+,un)+H(un+,un)+g\prime \left(un\right)+KId)\hfill \\ & & \times (un+-un),\hfill \end{array}$$

(492)

so that

$$\begin{array}{ccc}& & \parallel un+-un+\parallel \hfill \\ & \le & \parallel ((-\epsilon \nabla +g\prime \left(un+\right)+KId)-(-H(un+,un)+H(un+,un)+g\prime \left(un\right)+KId)\parallel \hfill \\ & & \times \parallel un+-un\parallel \hfill \\ & \le & \alpha \parallel un+-un\parallel .\hfill \end{array}$$

(493)

Summarizing, we have got

$$\parallel uj+-uj+\parallel \le \alpha \parallel uj+-uj\parallel ,\forall j\in \{,\dots ,n\}.$$

Therefore

$$\begin{array}{ccc}\hfill \parallel uj+-uj+\parallel & \le & \alpha \parallel uj+-uj\parallel \hfill \\ & \le & \alpha 02\parallel uj-uj-\parallel \hfill \\ & ⋮& \hfill \\ & \le & \alpha j+\parallel u-u\parallel .\hfill \end{array}$$

(494)

Thus,

$$\begin{array}{ccc}\hfill \parallel un+-u\parallel & =& \parallel un+-un++un+-\dots -u+u-u\parallel \hfill \\ & \le & \parallel un+-un+\parallel +\parallel un+-un\parallel +\dots +\parallel u-u\parallel \hfill \\ & \le & \sum j=n+\alpha j\parallel u-u\parallel \hfill \\ & \le & \sum j=\infty \alpha j\parallel u-u\parallel \hfill \\ & =& {\displaystyle \alpha -\alpha }\parallel u-u\parallel .\hfill \end{array}$$

(495)

Therefore

$$\begin{array}{ccc}\hfill \parallel un+-u\parallel & \le & \parallel un+-u+u-u\parallel \hfill \\ & \le & \parallel un+-u\parallel +\parallel u-u\parallel \hfill \\ & \le & {\displaystyle \alpha -\alpha }\parallel u-u\parallel +\parallel u-u\parallel \hfill \\ & =& {\displaystyle -\alpha }\parallel u-u\parallel \hfill \\ & <& {\displaystyle -\alpha }(-\alpha )r\hfill \\ & =& r.\hfill \end{array}$$

(496)

Summarizing,

$$\parallel un+-u\parallel <r,$$

so that

$$un+\in Br\left(u\right).$$

The induction is complete, so that

$$un\in Br\left(u\right),\forall n\in \mathbb{N}.$$

From such results we have also obtained

$$\parallel un+-un+\parallel \le \alpha \parallel un+-un\parallel ,\forall n\in \mathbb{N}.$$

Thus, from these results and the Banach fixed point theorem, there exists $u^\in B¯r\left(u\right)$ such that

$$\underset{n\to \infty }{lim}un=u^.$$

From such results we obtain

$$\begin{array}{ccc}\hfill \mathbf{0}& =& \underset{n\to \infty }{lim}\left(-\epsilon \nabla un++g\left(un\right)+g\prime \left(un\right)(un+-un)\right.\hfill \\ & & \left.-{\displaystyle un\parallel un\parallel }+K(un+-un)\right)\hfill \\ & =& -\epsilon \nabla u^+g(u^)-{\displaystyle u^\parallel u^\parallel }.\hfill \end{array}$$

(497)

Summarizing, we have got

$$-\epsilon \nabla u^+g(u^)-{\displaystyle u^\parallel u^\parallel }=\mathbf{0}.$$

Consequently, defining

$$u˜={\displaystyle u^\parallel u^\parallel ,,\Omega },$$

$$\lambda ={\displaystyle \parallel u^\parallel }$$

and recalling that

$$g(tu^)=tg(u^),\forall t>,$$

we have obtained

$$-\epsilon \nabla u˜+g(u˜)-\lambda u˜=\mathbf{0},$$

and

$$\parallel u˜\parallel ,,\Omega =.$$

The objective of this section is complete.

Remark 28.

For the general case we may drop the hypotheses of g being linear or $g\left(tu\right)=tg\left(u\right),\forall t>,\forall u\in \mathbb{R},$ by defining the following iterations:

$$-\epsilon \nabla un++\parallel un\parallel g\left({\displaystyle un+\parallel un\parallel }\right)-{\displaystyle un\parallel un\parallel }+K(un+-un)=\mathbf{0},\forall n\in \mathbb{N}\cup \left\{\right\}.$$

However in such a case some changes on the hypotheses are necessary in order to obtain the related theoretical results.

62.2. A numerical example

For $\Omega =[,]\subset \mathbb{R},$ we have obtained numerical results for the following eigenvalue equation

$$\left\{\begin{array}{cc}-\epsilon u\prime \prime +Au-\lambda u=,\hfill & \mathrm{in}\Omega ,\hfill \\ u=,\mathrm{on}\partial \Omega ,\hfill & \\ \int \Omega udx=\parallel u\parallel ,,\Omega =,\hfill \end{array}\right.$$

(498)

where $A=.$, $\epsilon =.01$ and $\epsilon =.001.$

Observe that for fixed $K\ge$ we may obtain for this last equation

$$-\epsilon u\prime \prime +Au+Ku-Ku-\lambda u=,\mathrm{in}\Omega ,$$

so that

$$-\epsilon u\prime \prime +Au+Ku-\lambda u=,\mathrm{in}\Omega ,$$

where $\lambda =K+\lambda .$ In this example we have fixed $K=$.

In order to obtain such numerical results we have used the following Newton method type algorithm:

Choose $u\in W,$, set $n=$, $b12=10-$ and $nmax=100.$

Calculate $un+\in W,$ solution of equation

$$-\epsilon un+\prime \prime +A{\displaystyle un+un\parallel un\parallel ,,\Omega }-A{\displaystyle un\parallel un\parallel ,,\Omega }-{\displaystyle un\parallel un\parallel ,,\Omega }=,\mathrm{in}\Omega ,$$

If $\parallel un+-un\parallel \infty \le b12$ or $n>nmax$, then stop. Otherwise $n:=n+$ and go to item 2.

For the optimal solutions u obtained, for $\epsilon =.01,$ please see Figure 43. For $\epsilon =.001$, please see Figure 44.

Here we present the software "EigenValueAugust2025A" in MAT-LAB through which we have obtained such numerical results.

********************

clear all

m8=1000;

d=1/m8;

K=0;

A=1;

e1=0.001;

for i=1:m8

uo(i,1)=5.0;

end;

b12=1.0;

k=1;

while $(b12>10-)$ && $(k<100)$

k

k=k+1;

S=0;

for i=1:m8-1

$S=S+uo(i,)*d;$

end;

i=1;

S=sqrt(S);

$m12=+K*d+*A*uo(i,)*d/e/S;$

m50(1)=1/m12;

$z\left(\right)=m50\left(\right)*(uo(i,)/S*d/e+*A*uo(i,)/S*d/e);$

for i=2:m8-1

$m12=+K*d+*A*uo(i,)*d/e/S-m50(i-);$

m50(i)=1/m12;

$z\left(i\right)=m50\left(i\right)*(uo(i,)/S*d/e+*A*uo(i,)/S*d/e+z(i-));$

end;

u(m8,1)=0;

for i=1:m8-1

u(m8-i,1)=m50(m8-i)*u(m8-i+1)+z(m8-i);

end;

b12=max(abs(uo-u));

uo=u;

end;

for i=1:m8

x(i,1)=i*d;

end;

plot(x,uo/S);

************************************

Remark 29.

With the value $K=$ we have obtained the following eigenvalue for this problem: For $\epsilon =.01$

$$\lambda =\left({\displaystyle S}\right)=.3254$$

and for $\epsilon =.001,$

$$\lambda =\left({\displaystyle S}\right)=.0932$$

The objective of this section is complete.

62.3. Conclusion

In this article we have presented a numerical procedure for solving a large class of eigenvalue problems involving partial differential equations.

Even though we have applied the method to a non-linear Ginzburg-Landau type equation, we emphasize such a method may be extended for a large class of other non-linear partial differential equations.

In a near future research we intend to address some other non-linear cases.

63. On the convergence of Newton’s method combined with a proximal approach for a general parabolic non-linear system

Let $\Omega \subset \mathbb{R}m$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider the parabolic non-linear system

$$\left\{\begin{array}{cc}{\displaystyle \partial uj\partial t}=\epsilon j\nabla uj+gj\left(u\right)+\sum k=r\sum l=mgjkl\left(u\right){\displaystyle \partial uk\partial xl}+fj,\hfill & \mathrm{in}\Omega \times (,T),\hfill \\ uj(x,)=(u^)j,\mathrm{in}\Omega ,\hfill & \\ uj=,\mathrm{on}\partial \Omega \times [,T],\forall j\in \{,\dots ,r\}.\hfill \end{array}\right.$$

(499)

Here

$$u=(u,\dots ,ur)=\left\{uj\right\}\in H01(\Omega ;\mathbb{R}r),$$

$\epsilon j>$, $f=\left\{fj\right\}\in L([,T],W,(\Omega ;\mathbb{R}r))\cap L\infty (\Omega \times [,T];\mathbb{R}r),$$u^=\left\{(u^)j\right\}\in H01(\Omega ;\mathbb{R}r)\cap L\infty (\Omega ;\mathbb{R}r),$ where t denotes time and $[,T]$ is a time interval.

Also $gj:\mathbb{R}\to \mathbb{R}$ and $gjkl:\mathbb{R}\to \mathbb{R}$ are $C$ class functions neither necessarily linear nor convex,$\forall j,k\in \{,\dots ,r\},l\in \{,\dots ,m\}$.

We define

$$Fj\left(u\right)=\epsilon j\nabla uj+gj\left(u\right)+\sum k=r\sum l=mgjkl\left(u\right){\displaystyle \partial uk\partial xl}+fj,$$

$\forall j\in \{,\dots ,r\}$, so that the system in question stands for

$${\displaystyle \partial ujdt}=Fj\left(u\right),\forall j\in \{,\dots ,r\}.$$

Fixing $N\in \mathbb{N}$ and defining $\Delta tN=T/N$, in a finite differences context we may define the following approximate system

$${\displaystyle un+-un\Delta tN}=Fj\left(un+\right),\forall j\in \{,\dots ,r\},\forall n\in \{,\dots ,N-\}.$$

Fix $n=$. In a Newton’s method context combined with a proximal approach, we shall obtain $u$ through the following iterations,

Define $u10=u^$ and having $u$ let $u$ be such that

$$uk+-u10=Fj\left(uk\right)\Delta tN+\Delta tN\left\{{\displaystyle \partial Fj\left(uk\right)\partial ul}\right\}(uk+-uk)-K(uk+-uk).$$

At this point we assume there exist $r>$ and $K>$, such that

$$-KId\le \left\{{\displaystyle \partial Fj\left(u\right)\partial ul}\right\}\le KId,$$

$\forall u\in Br(u^).$

Moreover, generically denoting $F\left(u\right)=\left\{Fj\left(u\right)\right\},$ we assume there exists a matrix operator $H(u,v),$ such that

$$F\left(u\right)-F\left(v\right)=H(u,v)(u-v),$$

and

$$-KId\le H(u,v)\le KId,$$

$\forall u,v\in Br(u^),$ for some appropriate real constant $K>.$

Now suppose $K>$ and $<\alpha <$ are such that there exists $N\in \mathbb{N}$ such that if $N>N$, then

$$(-\alpha )(Id-KId\Delta tN)-KId\Delta tN-KId\Delta tN)\ge \alpha KId=KId-(-\alpha )KId,$$

so that

$$(Id-KId\Delta tN+KId)-(KId\Delta tN+KId\Delta tN+KId)\le (-\alpha )KId.$$

Observe that such an $N$ may be such that

$$Id-\left\{{\displaystyle \partial Fj\left(u\right)\partial ul}\right\}\Delta tN+KId\ge Id-KId\Delta tN+KId>Id,$$

and

$$\le H(u,v)\Delta tN-\left\{{\displaystyle \partial Fj\left(v\right)\partial ul}\right\}\Delta tN+KId\le KId\Delta tN+KId\Delta tN+KId,$$

so that

$$\begin{array}{ccc}& & \left(Id-\left\{{\displaystyle \partial Fj\left(u\right)\partial ul}\right\}\Delta tN+KId\right)-\left(H(u,v)\Delta tN-\left\{{\displaystyle \partial Fj\left(v\right)\partial ul}\right\}\Delta tN+KId\right)\hfill \\ & \le & (Id-KId\Delta tN+KId)-(KId\Delta tN+KId\Delta tN+KId)\hfill \\ & \le & (-\alpha )Id,\hfill \end{array}$$

(500)

$\forall u,v\in Br(u^),\forall N>N.$

Hence, denoting $\alpha =(-\alpha ),$ we have got

$$\begin{array}{ccc}& & ∥\left(Id-\left\{{\displaystyle \partial Fj\left(u\right)\partial ul}\right\}\Delta tN+KId\right)-\left(H(u,v)\Delta tN-\left\{{\displaystyle \partial Fj\left(v\right)\partial ul}\right\}\Delta tN+KId\right)∥\hfill \\ & \le & ∥(Id-KId\Delta tN+KId)-(KId\Delta tN+KId\Delta tN+KId)∥\hfill \\ & \le & \alpha .\hfill \end{array}$$

(501)

$\forall u,v\in Br(u^),$$\forall N>N$.

Fix now a new $N>N$.

Suppose now $K=KnN>$ and $<\alpha =\left(\alpha \right)nN<$ are such that

$$u11\in B{\displaystyle rN}(-\alpha )(u^).$$

Reasoning inductively, suppose $u10,u11,\dots ,uk+\in B{\displaystyle rN}\left(u\right),$ and observe that

$$uk+-u01=Fj\left(uk\right)\Delta tN+\Delta tN\left\{{\displaystyle \partial Fj\left(uk\right)\partial ul}\right\}(uk+-uk)-K(uk+-uk).$$

and

$$uk+-u01=Fj\left(uk+\right)\Delta tN+\Delta tN\left\{{\displaystyle \partial Fj\left(uk+\right)\partial ul}\right\}(uk+-uk+)-K(uk+-uk+),$$

so that

$$\begin{array}{ccc}& & \left(Id-\left\{{\displaystyle \partial Fj\left(uk+\right)\partial ul}\right\}\Delta tN+KId\right)(uk+-uk+)\hfill \\ & =& \left((F\left(uk+\right)-F\left(uk\right))\Delta tN-\left\{{\displaystyle \partial Fj\left(uk\right)\partial ul}\right\}\Delta tN+KId\right)(uk+-uk)\hfill \\ & =& \left(H(uk+,uk)\Delta tN-\left\{{\displaystyle \partial Fj\left(uk\right)\partial ul}\right\}\Delta tN+KId\right)(uk+-uk).\hfill \end{array}$$

(502)

Thus,

$$\begin{array}{ccc}& & uk+-uk+\hfill \\ & =& \left(Id-\left\{{\displaystyle \partial Fj\left(uk+\right)\partial ul}\right\}\Delta tN+KId\right)-\left(H(uk+,uk)\Delta tN-\left\{{\displaystyle \partial Fj\left(uk\right)\partial ul}\right\}\Delta tN+KId\right)\hfill \\ & & \times (uk+-uk).\hfill \end{array}$$

(503)

Summarizing, we have got

$$\begin{array}{ccc}& & ∥uk+-uk+∥\hfill \\ & =& ∥\left(Id-\left\{{\displaystyle \partial Fj\left(uk+\right)\partial ul}\right\}\Delta tN+KId\right)-\left(H(uk+,uk)\Delta tN-\left\{{\displaystyle \partial Fj\left(uk\right)\partial ul}\right\}\Delta tN+KId\right)∥\hfill \\ & & \times \parallel uk+-uk\parallel \hfill \\ & \le & \alpha \parallel uk+-uk\parallel .\hfill \end{array}$$

(504)

Thus, we have got

$$\parallel uj+-uj+\parallel \le \alpha \parallel uj+-uj\parallel ,\forall j\in \{,\dots ,k\}.$$

Therefore

$$\begin{array}{ccc}\hfill \parallel uj+-uj+\parallel & \le & \alpha \parallel uj+-uj\parallel \hfill \\ & \le & \alpha 02\parallel uj-uj-\parallel \hfill \\ & ⋮& \hfill \\ & \le & \alpha j+\parallel u11-u10\parallel .\hfill \end{array}$$

(505)

Thus,

$$\begin{array}{ccc}\hfill \parallel uk+-u11\parallel & =& \parallel uk+-uk++uk+-\dots -u12+u12-u11\parallel \hfill \\ & \le & \parallel uk+-uk+\parallel +\parallel uk+-uk\parallel +\dots +\parallel u12-u11\parallel \hfill \\ & \le & \sum j=k+\alpha j\parallel u11-u10\parallel \hfill \\ & \le & \sum j=\infty \alpha j\parallel u11-u10\parallel \hfill \\ & =& {\displaystyle \alpha -\alpha }\parallel u11-u10\parallel .\hfill \end{array}$$

(506)

Therefore

$$\begin{array}{ccc}\hfill \parallel uk+-u10\parallel & \le & \parallel uk+-u11+u11-u10\parallel \hfill \\ & \le & \parallel uk+-u11\parallel +\parallel u11-u10\parallel \hfill \\ & \le & {\displaystyle \alpha -\alpha }\parallel u11-u10\parallel +\parallel u11-u10\parallel \hfill \\ & =& {\displaystyle -\alpha }\parallel u11-u10\parallel \hfill \\ & <& {\displaystyle -\alpha }(-\alpha ){\displaystyle rN}\hfill \\ & =& {\displaystyle rN}.\hfill \end{array}$$

(507)

Summarizing,

$$\parallel uk+-u10\parallel <{\displaystyle rN},$$

so that

$$uk+\in B{\displaystyle rN}\left(u01\right).$$

The induction is complete, so that

$$uk\in B{\displaystyle rN}\left(u01\right),\forall k\in \mathbb{N}.$$

From such results we have also obtained

$$\parallel uk+-uk+\parallel \le \alpha \parallel uk+-uk\parallel ,\forall k\in \mathbb{N}.$$

Thus, from these results and the Banach fixed point theorem, there exists $u\in B{\displaystyle rN}\left(u01\right)$ such that

$$\underset{k\to \infty }{lim}uk=u.$$

$$\begin{array}{ccc}\hfill \mathbf{0}& =& \underset{k\to \infty }{lim}\left(uk+-u^\right.\hfill \\ & & -F\left(uk\right)\Delta tN-\Delta tN\left\{{\displaystyle \partial Fj\left(uk\right)\partial ul}\right\}(uk+-uk)\hfill \\ & & \left.+K(uk+-uk)\right)\hfill \\ & =& u-u^-F\left(u\right)\Delta tN\hfill \end{array}$$

(508)

so that

$${\displaystyle u-u^\Delta tN}=F\left(u\right)\mathrm{in}\Omega ,$$

Reasoning inductively again having $u\in B{\displaystyle rN}(u^)$ and $uj\in B{\displaystyle rN}\left(uj-\right),\forall j\in \{,\dots ,n\}$ similarly as we have obtained $u$ in the last lines, we may obtain

$$un+\in B{\displaystyle rN}\left(un\right),$$

such that

$${\displaystyle un+-un\Delta tN}=F\left(un+\right),\mathrm{in}\Omega .$$

The induction on n is also complete.

Fix $n\in \{,\dots ,N-\}$.

Observe that

$$\begin{array}{ccc}\hfill \parallel un-u\parallel & =& \parallel un-un-+un--un-+\dots -u+u-u\parallel \hfill \\ & \le & \parallel un-un-\parallel +\dots +\parallel u-u\parallel \hfill \\ & \le & {\displaystyle nN}r\hfill \\ & <& r.\hfill \end{array}$$

(509)

Summarizing $un\in Br(u^),\forall n\in \{,,\dots ,N-\}.$

From these results, denoting now more generically $un\equiv unN$, we may infer that there exists $K^>$ such that

$$\parallel ujN\parallel ,,\Omega \le K^,\forall j\in \{,,\dots ,N\},\forall N\in \mathbb{N}.$$

Define now

$$uN(x,t)=unN\left(x\right)\left(n+-{\displaystyle t\Delta tN}\right)+un+N\left(x\right)\left({\displaystyle t\Delta tN}-n\right),$$

if $t\in [n\Delta tN,(n+)\Delta tN],\forall (x,t)\in \Omega \times [,T].$

Observe that

$$uN(x,t)=unN\left(x\right),\mathrm{if}t=n\Delta tN,\forall n\in \{,,\dots ,N\},$$

and

$$\begin{array}{ccc}\hfill {\displaystyle \partial uN(x,t)\partial t}& =& {\displaystyle un+N-unN\Delta tN}\hfill \\ & =& F\left(un+N\right),\hfill \end{array}$$

(510)

if $t\in [n\Delta tN,(n+)\Delta tN],\forall (x,t)\in \Omega \times [,T].$

Fix $\phi \in Cc\infty (\Omega ;\mathbb{R}r).$

Thus, fixing $t\in [n\Delta tN,(n+)\Delta tN],$ we have

$$\begin{array}{ccc}\hfill \left|〈{\displaystyle \partial uN\partial t},\phi 〉L\right|& \le & \left|\langle F\left(un+N\right),\phi \rangle L\right|\hfill \\ & \le & K\parallel \phi \parallel ,,\Omega ,\forall \phi \in Cc\infty (\Omega ;\mathbb{R}r),\hfill \end{array}$$

(511)

for some appropriate $K>.$

Since $\phi \in Cc\infty (\Omega ;\mathbb{R}r)$ is arbitrary, we may conclude that

$$∥{\displaystyle \partial uN\partial t}∥H-(\Omega ;\mathbb{R}r)\le K,\forall N>N,$$

uniformly in t on $[,T],$ for some appropriate constant $K>.$

Also, from the definition of $u$ we have that there exists $K>$ such that

$$\parallel uN\parallel ,,\Omega \le K,\forall N\in \mathbb{N}$$

also uniformly in t on $[,T].$

From such results, there exist $u\in L([,T],H01(\Omega ;\mathbb{R}r))$ and $v\in L([,T];H-(\Omega ;\mathbb{R}r))$ such that

$$uN\rightharpoonup u,\mathrm{weakly}\mathrm{in}L((,T);W,(\Omega ;\mathbb{R}r)),$$

and

$${\displaystyle \partial uN\partial t}\rightharpoonup v,\mathrm{weakly}-\mathrm{star}\mathrm{in}L([,T],H-(\Omega ;\mathbb{R}r)),$$

so that we may easily obtain

$$v={\displaystyle \partial u\partial t}$$

in a distributional sense. At this point, we provide more details about this last result.

Fix $t\in (,T)$. Thus, there exists $n\in \{,,\dots ,N-\}$ such that $t\in [n\Delta tN,(n+)\Delta tN]$.

Let $\phi \in Cc\infty (\Omega \times (,T);\mathbb{R}r).$

From this, we may infer that

$$\begin{array}{ccc}& & \int \Omega {\displaystyle \partial uN\partial t}\phi (x,t)dx\hfill \\ & =& \int \Omega {\displaystyle un+N-unN\Delta tN}\phi (x,t)dx\hfill \\ & \le & \langle F\left(un+N\right),\phi \rangle L\hfill \\ & \le & K\parallel \phi \parallel ,,\Omega ,\hfill \end{array}$$

(512)

for some appropriate constant $K>.$

Hence,

$$\begin{array}{ccc}& & \int T\int \Omega {\displaystyle \partial uN\partial t}\phi (x,t)dxdt\hfill \\ & \le & K\int \Omega \parallel \phi \parallel ,,\Omega dt\hfill \\ & \le & K19\parallel \phi \parallel ,,\Omega \times (,T),\hfill \end{array}$$

(513)

for some appropriate $K19>$.

Since such a $\phi \in Cc\infty (\Omega \times (,T);\mathbb{R}r)$ is arbitrary, we may infer that

$$∥{\displaystyle \partial uN\partial t}∥H-(\Omega \times (,T);\mathbb{R}r)\le K15,$$

for $N\in \mathbb{N}$, for some $K15>.$

From such a result and from the Banach-Alaoglu Theorem, there exists $v\in H-(\Omega \times (,T);\mathbb{R}r)$ such that, up to a not relabeled subsequence

$${\displaystyle \partial uN\partial t}\rightharpoonup v,\mathrm{weakly}-\mathrm{star}\mathrm{in}H-(\Omega \times (,T);\mathbb{R}r).$$

Therefore,

$$\int T\int \Omega {\displaystyle \partial uN\partial t}\phi dxdt\to \int T\int \Omega v\phi dxdt,$$

as $N\to \infty ,$$\forall \phi \in H01(\Omega \times (,T);\mathbb{R}r)$.

On the other hand

$$\parallel uN\parallel ,,\Omega \times (,T)\le K16,$$

$\forall N>N$, for some $K16>.$

From this and the Kakutani Theorem, there exists $u\in L(\Omega \times (,T);\mathbb{R}r)$ such that, up to a not relabeled subsequence,

$$uN\rightharpoonup u,\mathrm{weakly}\mathrm{in}L(\Omega \times (,T);\mathbb{R}r).$$

Now fix again $\phi \in Cc\infty (\Omega \times (,T);\mathbb{R}r).$

Observe that

$$\begin{array}{ccc}\hfill \int T\int \Omega u\phi tdxdt& =& \underset{N\to \infty }{lim}\int T\int \Omega uN\phi tdxdt\hfill \\ & =& -\underset{N\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uN\partial t}\phi dxdt\hfill \\ & =& -\int T\int \Omega v\phi dxdt,\hfill \end{array}$$

(514)

Since such a $\phi \in Cc\infty (\Omega \times (,T);\mathbb{R}r)$ is arbitrary, we may infer that

$$v={\displaystyle \partial u\partial t}$$

in a distributional sense.

Moreover, from such results we may also obtain, again up to a subsequence,

$$\underset{N\to \infty }{lim}\int \Omega {\displaystyle \partial uN\partial t}\phi dx=\int \Omega {\displaystyle \partial u\partial t}\phi dx,$$

$\forall \phi \in H01(\Omega ;\mathbb{R}r).$

Observe also that, as a consequence of the Rellich-Kondrashov theorem, through appropriate subsequences, we have

$$uNk\left(t\right)\to u(x,t),\mathrm{strongly}\mathrm{in}L(\Omega ;\mathbb{R}r),\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

so that, up to subsequences,

$$uNk\left(t\right)(x,t)\to u(x,t),\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

Here we emphasise the sequence $\left\{Nk\left(t\right)\right\}\subset \mathbb{N}$ may depends on t.

Fix $j\in \{,\dots ,r\}.$

Since $g$ is continuous we have that

$$gj\left(uNk\left(t\right)(x,t)\right)\to gj\left(u(x,t)\right),\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

Fix $t\in (,T).$

Let $\epsilon >$. From the Egorov Theorem, there exists a closed set F such that $m(\Omega \setminus F)\le \epsilon$ and $k\in \mathbb{N}$ such that if $k>k$, then

$$|gj\left(uNk\left(t\right)(x,t)\right)-gj\left(u(x,t)\right)|\le \epsilon ,\mathrm{for}\mathrm{almost}\mathrm{all}x\in F.$$

Let $\phi \in Cc\infty \left(\Omega \right).$ Observe now that

$$\begin{array}{ccc}& & \left|\int \Omega (gj\left(uNk\left(t\right)(x,t)\right)-gj\left(u(x,t)\right))\phi dx\right|\hfill \\ & \le & \int \Omega |gj\left(uNk\left(t\right)(x,t)\right)-gj\left(u(x,t)\right)\left|\right|\phi |dx\hfill \\ & =& \int F|gj\left(uNk\left(t\right)(x,t)\right)-gj\left(u(x,t)\right)\left|\right|\phi |dx+\int \Omega \setminus F|gj\left(uNk\left(t\right)(x,t)\right)-gj\left(u(x,t)\right)\left|\right|\phi |dx\hfill \\ & \le & \int F\epsilon \parallel \phi \parallel \infty dx+\int \Omega |gj\left(uNk\left(t\right)(x,t)\right)-gj\left(u(x,t)\right)\left|\right|\phi |\chi \Omega \setminus Fdx\hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+(\parallel gj\left(uNk\left(t\right)\right)\parallel ,,\Omega +\parallel gj\left(u\right)\parallel ,,\Omega )\parallel \phi \parallel ,,\Omega \parallel \chi \Omega \setminus F\parallel ,,\Omega \hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+K21\parallel \phi \parallel ,,\Omega m(\Omega \setminus F)/\hfill \\ & \le & \epsilon \parallel \phi \parallel \infty m\left(\Omega \right)+K21\parallel \phi \parallel ,,\Omega \epsilon /,\forall k>k,\hfill \end{array}$$

(515)

for some appropriate constant $K21>$ which does not depend on t.

Since such a $\epsilon >$ is arbitrary, we may infer that

$$\int \Omega gj\left(uNk\left(t\right)\right)\phi dx\to \int \Omega gj\left(u\right)\phi dx,\mathrm{as}k\to \infty ,$$

$\forall \phi \in Cc\infty \left(\Omega \right),$$\forall j\in \{,\dots ,r\}.$

Similarly, fixing $j,p\in \{,\dots ,n\},$ and $l\in \{,\dots ,m\},$ since $g$ is continuous we have that

$$gjpl\left(uNk\left(t\right)(x,t)\right)\to gjpl\left(u(x,t)\right),\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\mathrm{for}\mathrm{almost}\mathrm{all}t\in [,T].$$

Fix again $t\in (,T)$

Let $\epsilon >$ (a new value). From the Egorov Theorem, there exists a closed set $F$ such that $m(\Omega \setminus F)\le \epsilon$ and $k\in \mathbb{N}$ such that if $k>k$, then

$$|gjpl\left(uNk\left(t\right)(x,t)\right)-gjpl\left(u(x,t)\right)|\le \epsilon ,\mathrm{for}\mathrm{almost}\mathrm{all}x\in F.$$

Observe now that

$$\begin{array}{ccc}& & \int \Omega |gjpl\left(uNk\left(t\right)(x,t)\right)-gjpl\left(u(x,t)\right)|dx\hfill \\ & \le & \int F|gjpl\left(uNk\left(t\right)(x,t)\right)-gjpl\left(u(x,t)\right)|dx+\int \Omega \setminus F|gjpl\left(uNk\left(t\right)(x,t)\right)-gjpl\left(u(x,t)\right)|dx\hfill \\ & \le & \int F\epsilon dx+\int \Omega |gjpl\left(uNk\left(t\right)(x,t)\right)-gjpl\left(u(x,t)\right)|\chi \Omega \setminus Fdx\hfill \\ & \le & \epsilon m\left(\Omega \right)+K12\int \Omega \chi \Omega \setminus Fdx\hfill \\ & \le & \epsilon m\left(\Omega \right)+K12\epsilon ,\forall k>k.\hfill \end{array}$$

(516)

Since such a $\epsilon >$ is arbitrary, we may infer that

$$\int \Omega |gjpl\left(uNk\left(t\right)\right)-gjpl\left(u\right)|dx\to ,\mathrm{as}k\to \infty ,$$

$\forall j,p\in \{,\dots ,r\},$$l\in \{,\dots ,m\}.$

Select again $\phi \in Cc\infty \left(\Omega \right).$ Since

$$\parallel gjpl\left(uNk\left(t\right)\right)-gjpl\left(u\right)\parallel ,,\Omega \to ,\mathrm{as}k\to \infty$$

and

$$\nabla uNk\left(t\right)\rightharpoonup \nabla u,\mathrm{weakly}\mathrm{in}L(\Omega ;\mathbb{R}r\times m),$$

we obtain,

$$\begin{array}{ccc}& & \left|\int \Omega gjpl\left(uNk\left(t\right)\right)\left(\left(u\right)pNk\left(t\right)\right)xl\phi dx-\int \Omega gjpl\left(u\right)\left(\left(u\right)p\right)xl\phi dx\right|\hfill \\ & \le & \left|\int \Omega gjpl\left(uNk\left(t\right)\right)\left(\left(u\right)pNk\left(t\right)\right)xl\phi dx-\int \Omega gjpl\left(u\right)\left(\left(u\right)pNk\left(t\right)\right)xl\phi dx\right|\hfill \\ & & +\left|\int \Omega gjpl\left(u\right)\left(\left(u\right)pNk\left(t\right)\right)xl\phi dx-\int \Omega gjpl\left(u\right)\left(\left(u\right)p\right)xl\phi dx\right|\hfill \\ & \le & \parallel gjpl\left(uNk\left(t\right)\right)-gjpl\left(u\right)\parallel ,,\Omega K\parallel \phi \parallel \infty \hfill \\ & & +\left|\int \Omega gjpl\left(u\right)\left(\left(u\right)pNk\left(t\right)\right)xl\phi dx-\int \Omega gjpl\left(u\right)\left(\left(u\right)p\right)xl\phi dx\right|\hfill \\ & \to & ,\mathrm{as}k\to \infty ,\hfill \end{array}$$

(517)

$\forall j,p\in \{,\dots ,r\},l\in \{,\dots ,m\}.$

From such results, for an arbitrary $\phi \in Cc\infty (\Omega ;\mathbb{R}r)$, we have

$$\begin{array}{ccc}\hfill & =& \underset{k\to \infty }{lim}\left(\int \Omega {\displaystyle \partial \left(u\right)jNk\left(t\right)\partial t}\phi jdx-\langle Fj\left(uNk\left(t\right)\right),\phi j\rangle L\right)\hfill \\ & =& \int \Omega {\displaystyle \partial \left(u\right)j\partial t}\phi jdx+\epsilon j\langle \nabla \left(u\right)j,\nabla \phi j\rangle L-\langle gj\left(u\right),\phi j\rangle L\hfill \\ & & -\sum p=r\sum l=m\langle gjpl\left(u\right)\left(\left(u\right)p\right)xl,\phi j\rangle L-\langle fj,\phi j\rangle L\hfill \end{array}$$

(518)

so that, from this and by the density of $Cc\infty (\Omega ;;\mathbb{R}r)$ in $H01(\Omega ;\mathbb{R}r)$, we have got

$$\begin{array}{ccc}& & \int \Omega {\displaystyle \partial \left(u\right)j\partial t}\phi jdx\hfill \\ & =& -\epsilon j\langle \nabla \left(u\right)j,\nabla \phi j\rangle L+\langle gj\left(u\right),\phi j\rangle L\hfill \\ & & +\sum p=r\sum l=m\langle gjpl\left(u\right)\left(\left(u\right)p\right)xl,\phi j\rangle L+\langle fj,\phi j\rangle L,\hfill \end{array}$$

(519)

$\forall j\in \{,\dots ,r\},\forall \phi \in H01(\Omega ;\mathbb{R}r),$$\mathrm{a}.\mathrm{e}.\mathrm{on}[,T],$

Observe now that

$$\partial (\Omega \times (,T))=\left(\partial \Omega \times [,T]\right)\cup \left(\partial [,T]\times \Omega ¯\right).$$

Let $\phi \in Cc\infty (\Omega \times (,T);\mathbb{R}r).$

Hence

$$\underset{N\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uN\partial t}\phi dxdt=\int T\int \Omega {\displaystyle \partial u\partial t}\phi dxdt.$$

From this, since $Cc\infty (\Omega \times (,T);\mathbb{R}r)$ is dense $L(\Omega \times (,T);\mathbb{R}r)$ we may infer that

$$\underset{N\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uN\partial t}\phi dxdt=\int T\int \Omega {\displaystyle \partial u\partial t}\phi dxdt,$$

$\forall \phi \in L(\Omega \times (,T);\mathbb{R}r).$

Let $\phi \in C\infty (\Omega \times [,T];\mathbb{R}r)$ such that

$$\phi (x,T)=,\mathrm{in}\Omega .$$

From such results, we may obtain

$$\begin{array}{ccc}& & \underset{N\to \infty }{lim}\int T\int \Omega {\displaystyle \partial uN\partial t}\phi dxdt\hfill \\ & =& \underset{N\to \infty }{lim}\left(-\int T\int \Omega uN{\displaystyle \partial \phi \partial t}dxdt-\int \Omega uN(x,)\phi (x,)dx\right)\hfill \\ & =& -\int T\int \Omega u{\displaystyle \partial \phi \partial t}dxdt-\int \Omega u(x,)\phi (x,)dx.\hfill \end{array}$$

(520)

However, since $uN\rightharpoonup u,\mathrm{weakly}\mathrm{in}L(\Omega \times (,T);\mathbb{R}r)$, we obtain

$$\underset{N\to \infty }{lim}\int T\int \Omega uN{\displaystyle \partial \phi \partial t}dxdt=\int T\int \Omega u{\displaystyle \partial \phi \partial t}dxdt.$$

From these last results, we may infer that

$$\begin{array}{ccc}\hfill \int \Omega u^\phi (x,)dx& =& \underset{N\to \infty }{lim}\int \Omega uN(x,)\phi (x,)dx\hfill \\ & =& \int \Omega u(x,)\phi (x,)dx,\hfill \end{array}$$

(521)

so that

$$\int \Omega u^\left(x\right)\phi (x,)dx=\int \Omega u(x,)\phi (x,)dx,$$

$\forall \phi \in C\infty (\Omega \times [,T];\mathbb{R}r)$ such that $\phi (x,T)=,\mathrm{in}\Omega .$

Therefore, we may infer that $u(x,)=u^\left(x\right)$ in this specified weak sense.

Similarly, it may be proven that

$$u=,\mathrm{on}\partial \Omega \times [,T],$$

in an appropriate weak sense.

Hence, we have obtained that $u$ is a solution, in a weak sense, of the parabolic non-linear system in question.

64. A note on the convergence of the finite elements method

In this section we develop some remarks on the convergence of the finite elements method.

This section is based on reference [18], Chapter 7.

For the proofs not presented here and for more details please see reference [18], Chapter 7.

We start by recalling the following classical result.

Theorem 12

(Lax-Milgram). Let V be a separable Hilbert space with a inner product

$$\langle ·,·\rangle :V\times V\to \mathbb{R},$$

and related norm

$$\parallel ·\parallel :V\to \mathbb{R}+$$

where

$$\parallel u\parallel =\sqrt{\langle u,u\rangle },\forall u\in V.$$

Let $a:V\times V\to \mathbb{R}$ be a bilinear form such that

a is continuous, that is, there exists $M>$ such that

$$\left|a\right(u,v\left)\right|\le M\parallel u\parallel \parallel v\parallel ,\forall u,v\in V,$$

a is coercive, that is, there exists $\alpha >$ such that

$$a(v,v)\ge \alpha \parallel v\parallel ,\forall u\in V.$$

Moreover, let $L:V\to \mathbb{R}$ be a linear and continuous functional.

Under such hypotheses, there exists a unique $u\in V$ such that

$$a(u,v)=L\left(v\right),\forall v\in V.$$

Definition 2.

Let V be a Banach space. We say that a sequence $\left\{Vn\right\}$ of finite dimensional subspaces of V is a Galerkin scheme for V if for each $v\in V$, there exists a sequence $\left\{vk\right\}\subset \cup n=\infty Vn$ where $vk\in Vk,\forall k\in \mathbb{N},$ such that

$$vk\to v,\mathrm{strongly}\mathrm{in}\mathrm{norm},\mathrm{as}k\to \infty .$$

Remark 30.

Let $\Omega \subset \mathbb{R}$ be a polygonal set. A triangulation T of Ω is a finite union of subsets of $\Omega ¯,$ such that

$$\Omega ¯=\cup K\in TK,$$

Each set $K\in T$ is a triangle,

For each pair $K,K\in T$, such sets are quasi-disjoints, that is, their interiors are disjoint.

We define

$$h\left(T\right)=\underset{K\in T}{max}diam\left(K\right)\equiv h,$$

where

$$diam\left(K\right)=sup\left\{\right|x,y|:x,y\in K\}.$$

In such a case we also denote $T=Th.$

Moreover, we define

$$Vh=\{v\in C(\Omega ¯):v\mathrm{is}\mathrm{affine}\mathrm{on}\mathrm{each}K\in Th\mathrm{and}v=,\mathrm{on}\partial \Omega \}.$$

We denote by $a$ the vertices in the triangulation $T$, where

$$j\in \{,\dots ,I(h\left)\right\}.$$

Let $\left\{\phi j\right\}\subset Vh$ be such that

$$\phi j\left(ak\right)=\delta jk,\forall \le j,k\le I\left(h\right).$$

Here

$$\delta jk=\left\{\begin{array}{c},\mathrm{if}j=k,\hfill \\ \\ ,\mathrm{if}j\ne k.\hfill \end{array}\right.$$

(522)

Observe that $\{\phi ,\dots ,\phi I\left(h\right)\}$ is a basis for $Vh.$

At this point we define

$$Ph\left(v\right)=\sum j=I\left(h\right)v\left(aj\right)\phi j,\forall v\in V.$$

Here we assume $\left\{Th\right\}$ be a regular family of triangulations of $\Omega .$

Let $\left\{hn\right\}\subset \mathbb{R}+$ be a sequence such that

$$<hn+<hn,\forall n\in \mathbb{N},$$

and

$$\underset{n\to \infty }{lim}hn=.$$

We denote $Vn=Vhn$ and $Pn=Phn,\forall n\in \mathbb{N}.$

Consider the Ginzburg-Landau type equation

$$\left\{\begin{array}{cc}-\gamma \nabla u+\alpha u-\beta u-f=,\hfill & \mathrm{in}\Omega ,\hfill \\ \\ u=,\mathrm{on}\partial \Omega .\hfill \end{array}\right.$$

(523)

Here $\gamma >,$$\alpha >,$$\beta >$ and $f\in L\left(\Omega \right).$

Assume $un\in Vn$ is a weak solution of this last equation, in the following sense,

$$\begin{array}{ccc}& & \gamma \langle \nabla un,\nabla \phi \rangle L+\alpha \langle un,\phi \rangle L\hfill \\ & & -\beta \langle un,\phi \rangle L-\langle f,\phi \rangle L,\hfill \\ & =& ,\forall \phi \in Vn.\hfill \end{array}$$

(524)

We assume there exist $u\in H01\cap W,\infty \left(\Omega \right)$, $r>$, $\alpha >$ and $M>$ such that

$$\begin{array}{ccc}& & \alpha \parallel u-v\parallel ,,\Omega \hfill \\ & \le & \gamma \langle \nabla (u-v),\nabla (u-v)\rangle L+\alpha \langle u˜(u,v)(u-v),(u-v)\rangle L-\beta \langle (u-v),(u-v)\rangle L,\hfill \end{array}$$

and

$$\begin{array}{ccc}& & \left|\gamma \langle \nabla (u-v),\nabla (u-w)\rangle L+\alpha \langle u˜(u,v)(u-v),(u-w)\rangle L-\beta \langle (u-v),(u-w)\rangle L\right|\hfill \\ & \le & M\parallel u-v\parallel ,,\Omega \parallel u-w\parallel ,,\Omega ,\hfill \end{array}$$

(525)

$\forall u,v,w\in Br\left(u\right).$

Here $u˜(u,v)$ is on the line connecting u and v so that

$$u-v=u˜(u-v).$$

Similarly as we have done in previous sections, we assume $u$ and $r>$ are such that we may obtain $un\in Br\left(u\right)$, $\forall n\in \mathbb{N}.$

Also similarly as in the previous section, we may consider such a ball either related to the $H01\left(\Omega \right)$ norm or the $W,\infty \left(\Omega \right)$ one.

Let $m,n\in \mathbb{N}$ be such that $m>n.$

Observe that $un,um\in Vm$ so that

$$\begin{array}{ccc}& & \gamma \langle \nabla um,\nabla (un-um)\rangle L+\alpha \langle um,(un-um)\rangle L\hfill \\ & & -\beta \langle um,(un-um)\rangle L-\langle f,(un-um)\rangle L\hfill \\ & =& ,\hfill \end{array}$$

(526)

so that, for $\phi \in Vn$, we obtain

$$\begin{array}{ccc}& & \gamma \langle \nabla (un-um),\nabla (un-um)\rangle L+\alpha \langle (un-um),(un-um)\rangle L\hfill \\ & & -\beta \langle (un-um),(un-um)\rangle L\hfill \\ & =& \gamma \langle \nabla (un-um),\nabla (un-\phi )\rangle L+\alpha \langle (un-um),(un-\phi )\rangle L\hfill \\ & & -\beta \langle (un-um),(un-\phi )\rangle L\hfill \\ & & +\gamma \langle \nabla (un-um),\nabla (\phi -um)\rangle L+\alpha \langle (un-um),(\phi -un)\rangle L\hfill \\ & & -\beta \langle (un-um),(\phi -um)\rangle L\hfill \\ & =& \gamma \langle \nabla (un-um),\nabla (\phi -um)\rangle L+\alpha \langle (un-um),(\phi -um)\rangle L\hfill \\ & & -\beta \langle (un-um),(\phi -um)\rangle L.\hfill \end{array}$$

(527)

Summarizing, we have got

$$\begin{array}{ccc}& & \gamma \langle \nabla (un-um),\nabla (un-um)\rangle L+\alpha \langle u˜n(un-um),(un-um)\rangle L\hfill \\ & & -\beta \langle (un-um),(un-um)\rangle L\hfill \\ & =& \gamma \langle \nabla (un-um),\nabla (\phi -um)\rangle L+\alpha \langle u˜n(un-um),(\phi -um)\rangle L\hfill \\ & & -\beta \langle (un-um),(\phi -um)\rangle L\hfill \end{array}$$

(528)

$\forall \phi \in Vn,$ where $u$ is on the line connecting $u$ and $u$.

Here we recall that $\alpha >$ and $M>$ are such that

$$\begin{array}{ccc}& & \gamma \langle \nabla (un-um),\nabla (un-um)\rangle L+\alpha \langle u˜n(un-um),(un-um)\rangle L\hfill \\ & & -\beta \langle (un-um),(un-um)\rangle L\hfill \\ & \ge & \alpha \parallel un-um\parallel ,,\Omega \hfill \end{array}$$

(529)

and

$$\begin{array}{ccc}& & \gamma \langle \nabla (un-um),\nabla (\phi -um)\rangle L+\alpha \langle u˜n(un-um),(\phi -um)\rangle L\hfill \\ & & -\beta \langle (un-um),(un-\phi )\rangle L\hfill \\ & \le & M\parallel um-un\parallel ,,\Omega \parallel \phi -um\parallel ,,\Omega ,\hfill \end{array}$$

(530)

where $\alpha$ and M does not depend on $m,n$.

From such results, we may infer that

$$\parallel um-un\parallel ,,\Omega \le {\displaystyle M\alpha }\parallel um-\phi \parallel ,,\Omega ,\forall \phi \in Vn$$

so that

$$\parallel um-un\parallel ,,\Omega \le {\displaystyle M\alpha }\parallel um-Pn\left(um\right)\parallel ,,\Omega ,\forall m>n.$$

Moreover, since $um\in H01\left(\Omega \right)$, there exists a sequence $\{vk=vkm\}\subset Cc\infty \left(\Omega \right)$ such that

$$\parallel vkm-um\parallel ,,\Omega \to ,\mathrm{as}k\to \infty .$$

From such results, for a not relabeled subsequence we have

$$vkm\to um,\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,$$

$$\nabla vkm\to \nabla um,\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega .$$

Let $\epsilon >$.

From the Egorov theorem, for each $m\in \mathbb{N}$ there exists a closed set $Fm\subset \Omega$ such that $m\left(Fm\right)<\epsilon /m$ and

$$vkm\to um,\mathrm{uniformly}\mathrm{in}\Omega \setminus Fm,$$

$$\nabla vkm\to \nabla um,\mathrm{uniformly}\mathrm{in}\Omega \setminus Fm.$$

Define $F=\cup m=\infty Fm$ so that

$$m\left(F\right)\le \sum m=\infty \left(Fm\right)\le \sum m=\infty \epsilon /m\le \epsilon .$$

Observe that there exists $k=km\in \mathbb{N}$ such that if $k>k=km$, then

$$\parallel vkm-um\parallel ,,\Omega <\epsilon ,$$

and

$$\parallel vkm-um\parallel \infty ,\Omega \setminus Fm<\epsilon ,$$

and

$$\parallel \nabla vkm-\nabla um\parallel \infty ,\Omega \setminus Fm<\epsilon .$$

Fixing $m\in \mathbb{N}$ we may find $j\in \mathbb{N}$ (which does not depend on m) and $lm\in \mathbb{N}$

$$\parallel Pj\left(vkl\right)-Pj\left(ul\right)\parallel ,,\Omega \setminus Fl<K\epsilon ,$$

and

$$\parallel \nabla Pj\left(vkl\right)-\nabla Pj\left(ul\right)\parallel ,,\Omega \setminus Fl<K\epsilon ,\forall j>j,\forall l>lm\in \mathbb{N},\forall k>kl,$$

for some appropriate real constants $K>,K>.$

At this point we highlight that concerning the finite elements method $\parallel um\parallel ,\infty$ is uniformly bounded in m so that

$$\{\parallel vkm\parallel ,\infty ,k>km,m\in \mathbb{N}\}$$

is also uniformly bounded in m and $k>km.$

With such results in mind, fix $n>j$ and select $mn>max\{n,kn,ln\}$ so that for $\forall m>mn$ and $k>km$, we have

$$\begin{array}{ccc}& & \parallel \nabla um-\nabla Pn\left(vkm\right)\parallel ,,\Omega /Fm\hfill \\ & \le & \parallel \nabla um-\nabla vkm+\nabla vkm-\nabla Pn\left(vkm\right)\parallel ,,\Omega /Fm\hfill \\ & \le & \parallel \nabla um-\nabla vkm\parallel ,,\Omega +\parallel \nabla vkm-\nabla Pn\left(vkm\right)\parallel ,,\Omega /Fm\hfill \\ & \le & \epsilon +K/n,\hfill \end{array}$$

(531)

for some appropriate $K>.$

From such results, we may infer that

$$\begin{array}{ccc}& & \parallel um-Pn\left(um\right))\parallel ,,\Omega \hfill \\ & \le & \parallel um-Pn\left(vkm\right)+Pn\left(vkm\right)-Pn\left(um\right)\left)\right)\parallel ,,\Omega \hfill \\ & \le & \parallel um-Pn\left(vkm\right)\parallel ,,\Omega +\parallel Pn\left(vkm\right)-Pn\left(um\right)\parallel ,,\Omega /Fm+\parallel Pn\left(vkm\right)-Pn\left(um\right)\parallel ,,Fm\hfill \\ & \le & K(\epsilon +K/n+\epsilon /),\hfill \end{array}$$

(532)

for some appropriate $K>,$ so that

$$\begin{array}{ccc}\hfill \parallel um-un\parallel ,,\Omega & \le & {\displaystyle M\alpha }\parallel um-Pn\left(um\right)\parallel ,,\Omega \hfill \\ & \le & K10(\epsilon +K/n+\epsilon /),\forall m>mn,\hfill \end{array}$$

(533)

where

$$K10=K{\displaystyle M\alpha }.$$

Therefore, if $p,l>mn$, then

$$\begin{array}{ccc}\hfill \parallel ul-un\parallel ,,\Omega & \le & {\displaystyle M\alpha }\parallel ul-Pn\left(ul\right)\parallel ,,\Omega \hfill \\ & \le & K10(\epsilon +K/n+\epsilon /)\hfill \end{array}$$

(534)

and

$$\begin{array}{ccc}\hfill \parallel up-un\parallel ,,\Omega & \le & {\displaystyle M\alpha }\parallel up-Pn\left(up\right)\parallel ,,\Omega \hfill \\ & \le & K10(\epsilon +K/n+\epsilon /),\hfill \end{array}$$

(535)

so that

$$\begin{array}{ccc}\hfill \parallel ul-up\parallel ,,,\Omega & =& \parallel ul-un+un-up\parallel ,,\Omega \hfill \\ & \le & \parallel ul-un\parallel ,,\Omega +\parallel up-un\parallel ,,\Omega \hfill \\ & \le & K10(\epsilon +K/n+\epsilon /).\hfill \end{array}$$

(536)

Consequently, from such results we may infer that $\left\{un\right\}$ is a Cauchy sequence in $H01\left(\Omega \right)$ so that there exists $u\in H01\left(\Omega \right)$ such that

$$un\to u,\mathrm{strongly}\mathrm{in}H01\left(\Omega \right).$$

Let $\phi \in \cup n\in \mathbb{N}Vn.$

Indeed, we have got

$$\begin{array}{ccc}\hfill & =& \underset{n\to \infty }{lim}\left(\gamma \langle \nabla un,\nabla \phi \rangle L+\alpha \langle un,\phi \rangle L\right.\hfill \\ & & -\beta \langle un,\phi \rangle L\left.-\langle f,\phi \rangle L\right)\hfill \\ & =& \gamma \langle \nabla u,\nabla \phi \rangle L+\alpha \langle u03,\phi \rangle L\hfill \\ & & -\beta \langle un,\phi \rangle L-\langle f,\phi \rangle L.\hfill \end{array}$$

(537)

Summarizing, we may infer that

$$\begin{array}{ccc}& & \gamma \langle \nabla u,\nabla \phi \rangle L+\alpha \langle u03,\phi \rangle L\hfill \\ & & -\beta \langle un,\phi \rangle L-\langle f,\phi \rangle L\hfill \\ & =& ,\forall \phi \in H01\left(\Omega \right).\hfill \end{array}$$

(538)

Therefore $u\in H01\left(\Omega \right)$ is a weak solution of the equation in question so that, under the indicated hypotheses, the finite element method is convergent.

65. A dual functional for a general weak primal variational formulation combined with the Newton’s Method

Let $\Omega =[,]\subset \mathbb{R}$ and consider a weak variational formulation for a Ginzburg-Landau type equation corresponding to a functional $J:V\times V\to \mathbb{R}$, where

$$J(u,v*)=\int \Omega v*(-\epsilon u\prime \prime +Au-Bu-f)dx,$$

where $\epsilon >$, $A>$, $B>$ and $f\in Y=Y*=L\left(\Omega \right).$

Moreover $u\in V=W,\left(\Omega \right).$

Observe that the variation in $v$ of J, which stands for

$${\displaystyle \partial J(u,v*)\partial v*}=\mathbf{0},$$

corresponds to the following Ginzburg-Landau type equation

$$-\epsilon u\prime \prime +Au-Bu-f=,\mathrm{in}\Omega .$$

In a Newton type approach context, we linearize such an equation about a initial solution $u\in V$, obtaining,

$$-\epsilon u\prime \prime +Au02u-Au03-Bu-f=,\mathrm{in}\Omega .$$

With such results in mind, we define the functional $J:\left[V\right]\to \mathbb{R}$, where

$$J(u,u,v*)=\int \Omega v*(-\epsilon u\prime \prime +Au02u-Au03-Bu-f)dx.$$

We also define the functionals $F:\left[V\right]\to \mathbb{R}$ and $F:V\to \mathbb{R}$, where

$$F(u,u,v*)=J(u,u,v*)+{\displaystyle K}\int \Omega udx,$$

and

$$F\left(u\right)={\displaystyle K}\int \Omega udx.$$

Moreover, we define the polar functionals $F*:\left[V\right]\times Y*\to \mathbb{R}$ and $F*:Y*\to \mathbb{R}$ as

$$F*(u,v*,z*)=\underset{u\in V}{sup}\{\langle u,z*\rangle L-F(u,u,v*)\},$$

and

$$\begin{array}{ccc}\hfill F*(z*)& =& \underset{v\in Y}{sup}\{\langle v,z*\rangle L-F\left(v\right)\}\hfill \\ & =& {\displaystyle K}\int \Omega (z*)dx.\hfill \end{array}$$

(539)

Finally, we define the dual functional $J*:\left[V\right]\times Y*\to \mathbb{R}$ by

$$J*(u,v*,z*)=-F*(u,v*,z*)+F*(z*).$$

Remark 31.

Observe that

$$F*(u,v*,z*)=\underset{u\in V}{sup}\{\langle u,z*\rangle L-F(u,u,v*)\},$$

and such a supremum is attained through the equation

$${\displaystyle \partial \partial u}\left(\langle u,z*\rangle L-F(u,u,v*)\right)=,$$

which stands for

$$z*-(-\epsilon (v*)\prime \prime +Au02v*-Bv*)-Ku=,$$

so that

$$u={\displaystyle \epsilon (v*)\prime \prime -Au02v*+Bv*+z*K}.$$

Consequently, we may obtain

$$\begin{array}{ccc}\hfill F*(u,v*,z*)& =& {\displaystyle K}\int \Omega (\epsilon (v*)\prime \prime -Au02v*+Bv*+z*)dx\hfill \\ & & +\int \Omega (Au03+f)v*dx.\hfill \end{array}$$

(540)

Hence, the variation in $v$ of $J$,

$${\displaystyle \partial J*\partial v*}=-{\displaystyle \partial F*(v*)\partial v*}=\mathbf{0},$$

stands for

$$-\epsilon u\prime \prime +Au02u-Au03-Bu-f=,\mathrm{in}\Omega ,$$

where, as above indicated,

$$u={\displaystyle \epsilon (v*)\prime \prime -Au02v*+Bv*+z*K}.$$

We have obtained a critical of $J$ through the following algorithm.

Set $n=$, $b12=10-$, $nmax=100$, $z*=$ and choose $\left(u\right)\in V.$

Calculate $(v*)n\in V$ such that

$${\displaystyle \partial J*(\left(u\right)n,(v*)n,zn*)\partial v*}=\mathbf{0},$$

Calculate $un\in V$ such that

$${\displaystyle \partial H(un,\left(u\right)n,(v*)n,zn*)\partial u}=\mathbf{0},$$

where

$$H(u,\left(u\right),v*,z*)=\langle u,z*\rangle -F(u,u,v*),$$

so that

$$un={\displaystyle \epsilon (v*)n\prime \prime -A\left(u\right)n(v*)n+B(v*)n+zn*K}.$$

Set $\left(u\right)n+=un$ and $zn+*=Kun.$

If $\parallel \left(u\right)n+-\left(u\right)n\parallel \infty <b12$ or $n>nmax$, then stop. Otherwise, $n:=n+,$ and go to item 2.

Here we highlight that if $u^=limn\to \infty un$ with corresponding limits $v$ and $z^*=Ku^$, the solution of equation indicated in the item 2, given by

$${\displaystyle \partial J*(u^,v^*,z^*)\partial v*}=\mathbf{0},$$

will stand for

$$-\epsilon u^\prime \prime +Au^03-Bu^-f=,\mathrm{in}\Omega .$$

We have obtained numerical results for $\epsilon =.$, $A=B=$ and $f\equiv ,\mathrm{in}\Omega .$

For such an optimal solution $u$ obtained please see Figure 45.

Here we present the software in MAT-LAB through which we have obtained such numerical results.

*******************************

clear all

global m8 d yo K uo u z A B e1 v1

m8=100;

d=1/m8;

K=10;

A=1;

B=1;

e1=0.1;

z(:,1)=0.1*ones(m8,1);

yo(:,1)=ones(m8,1);

uo(:,1)=1.2*ones(m8,1);

for i=1:m8

xo(i,1)=1.2;

end

b12=1.0;

k=1;

while $(b12>10-)$ &&$(k<100)$

k

k=k+1;

b14=1.0;

k1=1;

while $(b14>10-)$ &&$(k<35)$

k1

k1=k1+1;

X=fminunc(’funJune2024C10’,xo);

b14=max(abs(X-xo));

xo=X;

u(m8/2,1)

end;

b12=max(abs(u-uo));

uo=u;

z=K*u;

end;

for i=1:m8

x(i,1)=i*d;

end;

plot(x,u);

*************************************

With the auxiliary function "funJune2024C10", where

*******************************

function S=funJune2024C10(x)

global m8 d yo K uo u z A B e1 v1

for i=1:m8

v1(i,1)=x(i,1);

end;

v1(m8,1)=0;

d2v1(1,1)=(-2*v1(1,1)+v1(2,1))/$d$;

for i=2:m8-1

d2v1(i,1)=(v1(i+1,1)-2*v1(i,1)+v1(i-1,1))/$d$;

end;

for i=1:m8-1

u(i,1)=$(e*dv(i,)+z(i,)-*A*uo(i,)*v(i,)+B*v(i,))/K;$

end;

u(m8,1)=0;

S=0;

for i=1:m8-1

S=S+$(-e*dv(i,)*u(i,))+v(i,)**A*uo(i,)*u(i,)$

$-B*v(i,)*u(i,)+K*u(i,)/-yo(i,)*v(i,)-*A*uo(i,)*v(i,);$

S=S $-z(i,)*u(i,)-v(i,)/$;

end;

S=-S;

**********************************

66. A New Convex Dual Variational Formulation for a Galerkin Type Non-Convex Primal One

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider a functional $J:V\times \left[Y\right]\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J(u,v,v)& =& {\displaystyle 12}\int \Omega \left(-\epsilon \nabla u+vu-f+v+K{\displaystyle u22}+K{\displaystyle v022}\right)dx\hfill \\ & & +{\displaystyle 12}\int \Omega \left(v+K{\displaystyle u22}+K{\displaystyle v022}\right)dx\hfill \\ & & +{\displaystyle 12}\int \Omega (v-A(u-B))dx.\hfill \end{array}$$

(541)

Here $\epsilon >,A>,B>,K>$, $f\in L\left(\Omega \right)\cap L\infty \left(\Omega \right),$$u\in V=W,\left(\Omega \right)$, $v,v\in Y=Y*=L\left(\Omega \right).$

Observe that the minimization of J corresponds to the solution of the following system of equations:

$$-\epsilon \nabla u+vu-f+v+K{\displaystyle u}+K{\displaystyle v02}=,\mathrm{in}\Omega ,$$

$$v+K{\displaystyle u22}+K{\displaystyle v022}=,\mathrm{in}\Omega ,$$

and

$$v-A(u-B)=,\mathrm{in}\Omega .$$

From such a solution we may obtain the solution of the following Ginzburg-Landau type equation:

$$-\epsilon \nabla u+A(u-B)u-f=,\mathrm{in}\Omega ,$$

which is our final objective in this section.

Define the approximate relaxed functional $J:V\times \left[Y\right]\to \mathbb{R}$ where

$$J(u,v,v)=J(u,v,v)+{\displaystyle \epsilon 12}\int \Omega v12dx,$$

where $\epsilon >$ is a small real constant.

Observe that

$$\begin{array}{ccc}\hfill J(u,v,v)& =& -〈-\epsilon \nabla u+vu-f+v+K{\displaystyle u22}+K{\displaystyle v022},v*〉L\hfill \\ & & +{\displaystyle 12}\int \Omega \left(-\epsilon \nabla u+vu-f+v+K{\displaystyle u22}+K{\displaystyle v022}\right)dx\hfill \\ & & -〈v+K{\displaystyle u22}+K{\displaystyle v022},v*〉L\hfill \\ & & +{\displaystyle 12}\int \Omega \left(v+K{\displaystyle u22}+K{\displaystyle v022}\right)dx\hfill \\ & & -〈v-A(u-B),v*〉L+{\displaystyle 12}\int \Omega (v-A(u-B))dx\hfill \\ & & +〈-\epsilon \nabla u+vu-f+v+K{\displaystyle u22}+K{\displaystyle v022},v*〉L\hfill \\ & & +〈v+K{\displaystyle u22}+K{\displaystyle v022},v*〉L\hfill \\ & & +〈v-A(u-B),v*〉L+{\displaystyle \epsilon }\int \Omega v12dx\hfill \\ & \ge & \underset{w\in Y}{inf}\left\{-\langle w,v*\rangle L+{\displaystyle 12}\int \Omega w12dx\right\}\hfill \\ & & +\underset{w\in Y}{inf}\left\{-\langle w,v*\rangle L+{\displaystyle 12}\int \Omega w22dx\right\}\hfill \\ & & +\underset{w\in Y}{inf}\left\{-\langle w,v*\rangle L+{\displaystyle 12}\int \Omega w32dx\right\}\hfill \\ & & +\underset{(u,v,v)\in V\times Y}{inf}\left\{〈-\epsilon \nabla u+vu-f+v+K{\displaystyle u}+K{\displaystyle v},v*〉L\right.\hfill \\ & & +〈v+K{\displaystyle u22}+K{\displaystyle v022},v*〉L\hfill \\ & & \left.+〈v-A(u-B),v*〉L+{\displaystyle \epsilon }\int \Omega v12dx\right\}\hfill \\ & =& -{\displaystyle 12}\int \Omega (v*)dx-{\displaystyle 12}\int \Omega (v*)dx-{\displaystyle 12}\int \Omega (v*)dx\hfill \\ & & -F*(v*,v*,v*),\forall v*=(v*,v*,v*)\in A*,\hfill \end{array}$$

(542)

where

$$A*=\{v*\in [Y*]:v*+v*\ge \mathrm{and}v*\le ,\mathrm{in}\Omega ,v*=,\mathrm{on}\partial \Omega \}.$$

Also

$$\begin{array}{ccc}& & F*(v*)=F*(v*,v*,v*)\hfill \\ & =& \underset{(u,v,v)\in V\times Y}{sup}\left\{-〈-\epsilon \nabla u+vu-f+v+K{\displaystyle u}+K{\displaystyle v},v*〉L\right.\hfill \\ & & -〈v+K{\displaystyle u22}+K{\displaystyle v022},v*〉L\hfill \\ & & \left.-〈v-A(u-B),v*〉L-{\displaystyle \epsilon 12}\int \Omega v12dx\right\}.\hfill \end{array}$$

(543)

Hence, defining $J*:A*\to \mathbb{R}$ by

$$J*(v*)=-{\displaystyle 12}\int \Omega (v*)dx-{\displaystyle 12}\int \Omega (v*)dx-{\displaystyle 12}\int \Omega (v*)dx-F*(v*),$$

we have obtained

$$\underset{(u,v,v)\in V\times \left[Y\right]}{inf}J(u,v,v)\ge \underset{v*\in A*}{sup}J*(v*).$$

Remark 32.

We highlight that for $K>$ sufficiently large, $J$ is concave on the convex set $A$. Moreover, this last inequality is in fact an equality so that there is no duality gap between such approximate primal and dual formulations.

66.1. A Numerical Example for a related similar functional

In order to obtain numerical results we proceed in following fashion.

Firstly we define $\Omega =[,]\subset \mathbb{R}$ and in a Newton’s method context, we linearize the Ginzburg-Landau equation in question namely,

$$-\epsilon u\prime \prime +Au-Bu-f=,\mathrm{in}\Omega$$

about an initial solution $u$, obtaining the following approximate equation

$$-\epsilon u\prime \prime +Au02u-Au03-Bu-f=,\mathrm{in}\Omega .$$

Now we define the functional $J*:\left[V\right]\times [Y*]\to \mathbb{R}$, where

$$\begin{array}{ccc}\hfill J*(u,u,w,v*,v*)& =& 〈-\epsilon u\prime \prime +Au02u-Au03-Bu-f+w+{\displaystyle K}u,v*〉L\hfill \\ & & +〈w+{\displaystyle K}u,v*〉L-{\displaystyle 12}\int \Omega (v*)dx-{\displaystyle 12}\int \Omega (v*)dx.\hfill \end{array}$$

(544)

Again, similarly as in the Newton’s method approach, we obtain a quadratic approximation for the non-quadratic terms

$${\displaystyle K}u(v*+v*),$$

expressed by

$$Ku((v*)+(v*))-{\displaystyle K}u02(v*+v*).$$

With replacements in mind, we define the functional $J*:\left[V\right]\times [Y*]\to \mathbb{R}$, where

$$\begin{array}{ccc}\hfill J*(u,u,w,v*,v*,(v*),(v*))& =& 〈-\epsilon u\prime \prime +Au02u-Au03-Bu-f,v*〉L\hfill \\ & & +〈Ku,(v*)+(v*)〉L-〈{\displaystyle K}u02,v*+v*〉L\hfill \\ & & +\langle w,v*+v*\rangle L-{\displaystyle 12}\int \Omega (v*)dx-{\displaystyle 12}\int \Omega (v*)dx.\hfill \end{array}$$

(545)

Let $u^\in V$ be such that

$$\left.{\displaystyle \partial J*(u,u,w,v*,v*,(v*),(v*))\partial u}\right|u=u^=\mathbf{0},$$

so that we define the functional $J*:V\times [Y*]\to \mathbb{R}$ by

$$J*(u,w,v*,v*,(v*),(v*))=J*(u^,u,w,v*,v*,(v*),(v*)).$$

The variation in $v$ of $J$ stands for

$$\begin{array}{ccc}\hfill {\displaystyle \partial J*(u,w,v*,v*,(v*),(v*))\partial v*}& =& {\displaystyle \partial J*(u^,u,w,v*,v*,(v*),(v*))\partial u}{\displaystyle \partial u^\partial v*}\hfill \\ & & +{\displaystyle \partial J*(u^,u,w,v*,v*,(v*),(v*))\partial v*}\hfill \\ & =& {\displaystyle \partial J*(u^,u,w,v*,v*,(v*),(v*))\partial v*}.\hfill \end{array}$$

(546)

Similarly, the variation of $J$ in $v$, stands for

$$\begin{array}{ccc}\hfill {\displaystyle \partial J*(u,w,v*,v*,(v*),(v*))\partial v*}& =& {\displaystyle \partial J*(u^,u,w,v*,v*,(v*),(v*))\partial u}{\displaystyle \partial u^\partial v*}\hfill \\ & & +{\displaystyle \partial J*(u^,u,w,v*,v*,(v*),(v*))\partial v*}\hfill \\ & =& {\displaystyle \partial J*(u^,u,w,v*,v*,(v*),(v*))\partial v*}.\hfill \end{array}$$

(547)

Summarizing, a critical point of $J$ must satisfy the following equations:

$${\displaystyle \partial J*(u^,u,w,v*,v*,(v*),(v*))\partial u}=\mathbf{0},$$

which stands for

$$-\epsilon (v*)\prime \prime +Au02v*-Bv*+Ku((v*)+(v*))=\mathbf{0},$$

$${\displaystyle \partial J*(u^,u,w,v*,v*,(v*),(v*))\partial v*}=\mathbf{0},$$

which stands for

$$-\epsilon u\prime \prime +Au02u-Au03-Bu-f+w-{\displaystyle K}u02-v*=\mathbf{0},$$

$${\displaystyle \partial J*(u^,u,w,v*,v*,(v*),(v*))\partial v*}=\mathbf{0},$$

which stands for

$$w-{\displaystyle K}u02-v*=\mathbf{0},$$

and

$${\displaystyle \partial J*(u^,u,w,v*,v*,(v*),(v*))\partial w}=\mathbf{0},$$

which stands for

$$v*+v*=\mathbf{0}.$$

It is worth highlighting such a system is linear in $(u,w,v*,v*)$ so that we have obtained numerical results, in a Newton’s method context, through the following algorithm.

Set $n=$, $b12=10-$, $nmax=100,$$(v*)\equiv .$, $(v*)\equiv .$ and $\left(u\right)\equiv ..$

Calculate $(un,\left(w\right)n,(v*)n,(v*)n)$ such that the following linear system of equations is satisfied

$${\displaystyle \partial J*(un,\left(u\right)n,\left(w\right)n,(v*)n,(v*)n,\left((v*)\right)n,\left((v*)\right)n)\partial u}=\mathbf{0},$$

$${\displaystyle \partial J*(un,\left(u\right)n,\left(w\right)n,(v*)n,(v*)n,\left((v*)\right)n,\left((v*)\right)n)\partial v*}=\mathbf{0},$$

$${\displaystyle \partial J*(un,\left(u\right)n,\left(w\right)n,(v*)n,(v*)n,\left((v*)\right)n,\left((v*)\right)n)\partial v*}=\mathbf{0},$$

$${\displaystyle \partial J*(un,\left(u\right)n,\left(w\right)n,(v*)n,(v*)n,\left((v*)\right)n,\left((v*)\right)n)\partial w}=\mathbf{0}.$$

Set $\left(u\right)n+=un$, $\left((v*)\right)n+=(v*)n,$ and $\left((v*)\right)n+=(v*)n.$

If $\parallel \left(u\right)n+-\left(u\right)n\parallel \infty <b12$ or $n>nmax$, then stop.

Otherwise $n:=n+$ and go to item 2.

We have obtained numerical results for $\epsilon =.01$, $A=B=$ and $f\equiv ,\mathrm{in}\Omega .$

For such an optimal solution $u$ obtained please see Figure 46.

Here we present the software in MAT-LAB through which we have obtained such numerical results.

***********************

clear all

global m8 d yo u K e1 A B v1 v2 uo vo1 vo2 K1

m8=200;

d=1/m8;

K=10;

K1=38;

e1=0.01;

A=1;

B=1;

uo(:,1)=1.2*ones(m8,1);

yo(:,1)=ones(m8,1);

vo1(:,1)=0.4*ones(m8,1);

vo2(:,1)=0.4*ones(m8,1);

xo=1.2*ones(4*m8,1);

b14=1;

k1=1;

while $(b14>10-)$&&$(k<100)$

k1

k1=k1+1;

b12=1;

k=1;

while $(b12>10-)$ &&$(k<25)$

k

k=k+1;

X=lsqnonlin(’funJune2024DC25’,xo);

b12=max(abs(X-xo));

xo=X;

u(m8/2,1)

end;

b14=max(abs(u-uo));

uo=u;

vo1=v1;

vo2=v2;

end;

for i=1:m8

x(i,1)=i*d;

end;

plot(x,u);

***********************

With the auxiliary function "funJune2024DC25",

*************************

function W=funJune2024DC25(x)

global m8 d yo u K e1 A B v1 v2 vo1 vo2 uo

for i=1:m8

u(i,1)=x(i,1);

v1(i,1)=x(i+m8,1);

v2(i,1)=x(i+2*m8,1);

w(i,1)=x(i+3*m8,1);

end;

v1(m8,1)=0;

u(m8,1)=0;

d2v(1,1)=$(-*v(,)+v(,))/d;$

d2u(1,1)=$(-*u(,)+u(,))/d;$

for i=2:m8-1

d2v1(i,1)=$(v(i+,)-*v(i,)+v(i-,))/d;$

d2u(i,1)=$(u(i+,)-*u(i,)+u(i-,))/d;$

end;

for i=1:m8-1

$W(i,)=-e*dv(i,)+*A*uo(i,)*v(i,)-B*v(i,)+*K*vo(i,)*u(i,)+*K*vo(i,)*u(i,);$

$W(i+m,)=-e*du(i,)+*A*uo(i,)*u(i,)-*uo(i,)*A-B*u(i,)-yo(i,)+w(i,)-K*uo(i,)/-v(i,);$

$W(i+*m,)=w(i,)-K*uo(i,)/-v(i,);$

$W(i+*m,)=v(i,)+v(i,);$

end;

********************************

67. A convex dual variational formulation for a Burger’s type equation

Let $\Omega =[,]\subset \mathbb{R}.$

Consider the Burger’s type equation

$$\left\{\begin{array}{cc}\nu uxx-uux=,\hfill & \mathrm{in}\Omega ,\hfill \\ u\left(\right)=,\hfill & u\left(\right)=.\hfill \end{array}\right.$$

(548)

Here $\nu >$ is a real constant.

Define the Galerkin type functional $J:V\to \mathbb{R}$ where

$$J\left(u\right)={\displaystyle 12}\int \Omega (\nu uxx-uux)dx,$$

and

$$V=\{u\in W,\left(\Omega \right):u\left(\right)=,\mathrm{and}u\left(\right)=\}.$$

Denoting $Y=Y*=L\left(\Omega \right)$, define $F:V\times Y*\to \mathbb{R}$ and $F:V\times Y*\to \mathbb{R}$ by

$$F(u,v*)={\displaystyle 12}\int \Omega (\nu uxx-uux+v*+Ku+Kux)dx,$$

and

$$F(u,v*)={\displaystyle 12}\int \Omega (v*+Ku+Kux)dx,$$

respectively. Here $K>$ is an appropriate large real constant.

Define also $J:V\times Y*\to \mathbb{R}$ by

$$J(u,v*)=F(u,v*)+F(u,v*),$$

Observe that

$$\begin{array}{ccc}\hfill J(u,v*)& =& F(u,v*)+F(u,v*)\hfill \\ & =& -\langle v*+\nu uxx,v*\rangle L-\langle u,v*\rangle L-\langle ux,v*\rangle L+F(u,v*)\hfill \\ & & -\langle v*,v*\rangle L-\langle u,v*\rangle L-\langle ux,v*\rangle L+F(u,v*)\hfill \\ & & +\langle v*+\nu uxx,v*\rangle L+\langle u,v*\rangle L+\langle ux,v*\rangle L\hfill \\ & & +\langle v*,v*\rangle L+\langle u,v*\rangle L+\langle ux,v*\rangle L\hfill \\ & \ge & \underset{(v,v,v)\in \left[Y\right]}{inf}\left\{-\langle v,v*\rangle L-\langle v,v*\rangle L-\langle v,v*\rangle L+F˜(v,v,v)\right\}\hfill \\ & & +\underset{(v,v,v)\in \left[Y\right]}{inf}\left\{-\langle v,v*\rangle L-\langle v,v*\rangle L-\langle v,v*\rangle L+F˜(v,v,v)\right\}\hfill \\ & & +\underset{(u,v*)\in V\times Y*}{inf}\left\{\langle v*+\nu uxx,v*\rangle L+\langle u,v*\rangle L+\langle ux,v*\rangle L\right.\hfill \\ & & \left.+\langle v*,v*\rangle L+\langle u,v*\rangle L+\langle ux,v*\rangle L\right\}\hfill \\ & =& -F˜*(v*,v*,v*)-F˜*(v*,v*,v*)\hfill \\ & & +\nu (v*)x\left(\right)u\left(\right),\forall (u,v*)\in V\times Y*,\forall v*\in A*\cap B*,\hfill \end{array}$$

(549)

where

$$A*=\{v*=(v*,v*,v*,v*,v*,v*)\in [Y*]:\nu (v*)xx+v*-(v*)x=,\mathrm{in}\Omega \},$$

$$B*=\{v*\in [Y*]:v*\ge ,v*\ge ,v*+v*=,\mathrm{in}\Omega \mathrm{and}v*\left(\right)=v*\left(\right)=\}.$$

Moreover, denoting

$$F˜(v,v,v)={\displaystyle 12}\int \Omega (v-vv+Kv22+Kv32)dx,$$

and

$$F˜(v,v,v)={\displaystyle 12}\int \Omega (v+Kv22+Kv32)dx,$$

for $v*\in B*$, we have

$$\begin{array}{ccc}& & F˜*(v*,v*,v*)\hfill \\ & =& \underset{(v,v,v)\in \left[Y\right]}{sup}\left\{\langle v,v*\rangle L+\langle v,v*\rangle L+\langle v,v*\rangle L-F˜(v,v,v)\right\}\hfill \\ & =& {\displaystyle (K-)}\int \Omega {\displaystyle (v*v*+K((v*)+(v*)))v*}dx+{\displaystyle 12}\int \Omega (v*)dx,\hfill \end{array}$$

(550)

$$\begin{array}{ccc}& & F˜*(v*,v*,v*)\hfill \\ & =& \underset{(v,v,v)\in \left[Y\right]}{sup}\left\{\langle v,v*\rangle L+\langle v,v*\rangle L+\langle v,v*\rangle L-F˜(v,v,v)\right\}\hfill \\ & =& {\displaystyle K}\int \Omega {\displaystyle (v*)v*}dx+{\displaystyle K}\int \Omega {\displaystyle (v*)v*}dx+{\displaystyle 12}\int \Omega (v*)dx.\hfill \end{array}$$

(551)

Here we define $J*:[Y*]\to \mathbb{R}$ by

$$J*(v*)=-F˜*(v*,v*,v*)-F˜*(v*,v*,v*)+\nu (v*)x\left(\right)u\left(\right).$$

It is worth highlighting we have got

$$\underset{(u,v*)\in V\times Y*}{inf}J(u,v*)\ge \underset{v*\in A*\cap B*}{sup}J*(v*).$$

Finally, we also emphasize that $J$ is convex (in fact concave) in the convex set $A*\cap B*$ so that we have obtained a convex dual formulation for an originally non-convex primal dual one.

Remark 33.

The conditions which define $B$ must be replaced by those concerning the regularized set

$$B\epsilon *=\{v*\in [Y*]:v*\ge \epsilon ,v*\ge \epsilon ,v*+v*=\epsilon ,\mathrm{in}\Omega \mathrm{and}v*\left(\right)=v*\left(\right)=\epsilon \}$$

for an appropriate real constant $<\epsilon \ll .$ Therefore, through $B$, we may define an approximate dual formulation so that will be particularly interested in the system behaviour as

$$\epsilon \to +.$$

68. A Convex Dual Variational Formulation for an Approximate Navier-Stokes System

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega =S.$

Consider the approximate incompressible and time independent Navier-Stokes system, where

$$\left\{\begin{array}{cc}\nu \nabla u-uux-vuy-Px=,\hfill & \\ \nu \nabla v-uvx-vvy-Py=,\hfill & \\ \nabla P+ux+vy+uyvx=,\hfill & \mathrm{in}\Omega ,\hfill \\ u=u,v=v,P=P,\hfill & \mathrm{on}\partial \Omega =S.\hfill \end{array}\right.$$

(552)

Here $\nu >$ is a real constant. Moreover, $\mathbf{n}$ denotes the outward normal field to $\partial \Omega =S$.

Define the Galerkin type functional $J:V\to \mathbb{R}$, where

$$\begin{array}{ccc}\hfill J(u,v,P)& =& {\displaystyle 12}\int \Omega (\nu \nabla u-uux-vuy-Px)dx\hfill \\ & & +{\displaystyle 12}\int \Omega (\nu \nabla v-uvx-vvy-Py)dx\hfill \\ & & +{\displaystyle 12}\int \Omega (\nabla P+ux+vy+uyvx)dx,\hfill \end{array}$$

(553)

and

$$V=\{\mathbf{u}=(u,v,P)\in W,(\Omega ;\mathbb{R}):u=u,v=v\mathrm{and}P=P\mathrm{on}\partial \Omega \}.$$

Denoting $Y=Y*=L\left(\Omega \right)$, define $F:V\times Y*\to \mathbb{R}$, $F:V\times Y*\to \mathbb{R}$, $F:V\times Y*\to \mathbb{R}$, $F:V\times Y*\to \mathbb{R}$, $F:V\times Y*\to \mathbb{R}$ and $F:V\times Y*\to \mathbb{R}$ by

$$F(\mathbf{u},v50*)={\displaystyle 12}\int \Omega (\nu \nabla u-uux-vuy-Px+Ku+Kux+Kv+Kuy+v50*)dx,$$

$$F(\mathbf{u},v60*)={\displaystyle 12}\int \Omega (\nu \nabla v-uvx-vvy-Py+Ku+Kvx+Kv+Kvy+v60*)dx,$$

$$F(\mathbf{u},v70*)={\displaystyle 12}\int \Omega (\nabla P+ux+vy+uyvx+Kux+Kvy+Kvx+Kuy+v70*)dx,$$

$$F(\mathbf{u},v50*)={\displaystyle 12}\int \Omega (Ku+Kux+Kv+Kuy+v50*)dx,$$

$$F(\mathbf{u},v60*)={\displaystyle 12}\int \Omega (Ku+Kvx+Kv+Kvy+v60*)dx,$$

and

$$F(\mathbf{u},v70*)={\displaystyle 12}\int \Omega (Kux+Kvy+Kvx+Kuy+v70*)dx,$$

respectively. Here $K>$ is an appropriate large real constant.

Define also $J:V\times [Y*]\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill J(\mathbf{u},v50*,v60*,v70*)& =& F(\mathbf{u},v50*)+F(\mathbf{u},v60*)\hfill \\ & & +F(\mathbf{u},v70*)+F(\mathbf{u},v50*)\hfill \\ & & +F(\mathbf{u},v60*)+F(\mathbf{u},v70*).\hfill \end{array}$$

(554)

Observe that

$$\begin{array}{ccc}\hfill J(\mathbf{u},v50*,v60*,v70*)& =& F(\mathbf{u},v50*)+F(\mathbf{u},v60*)\hfill \\ & & +F(\mathbf{u},v70*)+F(\mathbf{u},v50*)\hfill \\ & & +F(\mathbf{u},v60*)+F(\mathbf{u},v70*)\hfill \\ & =& -\langle v50*+\nu \nabla u-Px,v*\rangle L-\langle u,v*\rangle L-\langle ux,v*\rangle L\hfill \\ & & -\langle v,v*\rangle L-\langle uy,v*\rangle L+F(\mathbf{u},v50*)\hfill \\ & & -\langle v60*+\nu \nabla v-Py,v*\rangle L-\langle u,v*\rangle L-\langle vx,v*\rangle L\hfill \\ & & -\langle v,v*\rangle L-\langle vy,v10*\rangle L+F(\mathbf{u},v60*)\hfill \\ & & -\langle v70*+\nabla P,v11*\rangle L-\langle ux,v12*\rangle L-\langle vy,v13*\rangle L\hfill \\ & & -\langle vx,v14*\rangle L-\langle uy,v15*\rangle L+F(\mathbf{u},v70*)\hfill \\ & & -\langle v50*,v16*\rangle L-\langle u,v17*\rangle L-\langle ux,v18*\rangle L\hfill \\ & & -\langle v,v19*\rangle L-\langle uy,v20*\rangle L+F(\mathbf{u},v50*)\hfill \\ & & -\langle v60*,v21*\rangle L-\langle u,v22*\rangle L-\langle vx,v23*\rangle L\hfill \\ & & -\langle v,v24*\rangle L-\langle vy,v25*\rangle L+F(\mathbf{u},v60*)\hfill \\ & & -\langle v70*,v26*\rangle L-\langle ux,v27*\rangle L-\langle vy,v28*\rangle L\hfill \\ & & -\langle vx,v29*\rangle L-\langle uy,v30*\rangle L+F(\mathbf{u},v70*)\hfill \\ & & +\langle v50*+\nu \nabla u-Px,v*\rangle L+\langle u,v*\rangle L+\langle ux,v*\rangle L\hfill \\ & & +\langle v,v*\rangle L+\langle uy,v*\rangle L\hfill \\ & & +\langle v60*+\nu \nabla v-Py,v*\rangle L+\langle u,v*\rangle L+\langle vx,v*\rangle L\hfill \\ & & +\langle v,v*\rangle L+\langle vy,v10*\rangle L\hfill \\ & & +\langle v70*+\nabla P,v11*\rangle L-\langle ux,v12*\rangle L-\langle vy,v13*\rangle L\hfill \\ & & +\langle vx,v14*\rangle L+\langle uy,v15*\rangle L\hfill \\ & & +\langle v50*,v16*\rangle L+\langle u,v17*\rangle L+\langle ux,v18*\rangle L\hfill \\ & & +\langle v,v19*\rangle L+\langle uy,v20*\rangle L\hfill \\ & & +\langle v60*,v21*\rangle L+\langle u,v22*\rangle L+\langle vx,v23*\rangle L\hfill \\ & & +\langle v,v24*\rangle L+\langle vy,v25*\rangle L\hfill \\ & & +\langle v70*,v26*\rangle L+\langle ux,v27*\rangle L+\langle vy,v28*\rangle L\hfill \\ & & +\langle vx,v29*\rangle L+\langle uy,v30*\rangle L.\hfill \end{array}$$

(555)

From such a result, we obtain

$$\begin{array}{ccc}& & J(\mathbf{u},v50*,v60*,v70*)\hfill \\ & \ge & \underset{(v,\dots ,v)\in \left[Y\right]}{inf}\{-\langle v,v*\rangle L-\langle v,v*\rangle L-\langle v,v*\rangle L\hfill \\ & & -\langle v,v*\rangle L-\langle v,v*\rangle L+F˜(v,\dots ,v)\}\hfill \\ & & \underset{(v,\dots ,v10)\in \left[Y\right]}{inf}\{-\langle v,v*\rangle L-\langle v,v*\rangle L-\langle v,v*\rangle L\hfill \\ & & -\langle v,v*\rangle L-\langle v10,v10*\rangle L+F˜(v,\dots ,v10)\}\hfill \\ & & \underset{(v11,\dots ,v15)\in \left[Y\right]}{inf}\{-\langle v11,v11*\rangle L-\langle v12,v12*\rangle L-\langle v13,v13*\rangle L\hfill \\ & & -\langle v14,v14*\rangle L-\langle v15,v15*\rangle L+F˜(v11,\dots ,v15)\}\hfill \\ & & \underset{(v16,\dots ,v20)\in \left[Y\right]}{inf}\{-\langle v16,v16*\rangle L-\langle v17,v17*\rangle L-\langle v18,v18*\rangle L\hfill \\ & & -\langle v19,v19*\rangle L-\langle v20,v20*\rangle L+F˜(v16,\dots ,v20)\}\hfill \\ & & \underset{(v21,\dots ,v25)\in \left[Y\right]}{inf}\{-\langle v21,v21*\rangle L-\langle v22,v22*\rangle L-\langle v23,v23*\rangle L\hfill \\ & & -\langle v24,v24*\rangle L-\langle v25,v25*\rangle L+F˜(v21,\dots ,v25)\}\hfill \\ & & \underset{(v26,\dots ,v30)\in \left[Y\right]}{inf}\{-\langle v26,v26*\rangle L-\langle v27,v27*\rangle L-\langle v28,v28*\rangle L\hfill \\ & & -\langle v29,v29*\rangle L-\langle v30,v30*\rangle L+F˜(v26,\dots ,v30)\}\hfill \\ & & +\underset{(\mathbf{u},v50*,v60*,v70*)\in V\times \left[Y\right]}{inf}\{\langle v50*+\nu \nabla u-Px,v*\rangle L+\langle u,v*\rangle L+\langle ux,v*\rangle L\hfill \\ & & +\langle v,v*\rangle L+\langle uy,v*\rangle L\hfill \\ & & -\langle v60*+\nu \nabla v-Py,v*\rangle L+\langle u,v*\rangle L+\langle vx,v*\rangle L\hfill \\ & & +\langle v,v*\rangle L+\langle vy,v10*\rangle L\hfill \\ & & +\langle v70*+\nabla P,v11*\rangle L+\langle ux,v12*\rangle L+\langle vy,v13*\rangle L\hfill \\ & & +\langle vx,v14*\rangle L+\langle uy,v15*\rangle L\hfill \\ & & +\langle v50*,v16*\rangle L+\langle u,v17*\rangle L+\langle ux,v18*\rangle L\hfill \\ & & +\langle v,v19*\rangle L+\langle uy,v20*\rangle L\hfill \\ & & +\langle v60*,v21*\rangle L+\langle u,v22*\rangle L+\langle vx,v23*\rangle L\hfill \\ & & +\langle v,v24*\rangle L+\langle vy,v25*\rangle L\hfill \\ & & +\langle v70*,v26*\rangle L+\langle ux,v27*\rangle L+\langle vy,v28*\rangle L\hfill \\ & & +\langle vx,v29*\rangle L+\langle uy,v30*\rangle L\}\hfill \\ & =& -F˜*(v*,\dots ,v*)-F˜*(v*,\dots ,v10*)-F˜*(v11*,\dots ,v15*)\hfill \\ & & -F˜*(v16*,\dots ,v20*)-F˜*(v21*,\dots ,v25*)-F˜*(v26*,\dots ,v30*)\hfill \\ & & +\nu \int \partial \Omega u(\nabla v*·\mathbf{n})dS+\nu \int \partial \Omega v(\nabla v*·\mathbf{n})dS+\int \partial \Omega P(\nabla v11*·\mathbf{n})dS,\hfill \end{array}$$

(556)

if $v*=(v*,\dots ,v30*)\in A*\cap B*$, where $A*=A*\cap A*\cap A*$,

$$\begin{array}{ccc}\hfill A*& =& \{v*\in [Y*]30:\nu \nabla v*+v*-(v*)x-(v*)y\hfill \\ & & v*-(v12*)x-(v14*)y+v17*\hfill \\ & & -(v18*)x-(v20*)y-v22*-(v30*)y=,\mathrm{in}\Omega \},\hfill \end{array}$$

(557)

$$\begin{array}{ccc}\hfill A*& =& \{v*\in [Y*]30:v*+\nu \nabla v*-(v*)x-v*\hfill \\ & & -(v10*)y-(v15*)x+v19*-(v23*)x\hfill \\ & & +v24*-(v25*)y-(v28*)y-(v29*)x=,\mathrm{in}\Omega \},\hfill \end{array}$$

(558)

$$\begin{array}{ccc}\hfill A*& =& \{v*\in [Y*]30:(v*)x+(v*)y+\nabla v11*=,\mathrm{in}\Omega \},\hfill \end{array}$$

(559)

$$\begin{array}{ccc}\hfill B*& =& \{v*\in [Y*]30:v*+v16*=,v*+v21*=,v11*+v26*=,\hfill \\ & & v*\ge ,v*\ge ,v11*\ge ,\hfill \\ & & v16*\ge ,v21*\ge ,v26*\ge ,\mathrm{in}\Omega ,\hfill \\ & & v*=v*=v11*=,\mathrm{on}\partial \Omega \}\hfill \end{array}$$

(560)

Moreover, denoting

$$F˜(v,\dots ,v)={\displaystyle 12}\int \Omega (v-vv-vv+Kv22+Kv32+Kv42+Kv52)dx,$$

$$F˜(v,\dots ,v10)={\displaystyle 12}\int \Omega (v-vv-vv10+Kv72+Kv82+Kv92+Kv10)dx,$$

$$F˜(v11,\dots ,v15)={\displaystyle 12}\int \Omega (v11+v12+v13+v14v15+Kv12+Kv13+Kv14+Kv15)dx,$$

$$F˜(v16,\dots ,v20)={\displaystyle 12}\int \Omega (v16+Kv17+Kv18+Kv19+Kv20)dx,$$

$$F˜(v21,\dots ,v25)={\displaystyle 12}\int \Omega (v21+Kv22+Kv23+Kv24+Kv25)dx,$$

$$F˜(v26,\dots ,v30)={\displaystyle 12}\int \Omega (v26+Kv27+Kv28+Kv29+Kv30)dx,$$

we have

$$\begin{array}{ccc}& & F˜*(v*,\dots ,v*)\hfill \\ & =& \underset{(v,\dots ,v)\in \left[Y\right]}{sup}\{\langle v,v*\rangle L+\langle v,v*\rangle L+\langle v,v*\rangle L\hfill \\ & & \langle v,v*\rangle L+\langle v,v*\rangle L-F˜(v,\dots ,v)\}\hfill \\ & =& {\displaystyle (K-)}\int \Omega {\displaystyle v*v*+v*v*+K((v*)+(v*)+(v*)+(v*))v*}dx\hfill \\ & & +{\displaystyle 12}\int \Omega (v*)dx,\hfill \end{array}$$

(561)

$$\begin{array}{ccc}& & F˜*(v*,\dots ,v10*)\hfill \\ & =& \underset{(v,\dots ,v10)\in \left[Y\right]}{sup}\{\langle v,v*\rangle L+\langle v,v*\rangle L+\langle v,v*\rangle L\hfill \\ & & \langle v,v*\rangle L+\langle v10,v10*\rangle L-F˜(v,\dots ,v10)\}\hfill \\ & =& {\displaystyle (K-)}\int \Omega {\displaystyle v*v*+v*v10*+K((v*)+(v*)+(v*)+(v10*))v*}dx\hfill \\ & & +{\displaystyle 12}\int \Omega (v*)dx,\hfill \end{array}$$

(562)

$$\begin{array}{ccc}& & F˜*(v11*,\dots ,v15*)\hfill \\ & =& \underset{(v11,\dots ,v15)\in \left[Y\right]}{sup}\{\langle v11,v11*\rangle L+\langle v12,v12*\rangle L+\langle v13,v13*\rangle L\hfill \\ & & \langle v14,v14*\rangle L+\langle v15,v15*\rangle L-F˜(v11,\dots ,v15)\}\hfill \\ & =& {\displaystyle (K-)}\int \Omega {\displaystyle (-+K)((v12*)+(v13*))+K(v14*)-v14*v15*+K(v15*)v11*}dx\hfill \\ & & +{\displaystyle 12}\int \Omega (v11*)dx,\hfill \end{array}$$

(563)

$$\begin{array}{ccc}& & F˜*(v16*,\dots ,v20*)\hfill \\ & =& \underset{(v16,\dots ,v20)\in \left[Y\right]}{sup}\{\langle v16,v16*\rangle L+\langle v17,v17*\rangle L+\langle v18,v18*\rangle L\hfill \\ & & \langle v19,v19*\rangle L+\langle v20,v20*\rangle L-F˜(v15,\dots ,v20)\}\hfill \\ & =& {\displaystyle K}\int \Omega {\displaystyle ((v17*)+(v18*)+(v19*)+(v20*))v16*}dx\hfill \\ & & +{\displaystyle 12}\int \Omega (v16*)dx,\hfill \end{array}$$

(564)

$$\begin{array}{ccc}& & F˜*(v21*,\dots ,v25*)\hfill \\ & =& \underset{(v21,\dots ,v25)\in \left[Y\right]}{sup}\{\langle v21,v21*\rangle L+\langle v22,v22*\rangle L+\langle v23,v23*\rangle L\hfill \\ & & \langle v24,v24*\rangle L+\langle v25,v25*\rangle L-F˜(v21,\dots ,v25)\}\hfill \\ & =& {\displaystyle K}\int \Omega {\displaystyle ((v22*)+(v23*)+(v24*)+(v25*))v21*}dx\hfill \\ & & +{\displaystyle 12}\int \Omega (v21*)dx,\hfill \end{array}$$

(565)

$$\begin{array}{ccc}& & F˜*(v26*,\dots ,v30*)\hfill \\ & =& \underset{(v26,\dots ,v30)\in \left[Y\right]}{sup}\{\langle v26,v26*\rangle L+\langle v27,v27*\rangle L+\langle v28,v28*\rangle L\hfill \\ & & \langle v29,v29*\rangle L+\langle v30,v30*\rangle L-F˜(v25,\dots ,v30)\}\hfill \\ & =& {\displaystyle K}\int \Omega {\displaystyle ((v27*)+(v28*)+(v29*)+(v30*))v26*}dx\hfill \\ & & +{\displaystyle 12}\int \Omega (v26*)dx.\hfill \end{array}$$

(566)

Here we define $J*:[Y*]30\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill J*(v*)& =& -F˜*(v*,\dots ,v*)-F˜*(v*,\dots ,v10*)-F˜*(v11*,\dots ,v15*)\hfill \\ & & -F˜*(v16*,\dots ,v20*)-F˜*(v21*,\dots ,v25*)-F˜*(v26*,\dots ,v30*)\hfill \\ & & +\nu \int \partial \Omega u(\nabla v*·\mathbf{n})dS+\nu \int \partial \Omega v(\nabla v*·\mathbf{n})dS+\int \partial \Omega P(\nabla v11*·\mathbf{n})dS,\hfill \end{array}$$

(567)

It is worth highlighting we have got

$$\underset{(\mathbf{u},v50*,v60*,v70*)\in V\times \left[Y\right]}{inf}J(\mathbf{u},v50*,v60*,v70*)\ge \underset{v*\in A*\cap B*}{sup}J*(v*).$$

Finally, we also emphasize that $J$ is convex (in fact concave) in the convex set $A*\cap B*$ so that we have obtained a convex dual formulation for an originally non-convex primal dual one.

Remark 34.

Here we highlight the conditions which define $B$ must be appropriately regularized through a small parameter

$$<\epsilon \ll ,$$

similarly as we have done in the previous section.

69. A D.C. Type Dual Variational Formulation for a Burger’s Type Equation

In this section we shall write a primal Galerkin type variational formulation for a Burger’s type equation as a difference of two convex functionals (the so called D.C. approach) and establish a related convex dual variational formulation.

Let $\Omega =[,]\subset \mathbb{R}.$

Consider the Burger’s type equation

$$\left\{\begin{array}{cc}\nu uxx-uux=,\hfill & \mathrm{in}\Omega ,\hfill \\ u\left(\right)=,\hfill & u\left(\right)=.\hfill \end{array}\right.$$

(568)

Here $\nu >$ is a real constant.

Define a Galerkin type functional $J:V\to \mathbb{R}$, where

$$J\left(u\right)={\displaystyle 12}\int \Omega (\nu uxx-uux)dx,$$

and

$$V=\{u\in W,\left(\Omega \right):u\left(\right)=,\mathrm{and}u\left(\right)=\}.$$

Denoting $Y=Y*=L\left(\Omega \right)$, define $F,F:V\times Y*\to \mathbb{R}$ and $F,F:V\to \mathbb{R}$ by

$$F(u,v50*)={\displaystyle 12}\int \Omega (\nu uxx-uux+v50*+Ku+Kux)dx+{\displaystyle K12}\int \Omega udx+{\displaystyle K12}\int \Omega uxdx,$$

$$F(u,v50*)={\displaystyle 12}\int \Omega (v50*+Ku+Kux)dx+{\displaystyle K12}\int \Omega udx+{\displaystyle K12}\int \Omega uxdx,$$

$$F\left(u\right)={\displaystyle K12}\int \Omega udx+{\displaystyle K12}\int \Omega uxdx$$

and

$$F\left(u\right)={\displaystyle K12}\int \Omega udx+{\displaystyle K12}\int \Omega uxdx,$$

respectively.

Here $K,K>$ are appropriate large real constants such that

$$K\gg K.$$

Define also $J:V\times Y*\to \mathbb{R}$ by

$$J(u,v50*)=F(u,v50*)+F(u,v50*)-F\left(u\right)-F\left(u\right),$$

Observe that

$$\underset{(u,v50*)\in V\times Y*}{inf}J(u,v50*)=,$$

so that, denoting

$$F˜(v,v,v)={\displaystyle 12}\int \Omega (v-vv+Kv22+Kv32)dx+{\displaystyle K12}\int \Omega \left(v\right)dx+{\displaystyle K12}\int \Omega \left(v\right)dx,$$

$$F˜(v,v,v)={\displaystyle 12}\int \Omega (v+Kv42+Kv52)dx+{\displaystyle K12}\int \Omega \left(v\right)dx+{\displaystyle K12}\int \Omega \left(v\right)dx,$$

$$F˜(z,z)={\displaystyle K12}\int \Omega \left(z\right)dx+{\displaystyle K12}\int \Omega \left(z\right)dx,$$

$$F˜(z,z)={\displaystyle K12}\int \Omega \left(z\right)dx+{\displaystyle K12}\int \Omega \left(z\right)dx,$$

we have

$$\begin{array}{ccc}\hfill \le J(u,v50*)& =& F(u,v50*)+F(u,v50*)-F\left(u\right)-F\left(u\right)\hfill \\ & =& -\langle u,z*,\rangle L-\langle ux,z*\rangle L+F(u,v50*)\hfill \\ & & -\langle u,z*\rangle L-\langle ux,z*\rangle L+F(u,v50*)\hfill \\ & & +\langle u,z*\rangle L+\langle ux,z*\rangle L-F\left(u\right)\hfill \\ & & +\langle u,z*\rangle L+\langle ux,z*\rangle L-F\left(u\right)\hfill \\ & \le & -\langle z*,u\rangle L-\langle ux,z*\rangle L+F(u,v50*)\hfill \\ & & -\langle u,z*\rangle L-\langle ux,z*\rangle L+F(u,v50*)\hfill \\ & & +\underset{(z,z)\in Y}{sup}\{\langle z,z*\rangle L+\langle z,z*\rangle L-F˜(z,z)\}\hfill \\ & & \underset{(z,z)\in Y}{sup}\{\langle z,z*\rangle L+\langle z,z*\rangle L-F˜(z,z)\}\hfill \\ & =& -\langle z*,u\rangle L-\langle ux,z*\rangle L+F(u,v50*)\hfill \\ & & -\langle u,z*\rangle L-\langle ux,z*\rangle L+F(u,v50*)\hfill \\ & & +F˜*(z*,z*)+F˜*(z*,z*),\forall u\in V,(z*,\dots ,z*)\in [Y*].\hfill \end{array}$$

(569)

From such results, similarly as obtained in [5], we may infer that

$$\begin{array}{ccc}\hfill & & =\underset{(u,v50*)\in V\times Y}{inf}J(u,v50*)\hfill \\ & \le & \underset{(u,v50*)\in V\times Y}{inf}\{-\langle u,z*,\rangle L-\langle ux,z*\rangle L+F(u,v50*)\hfill \\ & & -\langle u,z*\rangle L-\langle ux,z*\rangle L+F(u,v50*)\}\hfill \\ & & +F˜*(z*,z*)+F*(z*,z*),\forall z*=(z*,\dots ,z*)\in [Y*].\hfill \end{array}$$

(570)

On the other hand, observe that

$$\begin{array}{ccc}& & -\langle u,z*,\rangle L-\langle ux,z*\rangle L\hfill \\ & & -\langle \nu uxx+v50*,v*\rangle L-\langle u,v*\rangle L-\langle ux,v*\rangle L+F(u,v50*)\hfill \\ & & -\langle u,z*\rangle L-\langle ux,z*\rangle L\hfill \\ & & -\langle v50*,v*\rangle L-\langle u,v*\rangle L-\langle ux,v*\rangle L+F(u,v50*)\hfill \\ & & +\langle \nu uxx+v50*,v*\rangle L+\langle u,v*\rangle L+\langle ux,v*\rangle L\hfill \\ & & +\langle v50*,v*\rangle L+\langle u,v*\rangle L+\langle ux,v*\rangle L\hfill \\ & \ge & \underset{(v,v,v)\in \left[Y\right]}{inf}\{-\langle v,z*\rangle L-\langle v,z*\rangle L\hfill \\ & & -\langle v,v*\rangle L-\langle v,v*\rangle L-\langle v,v*\rangle L+F˜(v,v,v)\}\hfill \\ & & +\underset{(v,v,v)\in \left[Y\right]}{inf}\{-\langle v,z*\rangle L-\langle v,z*\rangle L\hfill \\ & & -\langle v,v*\rangle L-\langle v,v*\rangle L-\langle v,v*\rangle L+F˜(v,v,v)\}\hfill \\ & & +\underset{(u,v50*)\in V\times Y}{inf}\{\langle \nu uxx+v50*,v*\rangle L+\langle u,v*\rangle L+\langle ux,v*\rangle L\hfill \\ & & +\langle v50*,v*\rangle L+\langle u,v*\rangle L+\langle ux,v*\rangle L\}\hfill \\ & =& -F˜*(v*,v*,v*,z*,z*)-F˜*(v*,v*,v*,z*,z*)-\nu (v*)x\left(\right)u\left(\right),\hfill \end{array}$$

(571)

if $v*=(v*,\dots ,v*)\in A*\cap B*,$ where

$$A*=A*\cap A*,$$

$$A*=\{v*\in [Y*]:\nu (v*)xx+v*-(v*)x+v*-(v*)x=,\mathrm{in}\Omega \},$$

$$A*=\{v*\in [Y*]:v*+v*=,v*\ge ,v*\ge ,\mathrm{in}\Omega \},$$

and

$$B*=\{v*\in [Y*]:v*\left(\right)=v*\left(\right)=\}.$$

At this point we recall that

$$\begin{array}{ccc}& & F˜*(v*,v*,v*,z*,z*)\hfill \\ & =& \underset{(v,v,v)\in \left[Y\right]}{sup}\left\{\langle v,z*\rangle L+\langle v,z*\rangle L\right.\hfill \\ & & \left.+\langle v,v*\rangle L+\langle v,v*\rangle L+\langle v,v*\rangle L-F˜(v,v,v)\right\}\hfill \\ & =& {\displaystyle K}\int \Omega {\displaystyle (v*+z*)+(v*+z*)(Kv*+K)-(v*)}dx\hfill \\ & & +\int \Omega {\displaystyle (v*)((v*+z*)(v*+z*)+K(v*+z*)+K(v*+z*))(Kv*+K)-(v*)}dx\hfill \\ & & +{\displaystyle 12}\int \Omega (v*)dx,\hfill \end{array}$$

(572)

$$\begin{array}{ccc}& & F˜*(v*,v*,v*,z*,z*)\hfill \\ & =& \underset{(v,v,v)\in \left[Y\right]}{sup}\left\{\langle v,z*\rangle L+\langle v,z*\rangle L\right.\hfill \\ & & \left.+\langle v,v*\rangle L+\langle v,v*\rangle L+\langle v,v*\rangle L-F˜(v,v,v)\right\}\hfill \\ & =& {\displaystyle 12}\int \Omega {\displaystyle (v*+z*)+(v*+z*)(K+Kv*)}dx+{\displaystyle 12}\int \Omega (v*)dx,\hfill \end{array}$$

(573)

$$\begin{array}{ccc}\hfill F˜*(z*,z*)& =& \underset{(z,z)\in \left[Y\right]}{sup}\left\{\langle z,z*\rangle L+\langle z,z*\rangle L-F(z,z)\right\}\hfill \\ & =& {\displaystyle K}\int \Omega (z*)dx+{\displaystyle K}\int \Omega (z*)dx,\hfill \end{array}$$

(574)

and

$$\begin{array}{ccc}\hfill F˜*(z*,z*)& =& \underset{(z,z)\in \left[Y\right]}{sup}\left\{\langle z,z*\rangle L+\langle z,z*\rangle L-F(z,z)\right\}\hfill \\ & =& {\displaystyle K}\int \Omega (z*)dx+{\displaystyle K}\int \Omega (z*)dx.\hfill \end{array}$$

(575)

Moreover, for $K>$ sufficiently large, up to a restriction for the dual variables related to a ball of radius proportional to $K$, from the standard results on convex analysis and duality theory, we have

$$\begin{array}{ccc}& & \underset{(u,v50*)\in V\times Y}{inf}\{-\langle u,z*,\rangle L-\langle ux,z*\rangle L\hfill \\ & & -\langle \nu uxx+v50*,v*\rangle L-\langle u,v*,\rangle L-\langle ux,v*\rangle L+F(u,v50*)\hfill \\ & & -\langle u,z*\rangle L-\langle ux,z*\rangle L\hfill \\ & & -\langle v50*,v*\rangle L-\langle u,v*,\rangle L-\langle ux,v*\rangle L+F(u,v50*)\}\hfill \\ & =& \underset{v*\in A*\cap B*}{sup}\{-F˜*(v*,v*,v*,z*,z*)-F˜*(v*,v*,v*,z*,z*)-\nu (v*)x\left(\right)u\left(\right)\}.\hfill \end{array}$$

(576)

Consequently, from such results and (570) we have got

$$\begin{array}{ccc}& & =\underset{(u,v50*)\in V\times Y}{inf}J(u,v50*)\hfill \\ & \le & \underset{z*\in Y*}{inf}\left\{\underset{v*\in A*\cap B*}{sup}\left\{-F˜*(v*,v*,v*,z*,z*)-F˜*(v*,v*,v*,z*,z*)-\nu (v*)x\left(\right)u\left(\right)\right\}\right.\hfill \\ & & \left.+F˜*(z*,z*)+F*(z*,z*)\right\}.\hfill \end{array}$$

(577)

Therefore, defining $J*:[Y*]10\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill J*(v*,z*)& =& -F˜*(v*,v*,v*,z*,z*)-F˜*(v*,v*,v*,z*,z*)-\nu (v*)x\left(\right)u\left(\right)\hfill \\ & & +F˜*(z*,z*)+F*(z*,z*),\hfill \end{array}$$

(578)

we have got

$$=\underset{(u,v50*)\in V\times Y}{inf}J(u,v50*)\le \underset{z\in [Y*]}{inf}\left\{\underset{v*\in A*\cap B*}{sup}J*(v*,z*)\right\}.$$

Finally, we also emphasize that $J$ is concave in $v$ on the convex set $A*\cap B*$ and convex in $z$, so that, after the supremum evaluation in $v$, we have obtained a final convex dual formulation in $z$ for an originally non-convex primal dual one.

70. A convex dual formulation for the rank-one approximation of a non-convex primal one

In this section, we develop a convex dual formulation for an approximate rank-one primal formulation found in some vectorial phase transition models.

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega$.

Define a functional $J:V\to \mathbb{R}$ by

$$J\left(u\right)={\displaystyle 12}\int \Omega \left(\alpha ijkl\left({\displaystyle \partial ui\partial xj}-\beta ij\right)\left({\displaystyle \partial uk\partial xl}-\beta kl\right)\right)dx-\langle ui,fi\rangle L,$$

where $\left\{\alpha ijkl\right\}$ is a a fourth order constant positive definite and symmetric tensor, $\left\{\beta ij\right\}\in \mathbb{R}N$, $f=(f,f,f)\in L(\Omega ;\mathbb{R}N)$ and

$$V=W,(\Omega ;\mathbb{R}N).$$

From now and on we denote $Y=Y*=L\left(\Omega \right)$ and

$$Y=Y*=\left[Y\right]N+N++.$$

Define also $F:Y\to \mathbb{R}$, $F:Y\to \mathbb{R}$ and $F:\left[Y\right]N++\to \mathbb{R}$ by

$$\begin{array}{ccc}& & F(w,\xi ,\eta ,v50)\hfill \\ & =& {\displaystyle 12}\int \Omega \left(\alpha ijkl(wij-\beta ij)(wkl-\beta kl)+K\left|\xi \right|+K\left|\eta \right|+v50\right)dx,\hfill \end{array}$$

(579)

$$F(w,\xi ,\eta ,v50)=\sum i=N\sum j={\displaystyle K12}\int \Omega \left(wij-\xi i\eta j+K\left|\xi \right|+K\left|\eta \right|+v50\right)dx,$$

and

$$F(\xi ,\eta ,v50)={\displaystyle K12}\int \Omega \left(K\left|\xi \right|+K\left|\eta \right|+v50\right)dx,$$

respectively.

Here $K,K>$ are real constants such that $K\gg K\gg .$

Moreover, define

$$\begin{array}{ccc}& & J(u,w,\xi ,\eta ,v50)\hfill \\ & =& -\langle \xi i\eta j,(v*)ij\rangle L+F(w,\xi ,\eta ,v50)\hfill \\ & & +F(w,\xi ,\eta ,v50)+F(\xi ,\eta ,v50)\hfill \\ & & +〈{\displaystyle \partial ui\partial xj},(v*)ij〉L-\langle ui,fi\rangle L.\hfill \end{array}$$

(580)

Observe that

$$\begin{array}{ccc}& & J(u,w,\xi ,\eta ,v50)\hfill \\ & \ge & \underset{(\xi ,\eta )\in \left[Y\right]+N}{inf}\left\{-\langle \xi i\eta j,(v*)ij\rangle L+F(w,\xi ,\eta ,v50)\right.\hfill \\ & & \left.+F(w,\xi ,\eta ,v50)+F(\xi ,\eta ,v50)\right\}\hfill \\ & & +\underset{u\in V}{inf}\left\{〈{\displaystyle \partial ui\partial xj},(v*)ij〉L-\langle ui,fi\rangle L\right\}\hfill \\ & =& -F˜12*(v*),\forall v*\in A*,\hfill \end{array}$$

(581)

where

$$F˜12*(v*)=\underset{(\xi ,\eta )\in \left[Y\right]+N}{sup}\left\{\langle \xi i\eta j,(v*)ij\rangle L-F(w,\xi ,\eta ,v50)-F(w,\xi ,\eta ,v50)-F(\xi ,\eta ,v50)\right\}$$

and

$$A*=\{v*\in [Y*]N:(v*)ij,j+fi=,\forall i\in \{,\dots ,N\},\mathrm{in}\Omega \}.$$

On the other hand

$$\begin{array}{ccc}& & W˜\hfill \\ & =& -\langle \xi i\eta j,(v*)ij\rangle L+F(w,\xi ,\eta ,v50)\hfill \\ & & +F(w,\xi ,\eta ,v50)+F(\xi ,\eta ,v50)\hfill \\ & =& -\langle \xi i\eta j,(v*)ij\rangle L-\langle wij,(w*)ij\rangle L-\langle \xi i,(v*)i\rangle L\hfill \\ & & -\langle \eta j,(v*)j\rangle L-\langle v50,v*\rangle L+F(w,\xi ,\eta ,v50)\hfill \\ & & -\langle wij,(w*)ij\rangle L-\langle \xi i,(v*)i\rangle L\hfill \\ & & -\langle \eta j,(v*)j\rangle L-\langle v50,v*\rangle L+F(w,\xi ,\eta ,v50)\hfill \\ & & -\langle \xi i,(v*)i\rangle L-\langle \eta j,(v*)j\rangle L\hfill \\ & & -\langle v50,v10*\rangle L+F(w,\xi ,\eta ,v50)\hfill \\ & & +\langle wij,(w*)ij\rangle L+\langle \xi i,(v*)i\rangle L\hfill \\ & & +\langle \eta j,(v*)j\rangle L+\langle v50,v*\rangle L\hfill \\ & & +\langle wij,(w*)ij\rangle L+\langle \xi i,(v*)i\rangle L\hfill \\ & & +\langle \eta j,(v*)j\rangle L+\langle v50,v*\rangle L\hfill \\ & & +\langle \xi i,(v*)i\rangle L+\langle \eta j,(v*)j\rangle L+\langle v50,v10*\rangle L\hfill \end{array}$$

(582)

Thus,

$$\begin{array}{ccc}& & W˜\hfill \\ & \ge & \underset{(w,\xi ,\eta ,v50)\in Y}{inf}\{-\langle \xi i\eta j,(v*)ij-\langle wij,(w*)ij\rangle L-\langle \xi i,(v*)i\rangle L\hfill \\ & & -\langle \eta j,(v*)j\rangle L-\langle v50,v*\rangle L+F(w,\xi ,\eta ,v50)\}\hfill \\ & & +\underset{(w,\xi ,\eta ,v50)\in Y}{inf}\{-\langle wij,(w*)ij\rangle L-\langle \xi i,(v*)i\rangle L\hfill \\ & & -\langle \eta j,(v*)j\rangle L-\langle v50,v*\rangle L+F(w,\xi ,\eta ,v50)\}\hfill \\ & & +\underset{(\xi ,\eta ,v50)\in \left[Y\right]+N+}{inf}\{-\langle \xi i,(v*)i\rangle L-\langle \eta j,(v*)j\rangle L\hfill \\ & & -\langle v50,v10*\rangle L+F(w,\xi ,\eta ,v50)\}\hfill \\ & & +\underset{(w,\xi ,\eta ,v50)\in Y}{inf}\{\langle wij,(w*)ij\rangle L+\langle \xi i,(v*)i\rangle L\hfill \\ & & +\langle \eta j,(v*)j\rangle L+\langle v50,v*\rangle L\hfill \\ & & +\langle wij,(w*)ij\rangle L+\langle \xi i,(v*)i\rangle L\hfill \\ & & +\langle \eta j,(v*)j\rangle L+\langle v50,v*\rangle L\hfill \\ & & +\langle \xi i,(v*)i\rangle L+\langle \eta j,(v*)j\rangle L+\langle v50,v10*\rangle L\}\hfill \\ & =& -F˜*(w*,v*,v*.v*,v*)-F˜*(w*,v*,v*,v*)-F˜*(v*,v*,v10*),\hfill \\ & & \forall (w*,v*)\in A*,\hfill \end{array}$$

(583)

where $w*=(w*,w*)\in \left[Y\right]N\equiv Y*,$

$$v*=(v*,v*,v*,v*,v*,v*,v*,v*,v*,v10*)\in \left[Y\right]12N+\equiv Y*,$$

$$\begin{array}{ccc}\hfill A*& =& \{(w*,v*)\in Y*\times Y*:(w*)ij+(w*)ij=,\forall i\in \{,\dots ,N\},\forall j\in \{,,\},\mathrm{in}\Omega ,\hfill \\ & & (v*)i+(v*)i+(v*)i=,\forall i\in \{,\dots ,N\},\mathrm{in}\Omega ,\hfill \\ & & (v*)j+(v*)j+(v*)j=,\forall j\in \{,,\},\mathrm{in}\Omega ,\hfill \\ & & v*+v*+v10*=,\mathrm{in}\Omega \},\hfill \end{array}$$

(584)

$$A*=\{(w*,v*)\in Y*\times Y*:v*\in A*\},$$

and

$$A*=A*\cap A*.$$

Furthermore,

$$\begin{array}{ccc}& & F˜*(w*,v*,v*,v*,v*)\hfill \\ & =& \underset{(w,\xi ,\eta ,v50)\in Y}{sup}\{\langle \xi i\eta j,(v*)ij\rangle L+\langle wij,(w*)ij\rangle L+\langle \xi i,(v*)i\rangle L\hfill \\ & & +\langle \eta j,(v*)j\rangle L+\langle v50,v*\rangle L-F(w,\xi ,\eta ,v50)\},\hfill \end{array}$$

(585)

$$\begin{array}{ccc}& & F˜*(w*,v*,v*,v*)\hfill \\ & =& \underset{(w,\xi ,\eta ,v50)\in Y}{sup}\{\langle wij,(w*)ij\rangle L+\langle \xi i,(v*)i\rangle L\hfill \\ & & +\langle \eta j,(v*)j\rangle L+\langle v50,v*\rangle L-F(w,\xi ,\eta ,v50)\},\hfill \end{array}$$

(586)

$$\begin{array}{ccc}& & F˜*(v*,v*,v10*)\hfill \\ & =& \underset{(\xi ,\eta ,v50)\in \left[Y\right]+N+}{sup}\{\langle \xi i,(v*)i\rangle L+\langle \eta j,(v*)j\rangle L\hfill \\ & & +\langle v50,v10*\rangle L-F(\xi ,\eta ,v50)\}.\hfill \end{array}$$

(587)

Denoting

$$J*(w*,v*)=-F˜*(w*,v*,v*,v*,v*)-F˜*(w*,v*,v*,v*)-F˜*(v*,v*,v10*),$$

we have got

$$\begin{array}{ccc}\hfill \underset{u\in V}{inf}J\left(u\right)& \ge & \underset{(u,w,\xi ,\eta ,v50)\in V\times Y}{inf}J(u,w,\xi ,\eta ,v50)\hfill \\ & \ge & \underset{(w*,v*)\in A*}{sup}J*(w*,v*).\hfill \end{array}$$

(588)

Finally, we emphasize $J$ is a convex (in fact concave) functional.

71. A Dual Variational Formulation for a General Non-Convex Primal One

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider a functional $J:V\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J\left(u\right)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx\hfill \\ & & +{\displaystyle \alpha }\int \Omega (u-\beta )dx-\langle u,f\rangle L,\hfill \end{array}$$

(589)

where $\gamma >$, $\alpha >,$$\beta >$ and $f\in L\left(\Omega \right).$

Here $u\in V=W,\left(\Omega \right)$ and we denote $Y=Y*=L\left(\Omega \right).$

At this point, we define the functionals $F:V\to \mathbb{R}$, $F:V\times Y\to \mathbb{R}$ and $F:V\to \mathbb{R},$ where

$$F\left(u\right)={\displaystyle \gamma }\int \Omega \nabla u·\nabla udx,$$

$$F(u,v)={\displaystyle \alpha }\int \Omega (u-\beta +v)dx+{\displaystyle K}\int \Omega udx-\langle u,f\rangle L;$$

and

$$F\left(u\right)={\displaystyle K}\int \Omega udx,$$

for some constant $K>$.

Moreover, we define

$$V=\left\{u\in V:\parallel u\parallel \infty <{\displaystyle 32}\right\}$$

and the following polar functionals

$$\begin{array}{ccc}\hfill F*(v*,z*)& =& \underset{u\in V}{sup}\{\langle u,v*+z*\rangle L-F\left(u\right)\}\hfill \\ & =& {\displaystyle 12}\int \Omega {\displaystyle (v*+z*)(-\gamma \nabla )}dx,\hfill \end{array}$$

(590)

$$\begin{array}{ccc}\hfill F*(v*,v*)& =& \underset{(u,v)\in V\times Y*}{sup}\{\langle u,-v*\rangle L+\langle v,v*\rangle L-F(u,v)\}\hfill \\ & =& {\displaystyle 12}\int \Omega {\displaystyle (-v*+f)v*+K}dx\hfill \\ & & +{\displaystyle \alpha }\int \Omega (v*)dx+\beta \int \Omega v*dx,\hfill \end{array}$$

(591)

if $v*\in B*$, where

$$B*=\left\{v*\in Y*:\parallel v*\parallel \infty \le {\displaystyle K}\right\}$$

and,

$$\begin{array}{ccc}\hfill F*(z*)& =& \underset{v\in Y}{sup}\{\langle v,z*\rangle L-F\left(v\right)\}\hfill \\ & =& {\displaystyle K}\int \Omega (z*)dx.\hfill \end{array}$$

(592)

Finally, denoting

$$D*=\left\{v*\in Y*:\parallel -v*+f\parallel \infty \le {\displaystyle 54}K\right\},$$

we also define $J*:D*\times B*\times Y*\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill J*(v*,v*,z*)& =& -F*(v*,z*)-F*(v*,v*)+F*(z*)\hfill \\ & & +{\displaystyle K}∥{\displaystyle v*+z*-\gamma \nabla }-{\displaystyle z*K}∥,\hfill \\ & & +{\displaystyle K}∥{\displaystyle -v*+fv*+K}-{\displaystyle z*K}∥,.\hfill \end{array}$$

(593)

Observe that if $K>$ is sufficiently large, then $J$ is convex in $(v*,z*)$, $\forall v*\in B*.$

Let $(v^*,v^*,z^*)\in D*\times B*\times Y*$ be such that

$$\delta J*(v^*,v^*,z^*)=\mathbf{0}.$$

From such a concerning convexity of $J$ in $(v*,z*)$ we may infer that

$$J*(v^*,v^*,z^*)\le J*(v*,v^,z*),\forall v*\in D*,z*\in Y*.$$

In particular fixing $u\in V$, for $v*=(v*+K)u$ and $z*=Ku$, we obtain

$$\begin{array}{ccc}\hfill J*(v^*,v^*,z^*)& \le & J*(v*,v^,z*)\hfill \\ & \le & -\langle u,v*+z*\rangle L+{\displaystyle \gamma }\int \Omega \nabla u·\nabla udx\hfill \\ & & +\langle u,v*\rangle L+\langle u,v^*\rangle L+{\displaystyle K}\int \Omega udx\hfill \\ & & -\langle u,f\rangle L+\langle u,z*\rangle L-{\displaystyle K}\int \Omega udx\hfill \\ & & -{\displaystyle \alpha }\int \Omega (v^*)dx-\beta \int \Omega v^*dx\hfill \\ & & +{\displaystyle K}∥{\displaystyle -\gamma \nabla u+v^*u-f-\gamma \nabla }∥,\hfill \\ & \le & \underset{v*\in Y*}{sup}\left\{{\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+\langle u,v*\rangle L-\langle u,f\rangle L\right.\hfill \\ & & -{\displaystyle \alpha }\int \Omega (v*)dx-\beta \int \Omega v*dx\hfill \\ & & \left.+{\displaystyle K}∥{\displaystyle -\gamma \nabla u+v^*u-f-\gamma \nabla }∥,\right\}\hfill \\ & =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle \alpha }\int \Omega (u-\beta )dx\hfill \\ & & -\langle u,f\rangle L+{\displaystyle K}∥{\displaystyle -\gamma \nabla u+v^*u-f-\gamma \nabla }∥,\hfill \\ & & =J\left(u\right)+{\displaystyle K}∥{\displaystyle -\gamma \nabla u+v^*u-f-\gamma \nabla }∥,.\hfill \end{array}$$

(594)

Summarizing, we have got

$$J*(v^*,v^*,z^*)\le J\left(u\right)+{\displaystyle K12}∥{\displaystyle -\gamma \nabla u+v^*u-f-\gamma \nabla }∥,,\forall u\in V.$$

Let $u\in V$ be such that

$$u={\displaystyle z^*K}.$$

Assume $u\in V.$

Similarly as in the previous sections, we may prove that

$$v^*=\alpha (u02-\beta ),$$

$$\delta J\left(u\right)=-\gamma \nabla u+v^u-f=\mathbf{0},$$

and

$$J\left(u\right)=J*(v^*,v^*,z^*),$$

so that

$$\begin{array}{ccc}\hfill J\left(u\right)& =& \underset{u\in V}{min}\left\{J\left(u\right)+{\displaystyle K12}∥{\displaystyle -\gamma \nabla u+v^*u-f-\gamma \nabla }∥,\right\}\hfill \\ & =& J*(v^*,v^*,z^*)\hfill \\ & =& \underset{(v*,z*)\in D*\times Y*}{inf}J*(v*,v^*,z*).\hfill \end{array}$$

(595)

The objective of this section is complete.

72. A D.C. Type Duality Principle Suitable for Non-Convex Variational Optimization

In this section we develop results concerning a D.C. approach inspired by the results of J.J. Telega, W.R. Bielski and co-workers, [1,2,3,4] and Toland, [5].

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider a functional $J:V\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J\left(u\right)& =& {\displaystyle \epsilon }\int \Omega \nabla u·\nabla udx\hfill \\ & & +{\displaystyle \alpha }\int \Omega (u-\beta )dx-\langle u,f\rangle L,\hfill \end{array}$$

(596)

where $\epsilon >$, $\alpha >,$$\beta >$ and $f\in L\left(\Omega \right).$

Here $u\in V=W,\left(\Omega \right)$ and we denote $Y=Y*=L\left(\Omega \right).$

At this point, for a large constant $K>$, we define the approximate functional $J:V\times Y\to \mathbb{R},$ by

$$\begin{array}{ccc}\hfill J(u,v)& =& {\displaystyle \epsilon }\int \Omega \nabla u·\nabla udx+{\displaystyle \alpha }\int \Omega (v-\beta )dx\hfill \\ & & +{\displaystyle K}\int \Omega (v-u)dx-\langle u,f\rangle L.\hfill \end{array}$$

(597)

We define also the functionals $F:V\to \mathbb{R}$, $F:Y\to \mathbb{R}$, $F:V\times Y\to \mathbb{R},$ and $F:V\to \mathbb{R}$, where

$$F\left(u\right)={\displaystyle \epsilon }\int \Omega \nabla u·\nabla udx-\langle u,f\rangle L,$$

$$F\left(v\right)={\displaystyle \alpha }\int \Omega (v-\beta )dx$$

$$F(u,v)={\displaystyle K12}\int \Omega (v-u)dx+{\displaystyle K}\int \Omega udx$$

and

$$F\left(u\right)={\displaystyle K}\int \Omega udx,$$

for some appropriate constant $K>$.

Moreover, we define the following polar functionals

$$\begin{array}{ccc}\hfill F*(v*)& =& \underset{u\in V}{sup}\{-\langle u,v*\rangle L-F\left(u\right)\}\hfill \\ & =& {\displaystyle 12}\int \Omega {\displaystyle (-v*+f)(-\epsilon \nabla )}dx,\hfill \end{array}$$

(598)

$$\begin{array}{ccc}\hfill F*(v*)& =& \underset{v\in Y*}{sup}\{\langle v,v*\rangle L-F\left(v\right)\}\hfill \\ & =& {\displaystyle \alpha }\int \Omega (v*)dx+\beta \int \Omega v*dx,\hfill \end{array}$$

(599)

$$\begin{array}{ccc}\hfill F*(v*,v*,z*)& =& \underset{(u,v)\in V\times Y}{sup}\{\langle u,v*+z*\rangle L-\langle v,v*\rangle L-F(u,v)\}\hfill \\ & =& {\displaystyle 12}\int \Omega {\displaystyle (v*+z*)v*+K}dx+{\displaystyle K}\int \Omega (v*)dx,\hfill \end{array}$$

(600)

if $v*\in B*$, where

$$B*=\left\{v*\in Y*:\parallel v*\parallel \infty \le {\displaystyle K}\right\}$$

and,

$$\begin{array}{ccc}\hfill F*(z*)& =& \underset{w\in Y}{sup}\{\langle w,z*\rangle L-F\left(w\right)\}\hfill \\ & =& {\displaystyle K}\int \Omega (z*)dx.\hfill \end{array}$$

(601)

Finally, we define

$$D*=\left\{z*\in Y*:\parallel z*\parallel \infty \le {\displaystyle 54}K\right\},$$

and $J*:D*\times B*\times Y*\to \mathbb{R}$ by

$$J*(v*,v*,z*)=-F*(v*)-F*(v*)-F*(v*,v*,z*)+F*(z*).$$

Let $\alpha \in \mathbb{R}$ be such that

$$\underset{(u,v)\in V\times Y}{inf}J(u,v)=\alpha .$$

Observe that

$$\begin{array}{ccc}\hfill \alpha & \le & J(u,v)\hfill \\ & =& F\left(u\right)+F\left(v\right)+F(u,v)-F\left(u\right)\hfill \\ & =& -\langle u,z*\rangle L+F\left(u\right)+F\left(v\right)+F(u,v)\hfill \\ & & +\langle u,z*\rangle L-F\left(u\right)\hfill \\ & \le & -\langle u,z*\rangle L+F\left(u\right)+F\left(v\right)+F(u,v)\hfill \\ & & +\underset{w\in Y}{sup}\{\langle w,z*\rangle L-F\left(w\right)\}\hfill \\ & =& -\langle u,z*\rangle L+F\left(u\right)+F\left(v\right)+F(u,v)+F*(z*),\forall u\in V,v\in Y,z*\in D*.\hfill \end{array}$$

(602)

From such results we may infer that

$$\alpha \le \underset{(u,v)\in V\times Y}{inf}\{-\langle u,z*\rangle L+F\left(u\right)+F\left(v\right)+F(u,v)\}+F*(z*).$$

On the other hand, for an appropriate value of $K>$ and $z*\in D*$, from standard results in convex analysis, we have

$$\begin{array}{ccc}& & \underset{(u,v)\in V\times Y}{inf}\{-\langle u,z*\rangle L+F\left(u\right)+F\left(v\right)+F(u,v)\}\hfill \\ & =& \underset{(v*,v*)\in Y*\times B*}{sup}\{-F*(v*)-F*(v*)-F*(v*,v*,z*)\}.\hfill \end{array}$$

(603)

Joining the pieces, we have got

$$\alpha \le \underset{(v*,v*)\in Y*\times B*}{sup}\{-F*(v*)-F*(v*)-F*(v*,v*,z*)\}+F*(z*),$$

so that

$$\alpha \le \underset{z*\in D*}{inf}\left\{\underset{(v*,v*)\in Y*\times B*}{sup}\{-F*(v*)-F*(v*)-F*(v*,v*,z*)\}+F*(z*)\right\},$$

that is,

$$\alpha =\underset{(u,v)\in V\times Y}{inf}J(u,v)\le \underset{z*\in D*}{inf}\left\{\underset{(v*,v*)\in Y*\times B*}{sup}J*(v*,v*,z*)\right\}.$$

Let $(v^*,v^*,z^*)\in D*\times B*\times Y*$ be such that

$$\delta J*(v^*,v^*,z^*)=\mathbf{0}.$$

Let $(u,v)\in V\times Y$ be such that

$$u={\displaystyle z^*K}$$

and

$$v={\displaystyle v^*\alpha }+\beta .$$

Similarly as in the previous sections, we may prove that

$$\delta J(u,v)=\mathbf{0},$$

and

$$J(u,v)=J*(v^*,v^*,z^*),$$

so that

$$\begin{array}{ccc}\hfill J(u,v)& =& J*(v^*,v^*,z^*)\hfill \\ & =& \underset{(v*,v*)\in Y*\times B*}{sup}J*(v*,v*,z^*).\hfill \end{array}$$

(604)

The main objective of this section is complete.

72.1. A numerical example

We have obtained numerical results for an one-dimensional case where, $\Omega =[,]\subset \mathbb{R},$$A=B=$, $f\equiv$ and

Case A: $\epsilon =.$

Case B: $\epsilon =.01$

Case C: $\epsilon =.001.$

For the optimal solutions $u\in V$ obtained for the cases A,B and C, please see Figure 47, Figure 48 and Figure 49, respectively.

Here we present the software in MAT-LAB through which we have obtained such numerical results.

clear all

global m8 d A B yo e1 K1 e2 u z K v2

m8=100;

d=1/m8;

yo(:,1)=2*ones(m8-1,1);

z(:,1)=1.2*ones(m8-1,1);

A=1;

B=1;

e1=0.001;

e2=0.00000001;

K1=1000000;

K=30;

for i=1:2*(m8-1)

xo(i,1)=0.7;

x1(i,1)=1.1;

end;

b14=1.0;

k7=1.0;

while $(b14>10-)$ && $(k<70)$

k7

k7=k7+1;

b12=1.0;

k=1;

while $(b12>10-)$ && $(k<10)$

k

k=k+1;

X=fminunc(’funJuly2024A1’,xo);

b12=max(abs(X-xo));

xo=X;

u(m8/2,1)

end;

z=K*u;

b14=max(abs(x1-xo));

x1=xo;

end;

for i=1:m8-1

x(i,1)=i*d;

end;

plot(x,u);

*****************************

With the auxiliary function "funJuly2024A1", where

********************************

function S=funJuly2024A1(x)

global m8 d A B yo e1 K1 e2 u z K v2

m2=zeros(m8-1,m8-1);

y1=ones(m8-1,1);

for i=2:m8-2

m2(i,i)=-2.0;

m2(i,i+1)=1.0;

m2(i,i-1)=1.0;

end;

m2(1,1)=-2.0;

m2(1,2)=1.0;

m2(m8-1,m8-1)=-2.0;

m2(m8-1,m8-2)=1.0;

for i=1:m8-1

v1(i,1)=x(i,1);

v2(i,1)=x(i+(m8-1),1);

end;

$S=/*(-v+yo)\prime *inv(-e*m/d)*(-v+yo)+v\prime *v//A+B*v\prime *y;$

for i=1:m8-1

$S=S+(v(i,)+z(i,))/(*v(i,)+K)/+v(i,)//K;$

end;

$u=inv(-e*m/d)*(-v+yo);$

********************************************

73. A Concave Dual Variational Formulation for an Originally Non-Convex Primal One

In this section we develop results also inspired by the approach found in the articles of J.J. Telega, W.R. Bielski and co-workers, [1,2,3,4] and Toland, [5].

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider a functional $J:V\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J\left(u\right)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx\hfill \\ & & +{\displaystyle \alpha }\int \Omega (u-\beta )dx-\langle u,f\rangle L,\hfill \end{array}$$

(605)

where $\gamma >$, $\alpha >,$$\beta >$ and $f\in L\left(\Omega \right).$

Here $u\in V=W,\left(\Omega \right)$ and we denote $Y=Y*=L\left(\Omega \right).$

Consider also the functionals $F:V\times Y\to \mathbb{R}$, $F:V\times Y\to \mathbb{R}$ and $F:V\to \mathbb{R},$ where

$$\begin{array}{ccc}\hfill F(u,v*)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle 12}\langle u,v*\rangle L\hfill \\ & & +{\displaystyle K}\int \Omega udx-\langle u,f\rangle L\hfill \\ & & -{\displaystyle \alpha }\int \Omega (v*)dx-\beta \int \Omega v*dx,\hfill \end{array}$$

(606)

$$\begin{array}{ccc}\hfill F(u,v*)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx\hfill \\ & & +{\displaystyle 12}\langle u,v*\rangle L+{\displaystyle K}\int \Omega udx,\hfill \end{array}$$

(607)

$$F\left(u\right)=K\int \Omega udx,$$

for some appropriate constant $K>$.

Moreover, we define the following polar functionals

$$\begin{array}{ccc}\hfill F*(v*,v*,z*)& =& \underset{u\in V}{sup}\left\{〈u,v*+{\displaystyle z*}〉L-F(u,v*)\right\}\hfill \\ & =& {\displaystyle 12}\int \Omega {\displaystyle (v*+z*/+f)(-\gamma \nabla +v*)/+K}dx\hfill \\ & & +{\displaystyle \alpha }\int \Omega (v*)dx+\beta \int \Omega v*dx,\hfill \end{array}$$

(608)

$$\begin{array}{ccc}\hfill F*(v*,v*,z*)& =& \underset{u\in V}{sup}\left\{〈u,-v*+{\displaystyle z*}〉L-F(u,v*)\right\}\hfill \\ & =& {\displaystyle 12}\int \Omega {\displaystyle (-v*+z*/)(-\gamma \nabla +v*)/+K}dx,\hfill \end{array}$$

(609)

if $v*\in B*$, where

$$B*=\left\{v*\in Y*:\parallel v*\parallel \infty \le {\displaystyle K}\right\},$$

and,

$$\begin{array}{ccc}\hfill F*(z*)& =& \underset{w\in Y}{sup}\{\langle w,z*\rangle L-F\left(w\right)\}\hfill \\ & =& {\displaystyle K}\int \Omega (z*)dx.\hfill \end{array}$$

(610)

Finally, we define

$$A+=\{z*\in Y*:fz*\ge ,\mathrm{in}\Omega \},$$

$$D*=\left\{z*\in A+:\parallel z*\parallel \infty \le {\displaystyle 52}K\right\},$$

and $J*:Y*\times B*\times D*\to \mathbb{R}$ by

$$J*(v*,v*,z*)=-F*(v*,v*,z*)-F*(v*,v*,z*)+F*(z*).$$

Observe that the variation of $J$ in $v$ stands for

$$-{\displaystyle v*+z*/+f(-\gamma \nabla +v*)/+K}+{\displaystyle -v*+z*/(-\gamma \nabla +v*)/+K}=\mathbf{0},$$

so that

$$v*+f=\mathbf{0}.$$

The variation of $J$ in $z$ stands for

$$-{\displaystyle 12}{\displaystyle v*+z*/+f(-\gamma \nabla +v*)/+K}-{\displaystyle 12}{\displaystyle -v*+z*/(-\gamma \nabla +v*)/+K}+{\displaystyle z*K}=\mathbf{0}.$$

Finally, the variation of $J$ in $v$ stands for,

$${\displaystyle 12}\left({\displaystyle v*+z*/+f(-\gamma \nabla +v*)/+K}\right)+{\displaystyle 12}\left({\displaystyle -v*+z*/(-\gamma \nabla +v*)/+K}\right)-{\displaystyle v*\alpha }-\beta =\mathbf{0}.$$

With such results in mind, we define the functional $J*:Y*\times B*\times D*\to \mathbb{R}$, by

$$\begin{array}{ccc}& & J*(v*,v*,z*)\hfill \\ & =& J*(v*,v*,z*)-{\displaystyle K}\parallel v*+f\parallel ,\hfill \\ & & +{\displaystyle K282}∥-{\displaystyle 12}{\displaystyle v*+z*/+f(-\gamma \nabla +v*)/+K}-{\displaystyle 12}{\displaystyle -v*+z*/(-\gamma \nabla +v*)/+K}+{\displaystyle z*K}∥,\hfill \\ & & +{\displaystyle K282}∥{\displaystyle 12}\left({\displaystyle v*+z*/+f(-\gamma \nabla +v*)/+K}\right)+{\displaystyle 12}\left({\displaystyle -v*+z*/(-\gamma \nabla +v*)/+K}\right)-{\displaystyle v*\alpha }-\beta ∥,.\hfill \end{array}$$

(611)

Observe that for $K>$ sufficiently large $J*:E*\to \mathbb{R}$ is concave in $v$ on $E$, where

$$J*(v*)=\mathrm{sta}(v*,z*)\in B*\times D*J*(v*,v*,z*),$$

and

$$E*=\{v*\in Y*:\parallel v*+f\parallel \infty \le \}.$$

Let $(v^*,v^*,z^*)\in E*\times B*\times D*$ be such that

$$\delta J*(v^*,v^*,z^*)=\mathbf{0}.$$

Let $u\in V$ be such that

$$u={\displaystyle z^*K}$$

Similarly as in the previous sections, we may prove that

$$\delta J\left(u\right)=\mathbf{0},$$

and

$$J\left(u\right)=J*(v^*,v^*,z^*),$$

so that

$$\begin{array}{ccc}\hfill J\left(u\right)& =& J*(v^*,v^*,z^*)\hfill \\ & =& \underset{v*\in E*}{sup}J*(v*,v^*,z^*)\hfill \\ & =& \underset{v*\in E*}{sup}J*(v*)\hfill \\ & =& J*(v^*).\hfill \end{array}$$

(612)

The main objective of this section is complete.

74. A dual variational formulation for an originally non-convex primal one

In this section we develop results also inspired by the approach found in the articles of J.J. Telega, W.R. Bielski and co-workers, [1,2,3,4] and Toland, [5].

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider a functional $J:V\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J\left(u\right)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx\hfill \\ & & +{\displaystyle \alpha }\int \Omega (u-\beta )dx-\langle u,f\rangle L,\hfill \end{array}$$

(613)

where $\gamma >$, $\alpha >,$$\beta >$ and $f\in L\left(\Omega \right).$

Here $u\in V=W,\left(\Omega \right)$ and we denote $Y=Y*=L\left(\Omega \right).$

Consider also the functionals $F:V\to \mathbb{R}$ and $F:V\times Y\to \mathbb{R}$ where,

$$\begin{array}{ccc}\hfill F\left(u\right)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle K}\int \Omega udx,\hfill \end{array}$$

(614)

$$\begin{array}{ccc}\hfill F(u,v*)& =& \langle u,v*\rangle L\hfill \\ & & -\langle u,f\rangle L-{\displaystyle K}\int \Omega udx\hfill \\ & & -{\displaystyle \alpha }\int \Omega (v*)dx-\beta \int \Omega v*dx\hfill \end{array}$$

(615)

for some appropriate constant $K>$.

Moreover, we define the following polar functionals

$$\begin{array}{ccc}\hfill F*(v*)& =& \underset{u\in V}{sup}\left\{\langle u,v*\rangle L-F\left(u\right)\right\}\hfill \\ & =& {\displaystyle 12}\int \Omega {\displaystyle (v*)-\gamma \nabla +K}dx\hfill \end{array}$$

(616)

and

$$\begin{array}{ccc}\hfill F*(v*,v*)& =& \underset{u\in V}{inf}\left\{\langle u,v*\rangle L-F(u,v*)\right\}\hfill \\ & =& {\displaystyle 12}\int \Omega {\displaystyle (-v*+f)v*-K}dx\hfill \\ & & +{\displaystyle \alpha }\int \Omega (v*)dx+\beta \int \Omega v*dx,\hfill \end{array}$$

(617)

if $v*\in B*$, where

$$B*=\left\{v*\in Y*:\parallel v*\parallel \infty \le {\displaystyle K}\right\}.$$

Finally, we define

$$A+=\{u\in V:fu\ge ,\mathrm{in}\Omega \},$$

$$V=\left\{u\in A+:\parallel u\parallel \infty \le {\displaystyle 32}\right\},$$

$$D*=\left\{v*\in Y*:∥{\displaystyle v*-\gamma \nabla +K}∥\infty \le {\displaystyle 32}\right\}$$

and $J*:D*\times B*\to \mathbb{R}$ by

$$J*(v*,v*)=-F*(v*)-F*(v*,v*).$$

Observe that the variation of $J$ in $v$ stands for

$$\phi =-{\displaystyle v*-\gamma \nabla +K}+{\displaystyle -v*+fv*-K}=\mathbf{0}.$$

On the other hand, the variation of $J$ in $v$ stands for

$$\phi =\left({\displaystyle -v*+fv*-K}\right)-{\displaystyle v*\alpha }-\beta =\mathbf{0},$$

so that

$$\phi =\left({\displaystyle v*-\gamma \nabla +K}\right)-{\displaystyle v*\alpha }-\beta =\mathbf{0}.$$

With such results in mind, we define the functional $J*:D*\times B*\to \mathbb{R}$, by

$$\begin{array}{ccc}& & J*(v*,v*)\hfill \\ & =& J*(v*,v*)\hfill \\ & & +{\displaystyle K22}∥-{\displaystyle v*-\gamma \nabla +K}+{\displaystyle -v*+fv*-K}∥,\hfill \\ & & +{\displaystyle K22}∥\left({\displaystyle -v*+fv*-K}\right)-{\displaystyle v*\alpha }-\beta ∥,\hfill \end{array}$$

(618)

Here we assume $K\gg K\gg max\{,\alpha ,\beta ,\gamma ,/\alpha ,/\gamma \}$.

Let $(v^*,v^*)\in D*\times B*$ be such that

$$\delta J*(v^*,v^*)=\mathbf{0}.$$

Let $u\in V$ be such that

$$u={\displaystyle v^*-\gamma \nabla +K}$$

Similarly as in the previous sections, we may prove that

$$\delta J\left(u\right)=\mathbf{0},$$

and

$$J\left(u\right)=J*(v^*,v^*).$$

Moreover, at a critical point of $J$ we have $\phi =\phi =\phi =\mathbf{0}$ so that,

$${\displaystyle \partial J*(v*,v*)\partial (v*)}=\mathcal{O}\left(K22\right),$$

$${\displaystyle \partial J*(v*,v*)\partial (v*)}=\mathcal{O}\left(K22\right),$$

and with the help of software MATHEMATICA, at a critical point, we may obtain

$$det\left\{{\displaystyle \partial J*(v*,v*)\partial v*\partial v*}\right\}={\displaystyle K22(-\gamma \nabla +\alpha u+v*)\alpha (-\gamma \nabla +K)(K-v*)}+\mathcal{O}\left(K\right)>.$$

With such results in mind, since

$$\delta J\left(u\right)=-\gamma \nabla u+\alpha (u02-\beta )u-f=\mathbf{0},$$

joining the pieces, assuming an approximate finite dimensional version for the model in question, if necessary, we may infer that there exist $r,r>$ such that

$$\begin{array}{ccc}\hfill J\left(u\right)& =& \underset{u\in Br\left(u\right)}{inf}\left\{J\left(u\right)+{\displaystyle K22}∥{\displaystyle -\gamma \nabla u+\alpha (u-\beta )u-f-\gamma \nabla +K}∥,\right\}\hfill \\ & =& J*(v^*,v^*)\hfill \\ & =& \underset{(v*,v*)\in Br(v^*,v^*)}{inf}J*(v*,v*).\hfill \end{array}$$

(619)

The main objective of this section is complete.

75. A Convex Dual Variational Formulation for an Originally Non-Convex Primal One

In this section we develop results again inspired by the approach found in the articles of J.J. Telega, W.R. Bielski and co-workers, [1,2,3,4] and Toland, [5].

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider a functional $J:V\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J\left(u\right)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx\hfill \\ & & +{\displaystyle \alpha }\int \Omega (u-\beta )dx-\langle u,f\rangle L,\hfill \end{array}$$

(620)

where $\gamma >$, $\alpha >,$$\beta >$ and $f\in L\left(\Omega \right).$

Here $u\in V=W,\left(\Omega \right)$ and we denote $Y=Y*=L\left(\Omega \right).$

Consider also the functionals $F:V\to \mathbb{R}$ and $F:V\times Y\to \mathbb{R}$ where,

$$\begin{array}{ccc}\hfill F\left(u\right)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle K}\int \Omega udx,\hfill \end{array}$$

(621)

$$\begin{array}{ccc}\hfill F(u,v*)& =& \langle u,v*\rangle L\hfill \\ & & -\langle u,f\rangle L-{\displaystyle K}\int \Omega udx\hfill \\ & & -{\displaystyle \alpha }\int \Omega (v*)dx-\beta \int \Omega v*dx\hfill \end{array}$$

(622)

for some appropriate constant $K>$.

Moreover, we define the following polar functionals $F*:Y*\to \mathbb{R}$ and $F*:Y*\times Y*\to \mathbb{R}$ where,

$$\begin{array}{ccc}\hfill F*(v*)& =& \underset{u\in V}{sup}\left\{\langle u,v*\rangle L-F\left(u\right)\right\}\hfill \\ & =& {\displaystyle 12}\int \Omega {\displaystyle (v*)-\gamma \nabla +K}dx,\hfill \end{array}$$

(623)

$$\begin{array}{ccc}\hfill F*(v*,v*)& =& \underset{u\in V}{inf}\left\{\langle u,v*\rangle L-F(u,v*)\right\}\hfill \\ & =& {\displaystyle 12}\int \Omega {\displaystyle (-v*+f)v*-K}dx\hfill \\ & & +{\displaystyle \alpha }\int \Omega (v*)dx+\beta \int \Omega v*dx,\hfill \end{array}$$

(624)

if $v*\in B*$, where

$$B*=\left\{v*\in Y*:\parallel v*\parallel \infty \le {\displaystyle K}\right\}.$$

Finally, we define

$$A+=\{u\in V:fu\ge ,\mathrm{in}\Omega \},$$

$$V=\left\{u\in A+:\parallel u\parallel \infty \le {\displaystyle 32}\right\},$$

$$D+=\left\{v*\in Y*:f{\displaystyle v*-\gamma \nabla +K}\ge ,\mathrm{in}\Omega \right\},$$

$$D*=\left\{v*\in D+:∥{\displaystyle v*-\gamma \nabla +K}∥\infty \le {\displaystyle 32}\right\}$$

and $J*:D*\times B*\to \mathbb{R}$ by

$$J*(v*,v*)=-F*(v*)-F*(v*,v*).$$

Observe that the variation of $J$ in $v$ stands for

$$-{\displaystyle v*-\gamma \nabla +K}+{\displaystyle -v*+fv*-K}=\mathbf{0}.$$

On the other hand, the variation of $J$ in $v$ stands for

$$\left({\displaystyle -v*+fv*-K}\right)-{\displaystyle v*\alpha }-\beta =\mathbf{0},$$

so that

$$\left({\displaystyle v*-\gamma \nabla +K}\right)-{\displaystyle v*\alpha }-\beta =\mathbf{0},$$

With such results in mind, we define the functional $J*:D*\times B*\to \mathbb{R}$, by

$$\begin{array}{ccc}& & J*(v*,v*)\hfill \\ & =& J*(v*,v*)\hfill \\ & & +{\displaystyle K}∥-{\displaystyle v*-\gamma \nabla +K}+{\displaystyle -v*+fv*-K}∥,\hfill \\ & & +{\displaystyle K}∥\left({\displaystyle -v*+fv*-K}\right)-{\displaystyle v*\alpha }-\beta ∥,\hfill \end{array}$$

(625)

Observe that for $K\gg K\gg max\{,\alpha ,\beta ,\gamma ,/\alpha ,/\gamma \}$ we have that $J$ is convex in $(v*,v*)$ on $D*\times B*$.

Let $(v^*,v^*)\in D*\times B*$ be such that

$$\delta J*(v^*,v^*)=\mathbf{0}.$$

Let $u\in V$ be such that

$$u={\displaystyle v^*-\gamma \nabla +K}$$

Similarly as in the previous sections, we may prove that

$$\delta J\left(u\right)=\mathbf{0},$$

and

$$J\left(u\right)=J*(v^*,v^*).$$

so that from a concerning convexity,

$$J\left(u\right)=J*(v^*,v^*)=\underset{(v*,v*)\in D*\times B*}{inf}J*(v*,v*).$$

Fix $v˜*\in D*$. Let $u\in V$ be such that

$$u={\displaystyle -v˜*+fv˜*-K}$$

where $v˜*\in B*$ is such that

$$v˜*=\alpha (u-\beta ).$$

From these results and definitions we may obtain

$$\begin{array}{ccc}\hfill J\left(u\right)& =& J*(v^*,v^*)\hfill \\ & =& \underset{(v*,v*)\in D*\times B*}{inf}J*(v*,v*)\hfill \\ & \le & J*(v˜*,v˜*)\hfill \\ & =& -F*(v˜*)+\langle u,v˜*\rangle L\hfill \\ & & +{\displaystyle \alpha }\int \Omega (u-\beta )dx-\langle u,f\rangle L-{\displaystyle K}\int \Omega udx\hfill \\ & & +{\displaystyle K}∥{\displaystyle -\gamma \nabla u+v˜*u-f-\gamma \nabla +K}∥,\hfill \\ & \le & F\left(u\right)+{\displaystyle \alpha }\int \Omega (u-\beta )dx-\langle u,f\rangle L-{\displaystyle K}\int \Omega udx\hfill \\ & & +{\displaystyle K}∥{\displaystyle -\gamma \nabla u+v˜*u-f-\gamma \nabla +K}∥,\hfill \\ & =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+{\displaystyle \alpha }\int \Omega (u-\beta )dx-\langle u,f\rangle L\hfill \\ & & +{\displaystyle K}∥{\displaystyle -\gamma \nabla u+v˜*u-f-\gamma \nabla +K}∥,\hfill \\ & =& J\left(u\right)+{\displaystyle K}∥{\displaystyle -\gamma \nabla u+\alpha (u-\beta )u-f-\gamma \nabla +K}∥,.\hfill \end{array}$$

(626)

Since

$$\delta J\left(u\right)=-\gamma \nabla u+\alpha (u02-\beta )u-f=\mathbf{0},$$

joining the pieces we have obtained

$$\begin{array}{ccc}\hfill J\left(u\right)& =& \underset{u\in V}{inf}\left\{J\left(u\right)+{\displaystyle K22}∥{\displaystyle -\gamma \nabla u+\alpha (u-\beta )u-f-\gamma \nabla +K}∥,\right\}\hfill \\ & =& J*(v^*,v^*)\hfill \\ & =& \underset{(v*,v*)\in D*\times B*}{inf}J*(v*,v*).\hfill \end{array}$$

(627)

The main objective of this section is complete.

76. A Duality Principle and a Related Convex Dual Functional Suitable for Non-Convex Local Optimization

In this section we develop a new duality principle with a related convex dual functional.

Let $\Omega \subset \mathbb{R}$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider a functional $J:V\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J\left(u\right)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx\hfill \\ & & +{\displaystyle \alpha }\int \Omega (u-\beta )dx-\langle u,f\rangle L,\hfill \end{array}$$

(628)

where $\gamma >$, $\alpha >,$$\beta >$ and $f\in L\left(\Omega \right).$

Here $u\in V=W,\left(\Omega \right)$ and we denote $Y=Y*=L\left(\Omega \right).$

Define the functionals $F:V\times Y*\to \mathbb{R}$ and $F:V\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill F(u,v*)& =& {\displaystyle \gamma }\int \Omega \nabla u·\nabla udx+\langle u,v*\rangle L\hfill \\ & & +{\displaystyle \epsilon }\int \Omega udx-\langle u,f\rangle L\hfill \\ & & -{\displaystyle \alpha }\int \Omega (v*)dx-\beta \int \Omega v*dx,\hfill \end{array}$$

(629)

and

$$F\left(u\right)={\displaystyle \epsilon }\int \Omega udx,$$

where $\epsilon >$ is small real constant such that $\epsilon \ll .$

Define also the polar functionals $F*:Y*\times Y*\to \mathbb{R}$ and $F*:Y*\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill F*(v*,v*)& =& \mathrm{sta}u\in V\{\langle u,v*\rangle L-F(u,v*)\}\hfill \\ & =& {\displaystyle 12}\int \Omega {\displaystyle (v*+f)-\gamma \nabla +v*+\epsilon }dx\hfill \\ & & +{\displaystyle \alpha }\int \Omega (v*)dx+\beta \int \Omega v*dx,\hfill \end{array}$$

(630)

and

$$\begin{array}{ccc}\hfill F*(v*)& =& \underset{w\in L}{sup}\{\langle w,v*\rangle L-F\left(w\right)\}\hfill \\ & =& {\displaystyle \epsilon }\int \Omega (v*)dx.\hfill \end{array}$$

(631)

Moreover, define

$$B*=\{v*\in Y*:\parallel v*\parallel \le K/\},$$

$$D+=\{v*\in Y*:v*f\ge ,\mathrm{in}\Omega \},$$

and

$$D*=\{v*\in D+:\parallel v*\parallel \infty \le \}.$$

Assuming $K\gg max\{\gamma ,\alpha ,\beta ,/\alpha ,K,\}$ and

$${\displaystyle \epsilon }\gg K,$$

define $J*:D*\times B*\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill J*(v*,v*)& =& -F*(v*,v*)+F*(v*)\hfill \\ & & -K∥-{\displaystyle v*\alpha }-\beta +\left({\displaystyle v*+f-\gamma \nabla +v*+\epsilon }\right)∥,.\hfill \end{array}$$

(632)

Clearly, we have

$${\displaystyle \partial J*(v*,v*)\partial (v*)}=\mathcal{O}(/\epsilon )>\mathbf{0},$$

and

$${\displaystyle \partial J*(v*,v*)\partial (v*)}=-\mathcal{O}(K/\alpha )<\mathbf{0},$$

on $D*\times B*$.

Let $(v^*,v^*)\in D*\times B*$ be such that

$$\delta J*(v^*,v^*)=\mathbf{0}.$$

Since $J$ is convex in $v$ and concave in $v$ on $D*\times B*$, from this and the Min-Max theorem we may infer that

$$J*(v^*,v^*)=\underset{v*\in D*}{inf}\left\{\underset{v*\in B*}{sup}J*(v*,v*)\right\}.$$

Let $u\in V$ be such that

$$u={\displaystyle v^*\epsilon }.$$

Similarly as in previous sections we may obtain

$$\delta J\left(u\right)=\mathbf{0},$$

and

$$J\left(u\right)=J*(v^*,v^*).$$

Joining the pieces, we have got

$$\begin{array}{ccc}\hfill J\left(u\right)& =& J*(v^*,v^*)\hfill \\ & =& \underset{v*\in D*}{inf}\left\{\underset{v*\in B*}{sup}J*(v*,v*)\right\}.\hfill \end{array}$$

(633)

Remark 35.

Defining $J*:D*\in \mathbb{R}$ by

$$J*(v*)=\underset{v*\in B*}{sup}J*(v*,v*),$$

we have that $J$ is convex in $D$ as a supremum of a family of convex functions in $v$.

In such case, we have

$$\begin{array}{ccc}\hfill J\left(u\right)& =& J*(v^*,v^*)\hfill \\ & =& \underset{v*\in D*}{inf}\left\{\underset{v*\in B*}{sup}J*(v*,v*)\right\}\hfill \\ & =& \underset{v*\in D*}{inf}J*(v*)\hfill \\ & =& J*(v^*).\hfill \end{array}$$

(634)

The objective of this section is complete.

77. Conclusion

In the first part of this article we have developed a relaxation proposal and duality principles suitable for a large class of models in physics and engineering.

In a second part we develop duality principles for the quasi-convex envelop of some vectorial models in the calculus of variations.

We highlight such dual variational formulations established are in general convex (in fact concave).

Finally, in the last sections, we develop mathematical models for some types of chemical reactions, including the hydrogen nuclear fusion and the water hydrolysis. Among such results, we highlight our proposal of modeling the Ginzburg-Landau theory in super-conductivity as a two-phase eigenvalue approach.

(1)

Conflict of interest declaration: The author declares no conflict of interest concerning this article.

(2)

Data Availability: Details on the software for numerical results avaialable upon request.

e-mail: fabio.botelho@ufsc.br