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 its critical points. These features are relevant since, from a concerning strict convexity, the standard Newton, Newton type and similar methods are in general convergent. Moreover, the dual variational formulations are relevant since in some situations, from a concerning convexity, it is possible to assure the global optimality of some critical points which satisfy certain specific constraints theoretically established.

Moreover, among the main results, 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 1.1

(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 ${\parallel f\parallel }_{U^{*}}$, is defined as

$$\begin{array}{c}{\parallel f\parallel }_{U^{*}}=\underset{u\in U}{sup}\left\{\right|{\langle u,u^{*}\rangle }_U{|:\parallel u\parallel }_U\le 1\}\equiv \parallel u^{*}{\parallel }_{U^{*}}.\end{array}$$

(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

$$J\left(u\right)=F(\nabla u_1,\cdots ,\nabla u_N)+G(u_1,\cdots ,u_N)-{\langle u_i,h_i\rangle }_{L^2},$$

and where

$$F(\nabla u_1,\cdots ,\nabla u_N)={\int }_{\Omega }f(\nabla u_1,\cdots ,\nabla u_N)dx$$

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

$$V=\{u=(u_1,\cdots ,u_N)\in W^{1,p}(\Omega ;{\mathbb{R}}^N):u=u_0\mathrm{on}\partial \Omega \},$$

$h=(h_1,\cdots ,h_N)\in L^2(\Omega ;{\mathbb{R}}^N),$ and $1<p<+\infty .$

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 $V_0=W_0^{1,2}(\Omega ;{\mathbb{R}}^N)$ and

$$\begin{array}{ccc}& & V_0\left(u\right)\hfill \\ & =& \{\varphi \in V_0:f^{**}(\nabla u\left(x\right)+\nabla \varphi \left(x\right))<f(\nabla u\left(x\right)+\nabla \varphi )\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega \mathrm{and}\mathrm{supp}\varphi \subset B\left(u\right)\}\hfill \end{array}$$

where

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

Moreover, $Y_1=Y_1^{*}=L^2(\Omega ;{\mathbb{R}}^{N\times n})$, $Y_2=Y_2^{*}=L^2(\Omega ;{\mathbb{R}}^{N\times n})$, $Y_3=Y_3^{*}=L^2(\Omega ;{\mathbb{R}}^N),$ so that at this point we define, $F_1:V\times V_0\to \mathbb{R}$, $G_1:V\to \mathbb{R}$, $G_2:V\to \mathbb{R}$, $G_3:V_0\to \mathbb{R}$ and $G_4:V\to \mathbb{R},$ by

$$\begin{array}{ccc}\hfill F_1(u,\varphi )& =& F(\nabla u_1+\nabla {\varphi }_1,\cdots ,\nabla u_N+\nabla {\varphi }_N)+\frac{K}{2}{\int }_{\Omega }\nabla u_j·\nabla u_{j}dx\hfill \\ & & +\frac{K_2}{2}{\int }_{\Omega }\nabla {\varphi }_j·\nabla {\varphi }_{j}dx\hfill \end{array}$$

(3)

and

$$G_1(u_1,\cdots ,u_n)=G(u_1,\cdots ,u_N)+\frac{K_1}{2}{\int }_{\Omega }{u}_j{u}_{j}dx-{\langle u_i,f_i\rangle }_{L^2},$$

$$G_2(\nabla u_1,\cdots ,\nabla u_N)=\frac{K_1}{2}{\int }_{\Omega }\nabla u_j·\nabla u_{j}dx,$$

$$G_3(\nabla {\varphi }_1,\cdots ,\nabla {\varphi }_N)=\frac{K_2}{2}{\int }_{\Omega }\nabla {\varphi }_j·\nabla {\varphi }_{j}dx,$$

and

$$G_4(u_1,\cdots ,u_N)=\frac{K_1}{2}{\int }_{\Omega }{u}_j{u}_{j}dx.$$

Define now $J_1:V\times V_0\to \mathbb{R}$,

$$J_1(u,\varphi )=F(\nabla u+\nabla \varphi )+G\left(u\right)-{\langle u_i,h_i\rangle }_{L^2}.$$

Observe that

$$\begin{array}{ccc}\hfill J_1(u,\varphi )& =& F_1(u,\varphi )+G_1\left(u\right)-G_2\left(\nabla u\right)-G_3\left(\nabla \varphi \right)-G_4\left(u\right)\hfill \\ & \le & F_1(u,\varphi )+G_1\left(u\right)-{\langle \nabla u,z_1^{*}\rangle }_{L^2}-{\langle \nabla \varphi ,z_2^{*}\rangle }_{L^2}-{\langle u,z_3^{*}\rangle }_{L^2}\hfill \\ & & +\underset{v_1\in Y_1}{sup}\{{\langle v_1,z_1^{*}\rangle }_{L^2}-G_2\left(v_1\right)\}\hfill \\ & & +\underset{v_2\in Y_2}{sup}\{{\langle v_2,z_2^{*}\rangle }_{L^2}-G_3\left(v_2\right)\}\hfill \\ & & +\underset{u\in V}{sup}\{{\langle u,z_3^{*}\rangle }_{L^2}-G_4\left(u\right)\}\hfill \\ & =& F_1(u,\varphi )+G_1\left(u\right)-{\langle \nabla u,z_1^{*}\rangle }_{L^2}-{\langle \nabla \varphi ,z_2^{*}\rangle }_{L^2}-{\langle u,z_3^{*}\rangle }_{L^2}\hfill \\ & & +G_2^{*}\left(z_1^{*}\right)+G_3^{*}\left(z_2^{*}\right)+G_4^{*}\left(z_3^{*}\right)\hfill \\ & =& J_1^{*}(u,\varphi ,z^{*}),\hfill \end{array}$$

(4)

$\forall u\in V,\varphi \in V_0\left(u\right),z^{*}=(z_1^{*},z_2^{*},z_3^{*})\in Y^{*}=Y_1^{*}\times Y_2^{*}\times Y_3^{*}$.

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

$$\begin{array}{ccc}\hfill \underset{(u,\varphi )\in V\times V_0\left(u\right)}{inf}J(u,\varphi )& =& \underset{(u,\varphi ,z^{*})\in V\times V_0\left(u\right)\times Y^{*}}{inf}{J}_1^{*}(u,\varphi ,z^{*}).\hfill \end{array}$$

(5)

On the other hand

$$\underset{u\in V}{inf}J\left(u\right)\ge \underset{(u,\varphi )\in V\times V_0\left(u\right)}{inf}{J}_1(u,\varphi ).$$

From these last two results we may obtain

$$\underset{u\in V}{inf}J\left(u\right)\ge \underset{(u,\varphi ,z^{*})\in V\times V_0\left(u\right)\times Y^{*}}{inf}{J}_1^{*}(u,\varphi ,z^{*}).$$

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

$$\begin{array}{ccc}\hfill \underset{u\in V}{inf}{J}_1^{*}(u,\varphi ,z^{*})& =& \underset{u\in V}{inf}\{F_1(u,\varphi )+G_1\left(u\right)\hfill \\ & & -{\langle \nabla u,z_1^{*}\rangle }_{L^2}-{\langle \nabla \varphi ,z_2^{*}\rangle }_{L^2}-{\langle u,z_3^{*}\rangle }_{L^2}\hfill \\ & & +G_2^{*}\left(z_1^{*}\right)+G_3^{*}\left(z_2^{*}\right)+G_4^{*}\left(z_3^{*}\right)\}\hfill \\ & =& \underset{(v_1^{*},v_2^{*})\in C^{*}}{sup}\{-F_1^{*}(v_1^{*}+z_1^{*},\varphi )-G_1^{*}(v_2^{*}+z_3^{*})-{\langle \nabla \varphi ,z_2^{*}\rangle }_{L^2}\hfill \\ & & +G_2^{*}\left(z_1^{*}\right)+G_3^{*}\left(z_2^{*}\right)+G_4^{*}\left(z_3^{*}\right)\},\hfill \end{array}$$

(6)

where

$$C^{*}=\{v^{*}=(v_1^{*},v_2^{*})\in Y_1^{*}\times Y_3^{*}:-\mathrm{div}{\left(v_1^{*}\right)}_i+{\left(v_2^{*}\right)}_i=\mathbf{0},\forall i\in \{1,\cdots ,N\}\},$$

$$F_1^{*}(v_1^{*}+z_1^{*},\varphi )=\underset{u\in V}{sup}\{{\langle u,-\mathrm{div}(z_1^{*}+v_1^{*})\rangle }_{L^2}-F_1(u,\varphi )\},$$

and

$$G_1^{*}(v_2^{*}+z_2^{*})=\underset{u\in V}{sup}\{{\langle u,v_2^{*}+z_2^{*}\rangle }_{L^2}-G_1\left(u\right)\}.$$

Thus, defining

$$J_2^{*}(\varphi ,z^{*},v^{*})=F_1^{*}(v_1^{*}+z_1^{*},\varphi )-G_1^{*}(v_2^{*}+z_3^{*})-{\langle \nabla \varphi ,z_2^{*}\rangle }_{L^2}+G_2^{*}\left(z_1^{*}\right)+G_3^{*}\left(z_2^{*}\right)+G_4^{*}\left(z_3^{*}\right),$$

we have got

$$\begin{array}{ccc}\hfill \underset{u\in V}{inf}J\left(u\right)& \ge & \underset{(u,\varphi )\in V\times V_0}{inf}{J}_1(u,\varphi )\hfill \\ & =& \underset{(u,\varphi ,z^{*})\in V\times V_0\left(u\right)\times Y^{*}}{inf}{J}_1^{*}(u,\varphi ,z^{*})\hfill \\ & =& \underset{z^{*}\in Y^{*}}{inf}\left\{\underset{\varphi \in V_0}{inf}\left\{\underset{v^{*}\in C^{*}}{sup}{J}_2^{*}(\varphi ,z^{*},v^{*})\right\}\right\}.\hfill \end{array}$$

(7)

Finally, observe that

$$\begin{array}{ccc}& & \underset{u\in V}{inf}J\left(u\right)\hfill \\ & \ge & \underset{z^{*}\in Y^{*}}{inf}\left\{\underset{\varphi \in V_0\left(u\right)}{inf}\left\{\underset{v^{*}\in C^{*}}{sup}{J}_2^{*}(\varphi ,z^{*},v^{*})\right\}\right\}\hfill \\ & \ge & \underset{v^{*}\in C^{*}}{sup}\left\{\underset{(z^{*},\varphi )\in Y^{*}\times V_0\left(u\right)}{inf}{J}_2^{*}(\varphi ,z^{*},v^{*})\right\}.\hfill \end{array}$$

(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 $\Omega =[0,1]\subset \mathbb{R}$ and consider a functional $J:V\to \mathbb{R}$ where

$$J\left(u\right)=\frac{1}{2}{\int }_{\Omega }{({\left(u^{\prime }\right)}^2-1)}^{2}dx+\frac{1}{2}{\int }_{\Omega }{u}^{2}dx-{\langle u,f\rangle }_{L^2},$$

and where

$$V=\{u\in W^{1,4}(\Omega ):u\left(0\right)=0\mathrm{and}u\left(1\right)=1/2\}$$

and $f\in L^2(\Omega ).$

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 $V_0=W_0^{1,4}(\Omega )$, at this point we define, $F:V\to \mathbb{R}$ and $F_1:V\times V_0\to \mathbb{R}$ by

$$F\left(u\right)=\frac{1}{2}{\int }_{\Omega }{({\left(u^{\prime }\right)}^2-1)}^{2}dx,$$

and

$$F_1(u,\varphi )=\frac{1}{2}{\int }_{\Omega }{({(u^{\prime }+{\varphi }^{\prime })}^2-1)}^{2}dx.$$

Observe that

$$F\left(u\right)\ge \underset{\varphi \in V_0}{inf}{F}_1(u,\varphi ),\forall u\in V.$$

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

$$V_0=\left\{\varphi \in W_0^{1,4}(\Omega ):{\left({\varphi }^{\prime }\right)}^2-1\le 0,\mathrm{in}\Omega \right\}$$

and define also

$$F_2:V\times V_0\to \mathbb{R},$$

$$F_3:V\times V_0\to \mathbb{R}$$

and

$$G:V\times V_0\to \mathbb{R}$$

by

$$F_2(u,\varphi )=\frac{1}{2}{\int }_{\Omega }{({(u^{\prime }+{\varphi }^{\prime })}^2-1)}^{2}dx+\frac{1}{2}{\int }_{\Omega }{u}^{2}dx-{\langle u,f\rangle }_{L^2},$$

$$\begin{array}{ccc}\hfill F_3(u,\varphi )& =& F_2(u,\varphi )+\frac{K}{2}{\int }_{\Omega }{\left(u^{\prime }\right)}^{2}dx\hfill \\ & & +\frac{K_1}{2}{\int }_{\Omega }{\left({\varphi }^{\prime }\right)}^{2}dx\hfill \end{array}$$

(9)

and

$$\begin{array}{ccc}\hfill G(u,\varphi )& =& \frac{K}{2}{\int }_{\Omega }{\left(u^{\prime }\right)}^{2}dx\hfill \\ & & +\frac{K_1}{2}{\int }_{\Omega }{\left({\varphi }^{\prime }\right)}^{2}dx\hfill \end{array}$$

(10)

Denoting $Y=Y^{*}=L^2(\Omega )$ we also define the polar functional $G^{*}:Y^{*}\times Y^{*}\to \mathbb{R}$ by

$$G^{*}(v^{*},v_0^{*})=\underset{(u,\varphi )\in V\times V_0}{sup}\{{\langle u,v^{*}\rangle }_{L^2}+{\langle \varphi ,v_0^{*}\rangle }_{L^2}-G(u,\varphi )\}.$$

Observe that

$$\underset{u\in U}{inf}J\left(u\right)\ge \underset{((u,\varphi ),(v^{*},v_0^{*}))\in V\times V_0\times {\left[Y^{*}\right]}^2}{inf}\{G^{*}(v^{*},v_0^{*})-{\langle u,v^{*}\rangle }_{L^2}-{\langle \varphi ,v_0^{*}\rangle }_{L^2}+F_3(u,\varphi )\}.$$

With such results in mind, we define a relaxed primal dual variational formulation for the primal problem, represented by $J_1^{*}:V\times V_0\times {\left[Y^{*}\right]}^2\to \mathbb{R}$, where

$$J_1^{*}(u,\varphi ,v^{*},v_0^{*})=G^{*}(v^{*},v_0^{*})-{\langle u,v^{*}\rangle }_{L^2}-{\langle \varphi ,v_0^{*}\rangle }_{L^2}+F_3(u,\varphi ).$$

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 ${\left({\varphi }^{\prime }\right)}^2-1\le 0\mathrm{in}\Omega$).

• Set $K\approx 0.1$ and $K_1=120.0$ and $0<\epsilon \ll 1.$
• Choose $(u_1,{\varphi }_1)\in V\times V_0,$ such that $\parallel u_1{\parallel }_{1,\infty }<1$ and $\parallel {\varphi }_1{\parallel }_{1,\infty }<1.$
• Set $n=1.$
• Calculate $(v_n^{*},{\left(v_0^{*}\right)}_n)$ solution of the system of equations:
  $$\frac{\partial J_1^{*}(u_n,{\varphi }_n,v_n^{*},{\left(v_0^{*}\right)}_n)}{\partial v^{*}}=\mathbf{0}$$
  and
  $$\frac{\partial J_1^{*}(u_n,{\varphi }_n,v_n^{*},{\left(v_0^{*}\right)}_n)}{\partial v_0^{*}}=\mathbf{0},$$
  that is
  $$\frac{\partial G^{*}(v_n^{*},{\left(v_0^{*}\right)}_n)}{\partial v^{*}}-u_n=0$$
  and
  $$\frac{\partial G^{*}(v_n^{*},{\left(v_0^{*}\right)}_n)}{\partial v_0^{*}}-{\varphi }_n=0$$
  so that
  $$v_n^{*}=\frac{\partial G(u_n,{\varphi }_n)}{\partial u}$$
  and
  $${\left(v_0^{*}\right)}_n^{*}=\frac{\partial G(u_n,{\varphi }_n)}{\partial \varphi }$$
• Calculate $(u_{n+1},{\varphi }_{n+1})$ by solving the system of equations:
  $$\frac{\partial J_1^{*}(u_{n+1},{\varphi }_{n+1},v_n^{*},{\left(v_0^{*}\right)}_n)}{\partial u}=\mathbf{0}$$
  and
  $$\frac{\partial J_1^{*}(u_{n+1},{\varphi }_{n+1},v_n^{*},{\left(v_0^{*}\right)}_n)}{\partial \varphi }=\mathbf{0}$$
  that is
  $$-v_n^{*}+\frac{\partial F_3(u_{n+1},{\varphi }_{n+1})}{\partial u}=\mathbf{0}$$
  and
  $$-{\left(v_0^{*}\right)}_n+\frac{\partial F_3(u_{n+1},{\varphi }_{n+1})}{\partial \varphi }=\mathbf{0}$$
• If $max\{\parallel u_n-u_{n+1}{\parallel }_{\infty },\parallel {\varphi }_{n+1}-{\varphi }_n{\parallel }_{\infty }\}\le \epsilon$, then stop, else set $n:=n+1$ and go to item d.

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
  $uo(i,1)=i^2*d/2;$
  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;
  $f1=6*{(duo(i,1)+dvo(i,1))}^2-2;$
  m80(1,1,i)=-f1-K;
  m80(1,2,i)=-f1;
  m80(2,1,i)=-f1;
  m80(2,2,i)=-f1-K1;
  $y11(1,i)=K*(uo(i+1,1)-2*uo(i,1))/d^2-yo(i,1);$
  $y11(2,i)=K1*(vo(i+1,1)-2*vo(i,1))/d^2;$
  $m12=2*m80(:,:,i)-m9*d^2;$
  m50(:,:,i)=m80(:,:,i)*inv(m12);
  z(:,i)=inv(m12)*y11(:,i)*$d^2$;
  for i=2:m8-1
  $f1=6*{(duo(i,1)+dvo(i,1))}^2-2$;
  m80(1,1,i)=-f1-K;
  m80(1,2,i)=-f1;
  m80(2,1,i)=-f1;
  m80(2,2,i)=-f1-K1;
  $y11(1,i)=K*(uo(i+1,1)-2*uo(i,1)+uo(i-1,1))/d^2-yo(i,1);$
  $y11(2,i)=K1*(vo(i+1,1)-2*vo(i,1)+vo(i-1,1))/d^2;$
  $m12=2*m80(:,:,i)-m9*d^2-m80(:,:,i)*m50(:,:,i-1);$
  m50(:,:,i)=inv(m12)*m80(:,:,i);
  $z(:,i)=inv\left(m12\right)*(y11(:,i)*d^2+m80(:,:,i)*z(:,i-1));$
  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 $f\left(x\right)=0$, we have obtained numerical results for $K=0.1$ and $K_1=120$. For such a concerning solution $u_0$ obtained, please see Figure 1. For the case in which $f\left(x\right)=sin\left(\pi x\right)/2$, we have obtained numerical results also for $K=0.1$ and $K_1=120$. For such a concerning solution $u_0$ obtained, please see Figure 2.

Remark 3.1.

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

$$J\left(u\right)=F\left(\nabla u\right)+G\left(u\right)-{\langle u,f_1\rangle }_{L^2},$$

where

$$V=\left\{u\in W^{1,4}(\Omega ;{\mathbb{R}}^N):u=u_0\mathrm{on}\partial \Omega \right\},$$

$$u_0\in C^1(\Omega ;{\mathbb{R}}^N),$$

$f_1\in L^2(\Omega ;{\mathbb{R}}^N)$, $G:V\to \mathbb{R}$ is convex and Fréchet differentiable, and

$$F\left(\nabla u\right)={\int }_{\Omega }f\left(\nabla u\right)dx,$$

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

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

$$W_h\equiv \left\{y\in {\mathbb{R}}^{N\times n}:f^{**}\left(y\right)<f\left(y\right)\right\}={\cup }_{j=1}^{\widehat{N}}{W}_j$$

where for each $j\in \{1,\cdots ,\widehat{N}\}$$W_j\subset {\mathbb{R}}^{N\times n}$ is an open connected set such that $\partial W_j$ is regular. We also suppose

$$\overline{W_j}\cap \overline{W_k}=\varnothing ,\forall j\ne k.$$

Define

$${\widehat{W}}_j=\left\{v_j\in W_0^{1,4}(\Omega ;{\mathbb{R}}^N);\nabla v_j\left(x\right)\in W_j,\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega \right\}$$

and define also

$$W=\left\{v=(v_1,\cdots ,v_{\widehat{N}}):v_j\in {\widehat{W}}_j\forall j\in \{1,\cdots ,\widehat{N}\}\mathrm{and}\mathrm{supp}{v}_j\cap \mathrm{supp}{v}_k=\varnothing ,\forall j\ne k\right\}.$$

At this point we define

$$h_5(u\left(x\right),v\left(x\right))=\left\{\begin{array}{cc}f(\nabla u\left(x\right)+\nabla v_j\left(x\right)),\hfill & \mathrm{if}\nabla u\left(x\right)\in W_j,\hfill \\ f\left(\nabla u\right(x\left)\right),\hfill & \mathrm{if}\nabla u\left(x\right)\notin W_h,\hfill \end{array}\right.$$

(11)

and

$$H\left(u\right)=\underset{v\in W_u}{inf}{\int }_{\Omega }{h}_5(u,v)dx,$$

where

$$W_u=\{v\in W:\nabla u\left(x\right)+\nabla v_j\left(x\right)\in W_j,\mathrm{if}\nabla u\left(x\right)\in W_j,\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\forall j\in \{1,\cdots ,\widehat{N}\}\}.$$

Moreover, we propose the relaxed functional

$$J_1\left(u\right)=H\left(u\right)+G\left(u\right)-{\langle u,f_1\rangle }_{L^2}.$$

Observe that clearly

$$\underset{u\in V}{inf}{J}_1\left(u\right)\le \underset{u\in V}{inf}J\left(u\right).$$