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

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

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Fabio Botelho^*

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Submitted:

29 October 2023

Posted:

02 November 2023

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Abstract

This article develops duality principles and numerical results for a large class of non-convex variational models. The main results are based on fundamental tools of convex analysis, duality theory and calculus of variations. More specifically the approach is established for a class of non-convex functionals similar as those found in some models in phase transition. Finally, in some sections we present concerning numerical examples and the respective softwares.

Keywords:

Subject: Computer Science and Mathematics - Applied Mathematics

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.

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

{〈 \cdot, \cdot 〉}_{U} : U \times U^{*} \to R

(here we are referring to standard representations of dual spaces of Sobolev and Lebesgue spaces). Thus, given

f : U \to R

linear and continuous, we assume the existence of a unique

u^{*} \in U^{*}

such that

\begin{matrix} f (u) = {〈 u, u^{*} 〉}_{U}, \forall u \in U . \end{matrix}

(1)

The norm of f , denoted by

{∥ f ∥}_{U^{*}}

, is defined as

\begin{matrix} {∥ f ∥}_{U^{*}} = sup_{u \in U} {| {〈 u, u^{*} 〉}_{U} {| : ∥ u ∥}_{U} \leq 1} \equiv ∥ u^{*} ∥_{U^{*}} . \end{matrix}

(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

Ω \subset R^{n}

be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Consider a functional

J : V \to R

where

J (u) = F (\nabla u_{1}, \dots, \nabla u_{N}) + G (u_{1}, \dots, u_{N}) - {〈 u_{i}, h_{i} 〉}_{L^{2}},

and where

F (\nabla u_{1}, \dots, \nabla u_{N}) = \int_{Ω} f (\nabla u_{1}, \dots, \nabla u_{N}) d x

f : R^{N \times n} \to R

is a three times Fréchet differentiable function not necessarily convex. Moreover,

V = {u = (u_{1}, \dots, u_{N}) \in W^{1, p} (Ω; R^{N}) : u = u_{0} on \partial Ω},

h = (h_{1}, \dots, h_{N}) \in L^{2} (Ω; R^{N}),

and

1 < p < + \infty .

We assume there exists

α \in R

such that

α = inf_{u \in V} J (u) .

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.

Denoting

V_{0} = {ϕ \in W_{0}^{1, p} (Ω; R^{N}) : \nabla ϕ (x) = (\nabla ϕ_{1} (x), \dots, \nabla ϕ_{N} (x)) \in A, a . e . in Ω},

where

A = {y \in R^{N \times n} : f^{* *} (y) < f (y)},

Y_{1} = Y_{1}^{*} = L^{2} (Ω; R^{N \times n})

Y_{2} = Y_{2}^{*} = L^{2} (Ω; R^{N \times n})

Y_{3} = Y_{3}^{*} = L^{2} (Ω; R^{N}),

at this point we define,

F_{1} : V \times V_{0} \to R

G_{1} : V \to R

G_{2} : V \to R

G_{3} : V_{0} \to R

and

G_{4} : V \to R,

\begin{matrix} F_{1} (\nabla u, \nabla ϕ) & = & F (\nabla u_{1} + \nabla ϕ_{1}, \dots, \nabla u_{N} + \nabla ϕ_{N}) + \frac{K}{2} \int_{Ω} \nabla u_{j} \cdot \nabla u_{j} d x \\ + \frac{K_{2}}{2} \int_{Ω} \nabla ϕ_{j} \cdot \nabla ϕ_{j} d x \end{matrix}

(3)

and

G_{1} (u_{1}, \dots, u_{n}) = G (u_{1}, \dots, u_{N}) + \frac{K_{1}}{2} \int_{Ω} u_{j} u_{j} d x - {〈 u_{i}, f_{i} 〉}_{L^{2}},

G_{2} (\nabla u_{1}, \dots, \nabla u_{N}) = \frac{K_{1}}{2} \int_{Ω} \nabla u_{j} \cdot \nabla u_{j} d x,

G_{3} (\nabla ϕ_{1}, \dots, \nabla ϕ_{N}) = \frac{K_{2}}{2} \int_{Ω} \nabla ϕ_{j} \cdot \nabla ϕ_{j} d x,

and

G_{4} (u_{1}, \dots, u_{N}) = \frac{K_{1}}{2} \int_{Ω} u_{j} u_{j} d x .

Define now

J_{1} : V \times V_{0} \to R

J_{1} (u, ϕ) = F (\nabla u + \nabla ϕ) + G (u) - {〈 u_{i}, h_{i} 〉}_{L^{2}} .

Observe that

\begin{matrix} J_{1} (u, ϕ) & = & F_{1} (\nabla u, \nabla ϕ) + G_{1} (u) - G_{2} (\nabla u) - G_{3} (\nabla ϕ) - G_{4} (u) \\ \leq & F_{1} (\nabla u, \nabla ϕ) + G_{1} (u) - {〈 \nabla u, z_{1}^{*} 〉}_{L^{2}} - {〈 \nabla ϕ, z_{2}^{*} 〉}_{L^{2}} - {〈 u, z_{3}^{*} 〉}_{L^{2}} \\ + sup_{v_{1} \in Y_{1}} {{〈 v_{1}, z_{1}^{*} 〉}_{L^{2}} - G_{2} (v_{1})} \\ + sup_{v_{2} \in Y_{2}} {{〈 v_{2}, z_{2}^{*} 〉}_{L^{2}} - G_{3} (v_{2})} \\ + sup_{u \in V} {{〈 u, z_{3}^{*} 〉}_{L^{2}} - G_{4} (u)} \\ = & F_{1} (\nabla u, \nabla ϕ) + G_{1} (u) - {〈 \nabla u, z_{1}^{*} 〉}_{L^{2}} - {〈 \nabla ϕ, z_{2}^{*} 〉}_{L^{2}} - {〈 u, z_{3}^{*} 〉}_{L^{2}} \\ + G_{2}^{*} (z_{1}^{*}) + G_{3}^{*} (z_{2}^{*}) + G_{4}^{*} (z_{3}^{*}) \\ = & J_{1}^{*} (u, ϕ, z^{*}), \end{matrix}

(4)

\forall u \in V, ϕ \in V_{0}, 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{matrix} inf_{(u, ϕ) \in V \times V_{0}} J (u, ϕ) & = & inf_{(u, ϕ, z^{*}) \in V \times V_{0} \times Y^{*}} J_{1}^{*} (u, ϕ, z^{*}) . \end{matrix}

(5)

On the other hand

inf_{u \in V} J (u) \geq inf_{(u, ϕ) \in V \times V_{0}} J_{1} (u, ϕ) .

From these last two results we may obtain

inf_{u \in V} J (u) \geq inf_{(u, ϕ, z^{*}) \in V \times V_{0} \times Y^{*}} J_{1}^{*} (u, ϕ, z^{*}) .

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

\begin{matrix} inf_{u \in V} J_{1}^{*} (u, ϕ, z^{*}) & = & inf_{u \in V} {F_{1} (\nabla u, \nabla ϕ) + G_{1} (u) \\ - {〈 \nabla u, z_{1}^{*} 〉}_{L^{2}} - {〈 \nabla ϕ, z_{2}^{*} 〉}_{L^{2}} - {〈 u, z_{3}^{*} 〉}_{L^{2}} \\ + G_{2}^{*} (z_{1}^{*}) + G_{3}^{*} (z_{2}^{*}) + G_{4}^{*} (z_{3}^{*})} \\ = & sup_{(v_{1}^{*}, v_{2}^{*}) \in C^{*}} {- F_{1}^{*} (v_{1}^{*} + z_{1}^{*}, \nabla ϕ) - G_{1}^{*} (v_{2}^{*} + z_{3}^{*}) - {〈 \nabla ϕ, z_{2}^{*} 〉}_{L^{2}} \\ + G_{2}^{*} (z_{1}^{*}) + G_{3}^{*} (z_{2}^{*}) + G_{4}^{*} (z_{3}^{*})}, \end{matrix}

(6)

where

C^{*} = {v^{*} = (v_{1}^{*}, v_{2}^{*}) \in Y_{1}^{*} \times Y_{3}^{*} : - div {(v_{1}^{*})}_{i} + {(v_{2}^{*})}_{i} = 0, \forall i \in {1, \dots, N}},

F_{1}^{*} (v_{1}^{*} + z_{1}^{*}, \nabla ϕ) = sup_{v_{1} \in Y_{1}} {{〈 v_{1}, z_{1}^{*} + v_{1}^{*} 〉}_{L^{2}} - F_{1} (v_{1}, \nabla ϕ)},

and

G_{1}^{*} (v_{2}^{*} + z_{2}^{*}) = sup_{u \in V} {{〈 u, v_{2}^{*} + z_{2}^{*} 〉}_{L^{2}} - G_{1} (u)} .

Thus, defining

J_{2}^{*} (ϕ, z^{*}, v^{*}) = F_{1}^{*} (v_{1}^{*} + z_{1}^{*}, \nabla ϕ) - G_{1}^{*} (v_{2}^{*} + z_{3}^{*}) - {〈 \nabla ϕ, z_{2}^{*} 〉}_{L^{2}} + G_{2}^{*} (z_{1}^{*}) + G_{3}^{*} (z_{2}^{*}) + G_{4}^{*} (z_{3}^{*}),

we have got

\begin{matrix} inf_{u \in V} J (u) & \geq & inf_{(u, ϕ) \in V \times V_{0}} J_{1} (u, ϕ) \\ = & inf_{(u, ϕ, z^{*}) \in V \times V_{0} \times Y^{*}} J_{1}^{*} (u, ϕ, z^{*}) \\ = & inf_{z^{*} \in Y^{*}} \{inf_{ϕ \in V_{0}} \{sup_{v^{*} \in C^{*}} J_{2}^{*} (ϕ, z^{*}, v^{*})\}\} . \end{matrix}

(7)

Finally, observe that

\begin{matrix} inf_{u \in V} J (u) \\ \geq & inf_{z^{*} \in Y^{*}} \{inf_{ϕ \in V_{0}} \{sup_{v^{*} \in C^{*}} J_{2}^{*} (ϕ, z^{*}, v^{*})\}\} \\ \geq & sup_{v^{*} \in C^{*}} \{inf_{(z^{*}, ϕ) \in Y^{*} \times V_{0}} J_{2}^{*} (ϕ, z^{*}, v^{*})\} . \end{matrix}

(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

Ω = [0, 1] \subset R

and consider a functional

J : V \to R

where

J (u) = \frac{1}{2} \int_{Ω} {({(u^{'})}^{2} - 1)}^{2} d x + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}},

and where

V = {u \in W^{1, 4} (Ω) : u (0) = 0 and u (1) = 1 / 2}

and

f \in L^{2} (Ω) .

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} (Ω)

, at this point we define,

F : V \to R

and

F_{1} : V \times V_{0} \to R

F (u) = \frac{1}{2} \int_{Ω} {({(u^{'})}^{2} - 1)}^{2} d x,

and

F_{1} (u, ϕ) = \frac{1}{2} \int_{Ω} {({(u^{'} + ϕ^{'})}^{2} - 1)}^{2} d x .

Observe that

F (u) \geq inf_{ϕ \in V_{0}} F_{1} (u, ϕ), \forall u \in V .

In order to restrict the action of

ϕ

on the region where the primal functional is non-convex, we redefine a not relabeled

V_{0} = \{ϕ \in W_{0}^{1, 4} (Ω) : {(ϕ^{'})}^{2} - 1 \leq 0, in Ω\}

and define also

F_{2} : V \times V_{0} \to R,

F_{3} : V \times V_{0} \to R

and

G : V \times V_{0} \to R

F_{2} (u, ϕ) = \frac{1}{2} \int_{Ω} {({(u^{'} + ϕ^{'})}^{2} - 1)}^{2} d x + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}},

\begin{matrix} F_{3} (u, ϕ) & = & F_{2} (u, ϕ) + \frac{K}{2} \int_{Ω} {(u^{'})}^{2} d x \\ + \frac{K_{1}}{2} \int_{Ω} {(ϕ^{'})}^{2} d x \end{matrix}

(9)

and

\begin{matrix} G (u, ϕ) & = & \frac{K}{2} \int_{Ω} {(u^{'})}^{2} d x \\ + \frac{K_{1}}{2} \int_{Ω} {(ϕ^{'})}^{2} d x \end{matrix}

(10)

Denoting

Y = Y^{*} = L^{2} (Ω)

we also define the polar functional

G^{*} : Y^{*} \times Y^{*} \to R

G^{*} (v^{*}, v_{0}^{*}) = sup_{(u, ϕ) \in V \times V_{0}} {{〈 u, v^{*} 〉}_{L^{2}} + {〈 ϕ, v_{0}^{*} 〉}_{L^{2}} - G (u, ϕ)} .

Observe that

inf_{u \in U} J (u) \geq inf_{((u, ϕ), (v^{*}, v_{0}^{*})) \in V \times V_{0} \times {[Y^{*}]}^{2}} {G^{*} (v^{*}, v_{0}^{*}) - {〈 u, v^{*} 〉}_{L^{2}} - {〈 ϕ, v_{0}^{*} 〉}_{L^{2}} + F_{3} (u, ϕ)} .

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 {[Y^{*}]}^{2} \to R

, where

J_{1}^{*} (u, ϕ, v^{*}, v_{0}^{*}) = G^{*} (v^{*}, v_{0}^{*}) - {〈 u, v^{*} 〉}_{L^{2}} - {〈 ϕ, v_{0}^{*} 〉}_{L^{2}} + F_{3} (u, ϕ) .

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

{(ϕ^{'})}^{2} - 1 \leq 0 in Ω

Set $K \approx 0.1$ and $K_{1} = 120.0$ and $0 < ε ≪ 1 .$
Choose $(u_{1}, ϕ_{1}) \in V \times V_{0},$ such that $∥ u_{1} ∥_{1, \infty} < 1$ and $∥ ϕ_{1} ∥_{1, \infty} < 1 .$
Set $n = 1 .$
Calculate $(v_{n}^{*}, {(v_{0}^{*})}_{n})$ solution of the system of equations:

$\frac{\partial J_{1}^{*} (u_{n}, ϕ_{n}, v_{n}^{*}, {(v_{0}^{*})}_{n})}{\partial v^{*}} = 0$

and

$\frac{\partial J_{1}^{*} (u_{n}, ϕ_{n}, v_{n}^{*}, {(v_{0}^{*})}_{n})}{\partial v_{0}^{*}} = 0,$

that is

$\frac{\partial G^{*} (v_{n}^{*}, {(v_{0}^{*})}_{n})}{\partial v^{*}} - u_{n} = 0$

and

$\frac{\partial G^{*} (v_{n}^{*}, {(v_{0}^{*})}_{n})}{\partial v_{0}^{*}} - ϕ_{n} = 0$

so that

$v_{n}^{*} = \frac{\partial G (u_{n}, ϕ_{n})}{\partial u}$

and

${(v_{0}^{*})}_{n}^{*} = \frac{\partial G (u_{n}, ϕ_{n})}{\partial ϕ}$
Calculate $(u_{n + 1}, ϕ_{n + 1})$ by solving the system of equations:

$\frac{\partial J_{1}^{*} (u_{n + 1}, ϕ_{n + 1}, v_{n}^{*}, {(v_{0}^{*})}_{n})}{\partial u} = 0$

and

$\frac{\partial J_{1}^{*} (u_{n + 1}, ϕ_{n + 1}, v_{n}^{*}, {(v_{0}^{*})}_{n})}{\partial ϕ} = 0$

that is

$- v_{n}^{*} + \frac{\partial F_{3} (u_{n + 1}, ϕ_{n + 1})}{\partial u} = 0$

and

$- {(v_{0}^{*})}_{n} + \frac{\partial F_{3} (u_{n + 1}, ϕ_{n + 1})}{\partial ϕ} = 0$
If $max {∥ u_{n} - u_{n + 1} ∥_{\infty}, ∥ ϕ_{n + 1} - ϕ_{n} ∥_{\infty}} \leq ε$ , then stop, else set $n : = n + 1$ 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

$u o (i, 1) = i^{2} * d / 2;$

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

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

end;

k=1;

b12=1.0;

while $(b 12 > 10^{- 4.3})$ 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;

$f 1 = 6 * {(d u o (i, 1) + d v o (i, 1))}^{2} - 2;$

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

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

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

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

$y 11 (1, i) = K * (u o (i + 1, 1) - 2 * u o (i, 1)) / d^{2} - y o (i, 1);$

$y 11 (2, i) = K 1 * (v o (i + 1, 1) - 2 * v o (i, 1)) / d^{2};$

$m 12 = 2 * m 80 (:, :, i) - m 9 * d^{2};$

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

z(:,i)=inv(m12)*y11(:,i)* $d^{2}$ ;

for i=2:m8-1

$f 1 = 6 * {(d u o (i, 1) + d v o (i, 1))}^{2} - 2$ ;

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

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

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

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

$y 11 (1, i) = K * (u o (i + 1, 1) - 2 * u o (i, 1) + u o (i - 1, 1)) / d^{2} - y o (i, 1);$

$y 11 (2, i) = K 1 * (v o (i + 1, 1) - 2 * v o (i, 1) + v o (i - 1, 1)) / d^{2};$

$m 12 = 2 * m 80 (:, :, i) - m 9 * d^{2} - m 80 (:, :, i) * m 50 (:, :, i - 1);$

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

$z (:, i) = i n v (m 12) * (y 11 (:, i) * d^{2} + m 80 (:, :, i) * z (:, i - 1));$

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 $f (x) = 0$ , 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 $f (x) = sin (π x) / 2$ , 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.

Figure 1. solution

u_{0} (x)

for the case

f (x) = 0

Figure 1. solution

u_{0} (x)

for the case

f (x) = 0

Figure 2. solution

u_{0} (x)

for the case

f (x) = sin (π x) / 2

Figure 2. solution

u_{0} (x)

for the case

f (x) = sin (π x) / 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

Ω \subset R^{n}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Consider a functional

J : V \to R

where

J (u) = F (\nabla u) + G (u) - {〈 u, f_{1} 〉}_{L^{2}},

where

V = \{u \in W^{1, 4} (Ω; R^{N}) : u = u_{0} on \partial Ω\},

u_{0} \in C^{1} (Ω; R^{N}),

f_{1} \in L^{2} (Ω; R^{N})

G : V \to R

is convex and Fréchet differentiable, and

F (\nabla u) = \int_{Ω} f (\nabla u) d x,

where

f : R^{N \times n} \to R

is also Fréchet differentiable.

Assume there exists

\hat{N} \in N

such that

W_{h} \equiv \{y \in R^{N \times n} : f^{* *} (y) < f (y)\} = \cup_{j = 1}^{\hat{N}} W_{j}

where for each

j \in {1, \dots, \hat{N}}

W_{j} \subset R^{N \times n}

is an open connected set such that

\partial W_{j}

is regular. We also suppose

\bar{W_{j}} \cap \bar{W_{k}} = \emptyset, \forall j \neq k .

Define

{\hat{W}}_{j} = \{v_{j} \in W_{0}^{1, 4} (Ω; R^{N}); \nabla v_{j} (x) \in W_{j}, a . e . in Ω\}

and define also

W = \{v = (v_{1}, \dots, v_{\hat{N}}) : v_{j} \in {\hat{W}}_{j} \forall j \in {1, \dots, \hat{N}} and supp v_{j} \cap supp v_{k} = \emptyset, \forall j \neq k\} .

At this point we define

h_{5} (u (x), v (x)) = \{\begin{matrix} f (\nabla u (x) + \nabla v_{j} (x)), & if \nabla u (x) \in W_{j}, \\ f (\nabla u (x)), & if \nabla u (x) \notin W_{h}, \end{matrix}

(11)

and

H (u) = inf_{v \in W_{u}} \int_{Ω} h_{5} (u, v) d x,

where

W_{u} = {v \in W : \nabla u (x) + \nabla v_{j} (x) \in W_{j}, if \nabla u (x) \in W_{j}, a . e . in Ω, \forall j \in {1, \dots, \hat{N}}} .

Moreover, we propose the relaxed functional

J_{1} (u) = H (u) + G (u) - {〈 u, f_{1} 〉}_{L^{2}} .

Observe that clearly

inf_{u \in V} J_{1} (u) \leq inf_{u \in V} J (u) .

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

Ω = [0, 1] \subset R

and consider a functional

J : V \to R

where

J (u) = \frac{1}{2} \int_{Ω} min {{(u^{'} - 1)}^{2}, {(u^{'} + 1)}^{2}} d x + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}},

and where

V = {u \in W^{1, 2} (Ω) : u (0) = 0 and u (1) = 1 / 2}

and

f \in L^{2} (Ω) .

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 R

and

G : V \to R

\begin{matrix} F (u) & = & \frac{1}{2} \int_{Ω} min {{(u^{'} - 1)}^{2}, {(u^{'} + 1)}^{2}} d x \\ = & \frac{1}{2} \int_{Ω} {(u^{'})}^{2} d x - \int_{Ω} | u^{'} | d x + 1 / 2 \\ \equiv & F_{1} (u^{'}), \end{matrix}

(12)

and

G (u) = \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}} .

Denoting

Y = Y^{*} = L^{2} (Ω)

we also define the polar functional

F_{1}^{*} : Y^{*} \to R

and

G^{*} : Y^{*} \to R

\begin{matrix} F_{1}^{*} (v^{*}) & = & sup_{v \in Y} {{〈 v, v^{*} 〉}_{L^{2}} - F_{1} (v)} \\ = & \frac{1}{2} \int_{Ω} {(v^{*})}^{2} d x + \int_{Ω} | v^{*} | d x, \end{matrix}

(13)

and

\begin{matrix} G^{*} ({(v^{*})}^{'}) & = & sup_{u \in V} {- {〈 u^{'}, v^{*} 〉}_{L^{2}} - G (u)} \\ = & \frac{1}{2} \int_{Ω} {({(v^{*})}^{'} + f)}^{2} d x - \frac{1}{2} v^{*} (1) . \end{matrix}

(14)

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

inf_{u \in V} J (u) = max_{v^{*} \in Y^{*}} {- F_{1}^{*} (v^{*}) - G^{*} (- {(v^{*})}^{'})} .

Indeed, from the direct method of the calculus of variations, the maximum for the dual formulation is attained at some

{\hat{v}}^{*} \in Y^{*}

Moreover, the corresponding solution

u_{0} \in V

is obtained from the equation

u_{0} = \frac{\partial G ({({\hat{v}}^{*})}^{'})}{\partial {(v^{*})}^{'}} = {({\hat{v}}^{*})}^{'} + f .

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

\{\begin{matrix} {(v^{*})}^{''} + f^{'} - v^{*} - sign (v^{*}) = 0, & in Ω, \\ {(v^{*})}^{'} (0) + f (0) = 0, {(v^{*})}^{'} (1) + f (1) = 1 / 2, \end{matrix}

(15)

where

sign (v^{*} (x)) = 1

v^{*} (x) > 0,

sign (v^{*} (x)) = - 1,

v^{*} (x) < 0

and

- 1 \leq sign (v^{*} (x)) \leq 1,

v^{*} (x) = 0 .

We have computed the solutions

v^{*}

and corresponding solutions

u_{0} \in V

for the cases in which

f (x) = 0

and

f (x) = sin (π x) / 2 .

For the solution

u_{0} (x)

for the case in which

f (x) = 0

, please see Figure 3.

For the solution

u_{0} (x)

for the case in which

f (x) = sin (π x) / 2

, please see Figure 4.

Remark 4.1.

Observe that such solutions

u_{0}

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

\{\begin{matrix} {(v^{*})}^{''} + f^{'} - v^{*} - sign (v^{*}) = 0, & in Ω, \\ {(v^{*})}^{'} (0) = 0, {(v^{*})}^{'} (1) = 1 / 2 . \end{matrix}

(16)

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

| v^{*} | \approx \sqrt{{(v^{*})}^{2} + e_{1}},

where a small value for

e_{1}

is specified in the next lines.

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

clear all
$m_{8} = 800;$ (number of nodes)
$d = 1 / m_{8};$
$e_{1} = 0.00001;$
$f o r i = 1 : m_{8}$

$y o (i, 1) = 0.01;$

$y_{1} (i, 1) = sin (π * i / m_{8}) / 2;$

$e n d;$
$f o r i = 1 : m_{8} - 1$

$d y_{1} (i, 1) = (y_{1} (i + 1, 1) - y_{1} (i, 1)) / d;$

$e n d;$
$f o r k = 1 : 3000$ (we have fixed the number of iterations)

$i = 1;$

$h_{3} = 1 / \sqrt{v o {(i, 1)}^{2} + e_{1}};$

$m_{12} = 1 + d^{2} * h_{3} + d^{2};$

$m_{50} (i) = 1 / m_{12};$

$z (i) = m_{50} (i) * (d y_{1} (i, 1) * d^{2});$
$f o r i = 2 : m_{8} - 1$

$h_{3} = 1 / \sqrt{v o {(i, 1)}^{2} + e_{1}};$

$m_{12} = 2 + h_{3} * d^{2} + d^{2} - m 50 (i - 1);$

$m 50 (i) = 1 / m_{12};$

$z (i) = m_{50} (i) * (z (i - 1) + d y_{1} (i, 1) * d^{2});$

$e n d;$
$v (m_{8}, 1) = (d / 2 + z (m_{8} - 1)) / (1 - m_{50} (m_{8} - 1));$
$f o r i = 1 : m_{8} - 1$

$v (m_{8} - i, 1) = m_{50} (m_{8} - i) * v (m_{8} - i + 1) + z (m_{8} - i);$

$e n d;$
$v (m_{8} / 2, 1)$
$v o = v;$

$e n d;$
$f o r i = 1 : m_{8} - 1$

$u (i, 1) = (v (i + 1, 1) - v (i, 1)) / d + y_{1} (i, 1);$

$e n d;$
$f o r i = 1 : m_{8} - 1$

$x (i) = i * d;$

$e n d;$

$p l o t (x, u (:, 1))$

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

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

Ω \subset R^{2}

be an open, bounded, connected set which represents the middle surface of a plate of thickness h. The boundary of

Ω

, which is assumed to be regular (Lipschitzian), is denoted by

\partial Ω

. The vectorial basis related to the cartesian system

{x_{1}, x_{2}, x_{3}}

is denoted by

(a_{α}, a_{3})

, where

α = 1, 2

(in general Greek indices stand for 1 or 2), and where

a_{3}

is the vector normal to

Ω

, whereas

a_{1}

and

a_{2}

are orthogonal vectors parallel to

Ω .

Also,

n

is the outward normal to the plate surface.

The displacements will be denoted by

\hat{u} = {{\hat{u}}_{α}, {\hat{u}}_{3}} = {\hat{u}}_{α} a_{α} + {\hat{u}}_{3} a_{3} .

The Kirchhoff-Love relations are

\begin{matrix} {\hat{u}}_{α} (x_{1}, x_{2}, x_{3}) = u_{α} (x_{1}, x_{2}) - x_{3} w {(x_{1}, x_{2})}_{, α} \\ and {\hat{u}}_{3} (x_{1}, x_{2}, x_{3}) = w (x_{1}, x_{2}) . \end{matrix}

(17)

Here

- h / 2 \leq x_{3} \leq h / 2

so that we have

u = (u_{α}, w) \in U

where

\begin{matrix} U & = & \{u = (u_{α}, w) \in W^{1, 2} (Ω; R^{2}) \times W^{2, 2} (Ω), \\ u_{α} = w = \frac{\partial w}{\partial n} = 0 on \partial Ω\} \\ = & W_{0}^{1, 2} (Ω; R^{2}) \times W_{0}^{2, 2} (Ω) . \end{matrix}

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

We also define the operator

Λ : U \to Y \times Y

, where

Y = Y^{*} = L^{2} (Ω; R^{2 \times 2})

, by

Λ (u) = {γ (u), κ (u)},

γ_{α β} (u) = \frac{u_{α, β} + u_{β, α}}{2} + \frac{w_{, α} w_{, β}}{2},

κ_{α β} (u) = - w_{, α β} .

The constitutive relations are given by

N_{α β} (u) = H_{α β λ μ} γ_{λ μ} (u),

(18)

M_{α β} (u) = h_{α β λ μ} κ_{λ μ} (u),

(19)

where:

{H_{α β λ μ}}

and

\{h_{α β λ μ} = \frac{h^{2}}{12} H_{α β λ μ}\}

, are symmetric positive definite fourth order tensors. From now on, we denote

{{\bar{H}}_{α β λ μ}} = {H_{α β λ μ}}^{- 1}

and

{{\bar{h}}_{α β λ μ}} = {h_{α β λ μ}}^{- 1}

Furthermore

{N_{α β}}

denote the membrane force tensor and

{M_{α β}}

the moment one. The plate stored energy, represented by

(G \circ Λ) : U \to R

is expressed by

(G \circ Λ) (u) = \frac{1}{2} \int_{Ω} N_{α β} (u) γ_{α β} (u) d x + \frac{1}{2} \int_{Ω} M_{α β} (u) κ_{α β} (u) d x

(20)

and the external work, represented by

F : U \to R

, is given by

F (u) = {〈 w, P 〉}_{L^{2}} + {〈 u_{α}, P_{α} 〉}_{L^{2}},

(21)

where

P, P_{1}, P_{2} \in L^{2} (Ω)

are external loads in the directions

a_{3}

a_{1}

and

a_{2}

respectively. The potential energy, denoted by

J : U \to R

is expressed by:

J (u) = (G \circ Λ) (u) - F (u)

Define now

J_{3} : \tilde{U} \to R

J_{3} (u) = J (u) + J_{5} (w) .

where

J_{5} (w) = 10 \int_{Ω} \frac{a^{K b w}}{ln (a) K^{3 / 2}} d x + 10 \int_{Ω} \frac{a^{- K (b w - 1 / 100)}}{ln (a) K^{3 / 2}} d x .

In such a case for

a = 2.71

K = 185

b = P / | P |

Ω

and

\tilde{U} = {{u \in U : ∥ w ∥}_{\infty} \leq 0.01 and P w \geq 0 a . e . in Ω},

we get

\begin{matrix} \frac{\partial J_{3} (u)}{\partial w} & = & \frac{\partial J (u)}{\partial w} + \frac{\partial J_{5} (u)}{\partial w} \\ \approx & \frac{\partial J (u)}{\partial w} + O (\pm 3.0), \end{matrix}

(22)

and

\begin{matrix} \frac{\partial^{2} J_{3} (u)}{\partial w^{2}} & = & \frac{\partial^{2} J (u)}{\partial w^{2}} + \frac{\partial^{2} J_{5} (u)}{\partial w^{2}} \\ \approx & \frac{\partial^{2} J (u)}{\partial w^{2}} + O (850) . \end{matrix}

(23)

This new functional

J_{3}

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

\frac{\partial^{2} J (u)}{\partial w^{2}},

which has increased of order

O (700 - 1000) .

Moreover the difference between the approximate and exact equation

\frac{\partial J (u)}{\partial w} = 0

is of order

O (\pm 3.0)

which corresponds to a small perturbation in the original equation for a load of

P = 1500 N / m^{2},

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_{1} = {ϕ = ϕ (x, y) \in W^{1, 2} (Ω \times Ω; R^{2}) : ϕ = 0 on Ω \times \partial Ω},

we define the functional

J_{1} : U \times V_{1} \to R

, where

\begin{matrix} J_{1} (u, ϕ) & = & G_{1} ({w_{, α β}}) + G_{2} (\{\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} + \frac{1}{2} w_{, α} w_{, β}\}) \\ - {〈 w, P 〉}_{L^{2}} - {〈 u_{α}, P_{α} 〉}_{L^{2}} . \end{matrix}

(24)

where

G_{1} ({w_{, α β}}) = \frac{1}{2} \int_{Ω} h_{α β λ μ} w_{, α β} w_{, λ μ} d x

and,

\begin{matrix} G_{2} (\{\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} + \frac{1}{2} w_{, α} w_{, β}\}) \\ = & \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} H_{α β λ μ} (\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} (x, y) + \frac{1}{2} w_{, α} w_{, β}) \\ \times (\frac{1}{2} (u_{λ, μ} + u_{λ, μ}) + ϕ_{λ, y_{μ}} (x, y) + \frac{1}{2} w_{, λ} w_{, μ}) d x d y \end{matrix}

We define also

J_{2} ({u_{α}}, ϕ) = inf_{w \in W_{0}^{2, 2} (Ω)} J_{1} (u, ϕ),

and

J_{3} ({u_{α}}) = inf_{ϕ \in V_{1}} J_{2} ({u_{α}}, ϕ) .

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

inf_{u \in U} J (u) = inf_{{u_{α}} \in W_{0}^{1, 2} (Ω; R^{2})} J_{3} ({u_{α}}) .

At this point we denote

Y_{1} = Y_{1}^{*} = Y_{3} = Y_{3}^{*} \equiv L^{2} (Ω \times Ω; R^{4})

and

Y_{2} = Y_{2}^{*} \equiv L^{2} (Ω \times Ω; R^{2}) .

Observe that

\begin{matrix} J (u) \\ = & G_{1} ({w_{, α β}}) + G_{2} (\{\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} + \frac{1}{2} w_{, α} w_{, β}\}) \\ - {〈 w, P 〉}_{L^{2}} - {〈 u_{α}, P_{α} 〉}_{L^{2}} \\ = & G_{1} ({w_{, α β}}) - {〈 w_{, α β}, M_{α β} 〉}_{L^{2}} + {〈 w_{, α β}, M_{α β} 〉}_{L^{2}} \\ + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} w_{, α} (x), Q_{α} (x, y) d x d y - {〈 w, P 〉}_{L^{2}} \\ - \frac{1}{| Ω |} \int_{Ω} \int_{Ω} w_{, α} (x), Q_{α} (x, y) d x d y + G_{2} (\{\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} + \frac{1}{2} w_{, α} w_{, β}\}) \\ - \frac{1}{| Ω |} \int_{Ω} \int_{Ω} (\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} + \frac{1}{2} w_{, α} w_{, β}), v_{α β}^{*} (x, y) d x d y \\ + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} (\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} + \frac{1}{2} w_{, α} w_{, β}), v_{α β}^{*} (x, y) d x d y - {〈 u_{α}, P_{α} 〉}_{L^{2}} \\ \geq & inf_{v_{3} \in Y_{3}} {- {〈 {(v_{3})}_{α β}, M_{α β} 〉}_{L^{2}} + G_{1} ({(v_{3})}_{α β})} \\ + inf_{w \in W_{0}^{2, 2} (Ω)} \{{〈 w_{, α β}, M_{α β} 〉}_{L^{2}} + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} w_{, α} (x) Q_{α} (x, y) d x d y - {〈 w, P 〉}_{L^{2}}\} \\ + inf_{v \in Y_{1}} \{- \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v_{α β} v_{α β}^{*} d x d y + G_{2} ({v_{α β}})\} \\ + inf_{(v_{2}, {u_{α}}) \in Y_{2} \times W_{0}^{1, 2} (Ω; R^{2})} \{\frac{1}{| Ω |} \int_{Ω} \int_{Ω} (\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} + \frac{1}{2} {(v_{2})}_{α} (x, y) {(v_{2})}_{β} (x, y)) \\ \times v_{α β}^{*} (x, y) d x d y - {〈 u_{α}, P_{α} 〉}_{L^{2}} + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} {(v_{2})}_{α} (x, y) Q_{α} (x, y) d x d y\} \\ \geq & - G_{1}^{*} (M) - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} (\bar{v_{α β}^{*}}) Q_{α} Q_{β} d x d y - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} {\bar{H}}_{α β λ μ} v_{α β}^{*} v_{λ μ}^{*} d x d y, \end{matrix}

(25)

\forall u \in U, (M, Q) \in C^{*}

v = {v_{α β}} \in A^{*}

where

A^{*} = A_{1}^{*} \cap A_{2}^{*} \cap B^{*},

A_{1}^{*} = {{v_{α β}^{*}} \in Y_{1}^{*} : {(v_{α β}^{*})}_{, y_{β}} = 0, in Ω},

A_{2}^{*} = \{{v_{α β}^{*}} \in Y_{1}^{*} : \frac{1}{| Ω |} {(\int_{Ω} v_{α β}^{*} d y)}_{, x_{β}} + P_{α} = 0, in Ω\},

B^{*} = \{{v_{α β}^{*}} \in Y_{1}^{*} : \{v_{α β}^{*} (x, y)\} is positive definite in Ω \times Ω\} .

and

C^{*} = \{(M, Q) \in Y_{3}^{*} \times Y_{2}^{*} : M_{α β, α β} - {(\int_{Ω} Q_{α} d y)}_{, x_{α}} - P = 0, in Ω\} .

Also

\{\bar{v_{α β}^{*}}\} = {\{v_{α β}^{*}\}}^{- 1},

and

{{\bar{H}}_{α β λ μ}} = {H_{α β λ μ}}

in an appropriate tensor sense.

Here it is worth highlighting we have denoted,

\begin{matrix} G_{1}^{*} (M) & = & sup_{v_{3} \in Y_{3}} {{〈 {(v_{3})}_{α β}, M_{α β} 〉}_{L^{2}} - G_{1} (v_{3})} \\ = & \frac{1}{2} \int_{Ω} {\bar{h}}_{α β λ μ} M_{α β} M_{λ μ} d x, \end{matrix}

(26)

where we recall that

{{\bar{h}}_{α β λ μ}} = {h_{α β λ μ}}^{- 1}

in an appropriate tensorial sense.

Summarizing, defining

J^{*} : C^{*} \times A^{*} \to R

\begin{matrix} J^{*} ((M, Q), v^{*}) & = & - G_{1}^{*} (M) - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} (\bar{v_{α β}^{*}}) Q_{α} Q_{β} d x d y \\ - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} {\bar{H}}_{α β λ μ} v_{α β}^{*} v_{λ μ}^{*} d x d y, \end{matrix}

(27)

we have got

inf_{u \in U} J (u) \geq sup_{((M, Q), v^{*}) \in C^{*} \times A^{*}} J^{*} ((M, Q), v^{*}) .

Remark 5.1.

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_{0}, Q_{0}), v_{0}^{*}) \in C^{*} \times A^{*}

is such that

δ J^{*} ((M_{0}, Q_{0}), v_{0}^{*}) = 0,

it is a well known result from the Legendre transform proprieties that the corresponding

(u_{0}, ϕ_{0}) \in V \times V_{1}

such that

{(w_{0})}_{, α β} = {\bar{h}}_{α β λ μ} {(M_{0})}_{λ μ},

and

{(v_{0}^{*})}_{α β} = H_{α β λ} (\frac{{(u_{0})}_{λ, μ} + {(u_{0})}_{μ, λ}}{2} + \frac{{(ϕ_{0})}_{λ, y_{μ}} + {(ϕ_{0})}_{μ, y_{λ}}}{2} + \frac{1}{2} {(v_{2_{0}})}_{λ} {(v_{2_{0}})}_{μ}),

{(v_{0}^{*})}_{α β, y_{β}} = 0,

is also such that

δ J_{1} (u_{0}, ϕ_{0}) = 0

and

J_{1} (u_{0}, ϕ_{0}) = J^{*} ((M_{0}, Q_{0}), v_{0}^{*}) .

From this and

inf_{u \in V} J (u) = inf_{(u, ϕ) \in V \times V_{1}} J_{1} (u, ϕ) \geq sup_{((M, Q), v^{*}) \in C^{*} \times A^{*}} J^{*} ((M, Q), v^{*}),

we obtain

\begin{matrix} J_{1} (u_{0}, ϕ_{0}) & = & inf_{(u, ϕ) \in V \times V_{1}} J_{1} (u, ϕ) \\ = & sup_{((M, Q), v^{*}) \in C^{*} \times A^{*}} J^{*} ((M, Q), v^{*}) \\ = & J^{*} ((M_{0}, Q_{0}), v_{0}^{*}) \\ = & inf_{u \in V} J (u) . \end{matrix}

(28)

Also, from the modern calculus of variations theory, there exists a sequence

{u_{n}} \subset V

such that

u_{n} ⇀ u_{0}, weakly in V,

and

J (u_{n}) \to J_{1} (u_{0}, ϕ_{0}) = inf_{u \in V} J (u) .

From this and the Ekeland variational principle, there exists

{v_{n}} \subset V

such that

∥ u_{n} - v_{n} ∥_{V} \leq 1 / n,

J (v_{n}) \leq inf_{u \in V} J (u) + 1 / n,

and

∥ δ J (v_{n}) ∥_{V^{*}} \leq 1 / n, \forall n \in N,

so that

v_{n} ⇀ u_{0}, weakly in V,

and

J (v_{n}) \to J_{1} (u_{0}, ϕ_{0}) = inf_{u \in V} J (u) .

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

v_{n} \to u_{0}, strongly in R^{N}

for an appropriate

N \in N .

From continuity we obtain

δ J (v_{n}) \to δ J (u_{0}) = 0,

J (v_{n}) \to J (u_{0}) .

Summarizing, we have got

J (u_{0}) = inf_{u \in V} J (u),

δ J (u_{0}) = 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

Ω \subset R^{n}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω = Γ .

For

1 < p < + \infty

, consider a functional

J : V \to R

where

J (u) = G (\nabla u) + F (u) - {〈 u, f 〉}_{L^{2}},

where

V = \{u \in W^{1, p} (Ω; R^{N}) : u = u_{0} on \partial Ω\},

u_{0} \in C^{1} (\bar{Ω}; R^{N})

and

f \in L^{2} (Ω; R^{N}) .

We assume

G : Y \to R

and

F : V \to R

are Fréchet differentiable and F is also convex.

Also

G (\nabla u) = \int_{Ω} g (\nabla u) d x,

where

g : R^{N \times n} \to R

it is supposed to be Fréchet differentiable. Here we have denoted

Y = L^{p} (Ω; R^{N \times n})

We define also

J_{1} : V \times Y_{1} \to R

J_{1} (u, ϕ) = G_{1} (\nabla u + \nabla_{y} ϕ) + F (u) - {〈 u, f 〉}_{L^{2}},

where

Y_{1} = W^{1, p} (Ω \times Ω; R^{N})

and

G_{1} (\nabla u + \nabla_{y} ϕ) = \frac{1}{| Ω |} \int_{Ω} \int_{Ω} g (\nabla u (x) + \nabla_{y} ϕ (x, y)) d x d y .

Moreover, we define the relaxed functional

J_{2} : V \to R

J_{2} (u) = inf_{ϕ \in V_{0}} J_{1} (u, ϕ),

where

V_{0} = {ϕ \in Y_{1} : ϕ (x, y) = 0, on Ω \times \partial Ω} .

Now observe that

\begin{matrix} J_{1} (u, ϕ) & = & G_{1} (\nabla u + \nabla_{y} ϕ) + F (u) - {〈 u, f 〉}_{L^{2}} \\ = & - \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot (\nabla u + \nabla_{y} ϕ (x, y)) d y d x + G_{1} (\nabla u + \nabla_{y} ϕ) \\ + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot (\nabla u + \nabla_{y} ϕ (x, y)) d y d x + F (u) - {〈 u, f 〉}_{L^{2}} \\ \geq & inf_{v \in Y_{2}} \{- \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot v (x, y) d y d x + G_{1} (v)\} \\ + inf_{(v, ϕ) \in V \times V_{0}} \{\frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot (\nabla u + \nabla_{y} ϕ (x, y)) d y d x + F (u) - {〈 u, f 〉}_{L^{2}}\} \\ = & - G_{1}^{*} (v^{*}) - F^{*} ({div}_{x} (\frac{1}{| Ω |} \int_{Ω} v^{*} (x, y) d y) + f) \\ + \frac{1}{| Ω |} \int_{\partial Ω} (\int_{Ω} v^{*} (x, y) d y) \otimes n u_{0} d Γ, \end{matrix}

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\forall (u, ϕ) \in V \times V_{0}, v^{*} \in A^{*}

, where

A^{*} = {v^{*} \in Y_{2}^{*} : {div}_{y} v^{*} (x, y) = 0, in Ω} .

Here we have denoted

G_{1}^{*} (v^{*}) = sup_{v \in Y_{2}} \{\frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot v (x, y) d y d x - G_{1} (v)\},

where

Y_{2} = L^{p} (Ω \times Ω; R^{N \times n}),

Y_{2}^{*} = L^{q} (Ω \times Ω; R^{N \times n}),

and where

\frac{1}{p} + \frac{1}{q} = 1 .

Furthermore, for

v^{*} \in A^{*}

, we have

\begin{matrix} F^{*} ({div}_{x} (\frac{1}{| Ω |} \int_{Ω} v^{*} (x, y) d y) + f) - \frac{1}{| Ω |} \int_{\partial Ω} (\int_{Ω} v^{*} (x, y) d y) \otimes n u_{0} d Γ \\ = & sup_{(v, ϕ) \in V \times V_{0}} \{- \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot (\nabla u + \nabla_{y} ϕ (x, y)) d y d x - F (u) + {〈 u, f 〉}_{L^{2}}\}, \end{matrix}

(30)

Therefore, denoting

J_{3}^{*} : Y_{2}^{*} \to R

J_{3}^{*} (v^{*}) = - G_{1}^{*} (v^{*}) - F^{*} ({div}_{x} (\int_{Ω} v^{*} (x, y) d y) + f) + \frac{1}{| Ω |} \int_{\partial Ω} (\int_{Ω} v^{*} (x, y) d y) \otimes n u_{0} d Γ,

we have got

inf_{u \in V} J_{2} (u) \geq sup_{v^{*} \in A^{*}} J_{3}^{*} (v^{*}) .

Finally, we highlight such a dual functional

J_{3}^{*}

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

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Concerning a standard model in non-linear elasticity, consider a functional

J : V \to R

where

\begin{matrix} J (u) \\ = & \frac{1}{2} \int_{Ω} H_{i j k l} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{1}{2} u_{m, i} u_{m, j}) (\frac{u_{k, l} + u_{j, i}}{2} + \frac{1}{2} u_{m, k} u_{m, l}) d x \\ - {〈 u_{i}, f_{i} 〉}_{L^{2}} \end{matrix}

(31)

where

f \in L^{2} (Ω; R^{3})

and

V = W_{0}^{1, 2} (Ω; R^{3}) .

Here

{H_{i j k l}}

is a fourth-order and positive definite symmetric tensor (in an appropriate standard sense). Moreover,

u = (u_{1}, u_{2}, u_{3}) \in V

is a field of displacements resulting from the f load field action on the volume comprised by

Ω

At this point, we define the functional

J_{1} : V \times V_{1} \to R

, where

\begin{matrix} J_{1} (u, ϕ) \\ = & \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} H_{i j k l} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2} \frac{1}{2} (u_{m, i} + ϕ_{m, y_{i}}) (u_{m, j} + ϕ_{m, y_{j}})) \\ \times (\frac{u_{k, l} + u_{l, k}}{2} + \frac{ϕ_{k, y_{l}} + ϕ_{l, y_{k}}}{2} + \frac{1}{2} (u_{m, k} + ϕ_{m, y_{k}}) (u_{m, l} + ϕ_{m, y_{l}})) d x d y \\ - {〈 u_{i}, f_{i} 〉}_{L^{2}}, \end{matrix}

(32)

where

V_{1} = {ϕ \in W^{1, 2} (Ω \times Ω; R^{3}) : ϕ = 0 on Ω \times \partial Ω} .

We define also the quasi-convex envelop of J, denoted by

Q_{J} : V \to R

, as

Q_{J} (u) = inf_{ϕ \in V_{1}} J_{1} (u, ϕ) .

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

inf_{u \in V} J (u) = inf_{u \in V} Q_{J} (u) .

Observe now that, denoting

Y_{1} = Y_{1}^{*} = L^{2} (Ω \times Ω; R^{9})

Y_{2} = Y_{2}^{*} = L^{2} (Ω \times Ω; R^{3}),

and

\begin{matrix} G_{1} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2} + \frac{1}{2} (u_{m, i} + ϕ_{m, y_{i}}) (u_{m, j} + ϕ_{m, y_{j}})) \\ = & \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} H_{i j k l} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2} + \frac{1}{2} (u_{m, i} + ϕ_{m, y_{i}}) (u_{m, j} + ϕ_{m, y_{j}})) \\ \times (\frac{u_{k, l} + u_{l, k}}{2} + \frac{ϕ_{k, y_{l}} + ϕ_{l, y_{k}}}{2} + \frac{1}{2} (u_{m, k} + ϕ_{m, y_{k}}) (u_{m, l} + ϕ_{m, y_{l}})) d x d y \end{matrix}

(33)

we have that

\begin{matrix} J_{1} (u, ϕ) \\ = & G_{1} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2} + \frac{1}{2} (u_{m, i} + ϕ_{m, y_{i}}) (u_{m, j} + ϕ_{m, y_{j}})) - {〈 u_{i}, f_{i} 〉}_{L^{2}} \\ = & - \frac{1}{| Ω |} \int_{Ω} \int_{Ω} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2} + \frac{1}{2} (u_{m, i} + ϕ_{m, y_{i}}) (u_{m, j} + ϕ_{m, y_{j}})) σ_{i j} d x d y \\ + G_{1} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2} + \frac{1}{2} (u_{m, i} + ϕ_{m, y_{i}}) (u_{m, j} + ϕ_{m, y_{j}})) \\ + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2} + \frac{1}{2} (u_{m, i} + ϕ_{m, y_{i}}) (u_{m, j} + ϕ_{m, y_{j}})) σ_{i j} d x d y - {〈 u_{i}, f_{i} 〉}_{L^{2}} \\ \geq & inf_{v \in Y_{1}} \{- \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v_{i j} σ_{i j} d x d y - G_{1} ({v_{i j}})\} \\ + inf_{v_{2} \in Y_{1}} \{- \frac{1}{| Ω |} \int_{Ω} \int_{Ω} {(v_{2})}_{i j} Q_{i j} d x d y + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} (σ_{i j} \frac{1}{2} ({(v_{2})}_{m i} {(v_{2})}_{m j})) d x d y\} \\ + inf_{(u, ϕ) \in V \times V_{1}} \{\frac{1}{| Ω |} \int_{Ω} \int_{Ω} (σ_{i j} + Q_{i j}) (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2}) d x d y - {〈 u_{i}, f_{i} 〉}_{L^{2}}\} \\ \geq & - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} {\bar{H}}_{i j k l} σ_{i j} σ_{k l} d x d y \\ - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} \bar{σ_{i j}} Q_{m i} Q_{m k} d x d y, \end{matrix}

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\forall (u, ϕ) \in V \times V_{1}, (σ, Q) \in A^{*}

, where

A^{*} = A_{1}^{*} \cap A_{2}^{*} \cap A_{3}^{*}

A_{1}^{*} = {(σ, Q) \in Y_{1}^{*} \times Y_{1}^{*} : σ_{i j, y_{j}} + Q_{i j, y_{j}} = 0, in Ω \times Ω} .

A_{2}^{*} = \{(σ, Q) \in Y_{1}^{*} \times Y_{1}^{*} : \frac{1}{| Ω |} {(\int_{Ω} (σ_{i j}) d y)}_{x_{j}} + \frac{1}{| Ω |} {(\int_{Ω} (Q_{i j}) d y)}_{x_{j}} + f_{i} = 0, in Ω\},

A_{3}^{*} = {(σ, Q) \in Y_{1}^{*} \times Y_{1}^{*} : {σ_{i j}} is positive definite in Ω \times Ω} .

Hence, denoting

J^{*} (σ, Q) = - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} {\bar{H}}_{i j k l} σ_{i j} σ_{k l} d x d y - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} \bar{σ_{i j}} Q_{m i} Q_{m k} d x d y,

we have obtained

inf_{u \in V} J (u) \geq sup_{(σ, Q) \in A^{*}} J^{*} (σ, Q) .

Remark 6.1.

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

(σ_{0}, Q_{0}) \in A^{*}

is such that

δ J^{*} (σ_{0}, Q_{0}) = 0,

it is a well known result from the Legendre transform proprieties that the corresponding

(u_{0}, ϕ_{0}) \in V \times V_{1}

such that

{(σ_{0})}_{i j} = H_{i j k l} (\frac{u_{k, l} + u_{l, k}}{2} + \frac{ϕ_{k, y_{l}} + ϕ_{l, y_{k}}}{2} + \frac{1}{2} (u_{m, k} + ϕ_{m, y_{k}}) (u_{m, l} + ϕ_{m, y_{l}}))

and

{(Q_{0})}_{i j} = {(σ_{0})}_{i m} {(v_{2_{0}})}_{m j},

is also such that

δ J_{1} (u_{0}, ϕ_{0}) = 0

and

J_{1} (u_{0}, ϕ_{0}) = J^{*} (σ_{0}, Q_{0}) .

From this and

inf_{u \in V} J (u) = inf_{(u, ϕ) \in V \times V_{1}} J_{1} (u, ϕ) \geq sup_{(σ, Q) A^{*}} J^{*} (σ, Q),

we obtain

\begin{matrix} J_{1} (u_{0}, ϕ_{0}) & = & inf_{(u, ϕ) \in V \times V_{1}} J_{1} (u, ϕ) \\ = & sup_{(σ, Q) \in A^{*}} J^{*} (σ, Q) \\ = & J^{*} (σ_{0}, Q_{0}) \\ = & inf_{u \in V} J (u) . \end{matrix}

(35)

Also, from the modern calculus of variations theory, there exists a sequence

{u_{n}} \subset V

such that

u_{n} ⇀ u_{0}, weakly in V,

and

J (u_{n}) \to J_{1} (u_{0}, ϕ_{0}) = inf_{u \in V} J (u) .

From this and the Ekeland variational principle, there exists

{v_{n}} \subset V

such that

∥ u_{n} - v_{n} ∥_{V} \leq 1 / n,

J (v_{n}) \leq inf_{u \in V} J (u) + 1 / n,

and

∥ δ J (v_{n}) ∥_{V^{*}} \leq 1 / n, \forall n \in N,

so that

v_{n} ⇀ u_{0}, weakly in V,

and

J (v_{n}) \to J_{1} (u_{0}, ϕ_{0}) = inf_{u \in V} J (u) .

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

v_{n} \to u_{0}, strongly in R^{N}

for an appropriate

N \in N .

From continuity we obtain

δ J (v_{n}) \to δ J (u_{0}) = 0,

J (v_{n}) \to J (u_{0}) .

Summarizing, we have got

J (u_{0}) = inf_{u \in V} J (u),

δ J (u_{0}) = 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

Ω \subset R^{2}

be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Consider a functional

J : V \to R

where

J (u) = F (u_{x}, u_{y}) - {〈 u, f 〉}_{L^{2}},

V = W_{0}^{1, 2} (Ω)

and

f \in L^{2} (Ω) .

Here we denote

Y = Y^{*} = L^{2} (Ω)

and

Y_{1} = Y_{1}^{*} = L^{2} (Ω) \times L^{2} (Ω) .

Defining

V_{1} = {{u \in V : ∥ u ∥}_{1, \infty} \leq K_{1}}

for some appropriate

K_{1} > 0

, suppose also F is twice Fréchet differentiable and

det \{\frac{\partial^{2} F (u_{x}, u_{y})}{\partial v_{1} \partial v_{2}}\} \neq 0,

\forall u \in V_{1} .

Define now

F_{1} : V \to R

and

F_{2} : V \to R

F_{1} (u_{x}, u_{y}) = F (u_{x}, u_{y}) + \frac{ε}{2} \int_{Ω} u_{x}^{2} d x + \frac{ε}{2} \int_{Ω} u_{y}^{2} d x,

and

F_{2} (u_{x}, u_{y}) = \frac{ε}{2} \int_{Ω} u_{x}^{2} d x + \frac{ε}{2} \int_{Ω} u_{y}^{2} d x,

where here we denote

d x = d x_{1} d x_{2} .

Moreover, we define the respective Legendre transform functionals

F_{1}^{*}

and

F_{2}^{*}

F_{1}^{*} (v^{*}) = {〈 v_{1}, v_{1}^{*} 〉}_{L^{2}} + {〈 v_{2}, v_{2}^{*} 〉}_{L^{2}} - F_{1} (v_{1}, v_{2}),

where

v_{1}, v_{2} \in Y

are such that

v_{1}^{*} = \frac{\partial F_{1} (v_{1}, v_{2})}{\partial v_{1}},

v_{2}^{*} = \frac{\partial F_{1} (v_{1}, v_{2})}{\partial v_{2}},

and

F_{2}^{*} (v^{*}) = {〈 v_{1}, v_{1}^{*} + f_{1} 〉}_{L^{2}} + {〈 v_{2}, v_{2}^{*} 〉}_{L^{2}} - F_{2} (v_{1}, v_{2}),

where

v_{1}, v_{2} \in Y

are such that

v_{1}^{*} + f_{1} = \frac{\partial F_{2} (v_{1}, v_{2})}{\partial v_{1}},

v_{2}^{*} = \frac{\partial F_{2} (v_{1}, v_{2})}{\partial v_{2}} .

Here

f_{1}

is any function such that

{(f_{1})}_{x} = f, in Ω .

Furthermore, we define

\begin{matrix} J^{*} (v^{*}) & = & - F_{1}^{*} (v^{*}) + F_{2}^{*} (v^{*}) \\ = & - F_{1}^{*} (v^{*}) + \frac{1}{2 ε} \int_{Ω} {(v_{1}^{*} + f_{1})}^{2} d x + \frac{1}{2 ε} \int_{Ω} {(v_{2}^{*})}^{2} d x . \end{matrix}

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Observe that through the target conditions

v_{1}^{*} + f_{1} = ε u_{x},

v_{2}^{*} = ε u_{y},

we may obtain the compatibility condition

{(v_{1}^{*} + f_{1})}_{y} - {(v_{2}^{*})}_{x} = 0 .

Define now

A^{*} = {v^{*} = (v_{1}^{*}, v_{2}^{*}) \in B_{r} (0, 0) \subset Y_{1}^{*} : {(v_{1}^{*} + f_{1})}_{y} - {(v_{2}^{*})}_{x} = 0, in Ω},

for some appropriate

r > 0

such that

J^{*}

is convex in

B_{r} (0, 0) .

Consider the problem of minimizing

J^{*}

subject to

v^{*} \in A^{*} .

Assuming

r > 0

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

J_{1}^{*} (v^{*}, φ) = J^{*} (v^{*}) + {〈 φ, {(v_{1}^{*} + f)}_{y} - {(v_{2}^{*})}_{x} 〉}_{L^{2}},

where

φ

is an appropriate Lagrange multiplier.

Therefore

\begin{matrix} J_{1}^{*} (v^{*}) & = & - F_{1}^{*} (v^{*}) + \frac{1}{2 ε} \int_{Ω} {(v_{1}^{*} + f_{1})}^{2} d x + \frac{1}{2 ε} \int_{Ω} {(v_{2}^{*})}^{2} d x \\ + {〈 φ, {(v_{1}^{*} + f)}_{y} - {(v_{2}^{*})}_{x} 〉}_{L^{2}} . \end{matrix}

(37)

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

J_{1}^{*} .

From the variation of

J_{1}^{*}

v_{1}^{*}

we obtain

- \frac{\partial F_{1}^{*} (v^{*})}{\partial v_{1}^{*}} + \frac{v_{1}^{*} + f}{ε} - \frac{\partial φ}{\partial y} = 0 .

(38)

From the variation of

J_{1}^{*}

v_{2}^{*}

we obtain

- \frac{\partial F_{1}^{*} (v^{*})}{\partial v_{2}^{*}} + \frac{v_{2}^{*}}{ε} + \frac{\partial φ}{\partial x} = 0 .

(39)

From the variation of

J_{1}^{*}

φ

we have

{(v_{1}^{*} + f)}_{y} - {(v_{2}^{*})}_{x} = 0 .

From this last equation, we may obtain

u \in V

such that

v_{1}^{*} + f = ε u_{x},

and

v_{2}^{*} = ε u_{y} .

From this and the previous extremal equations indicated we have

- \frac{\partial F_{1}^{*} (v^{*})}{\partial v_{1}^{*}} + u_{x} - \frac{\partial φ}{\partial y} = 0,

and

- \frac{\partial F_{1}^{*} (v^{*})}{\partial v_{2}^{*}} + u_{y} + \frac{\partial φ}{\partial x} = 0 .

so that

v_{1}^{*} + f = \frac{\partial F_{1} (u_{x} - φ_{y}, u_{y} + φ_{x})}{\partial v_{1}},

and

v_{2}^{*} = \frac{\partial F_{1} (u_{x} - φ_{y}, u_{y} + φ_{x})}{\partial v_{2}} .

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

\begin{matrix} - ε {(\frac{\partial F_{1}^{*} (v^{*})}{\partial v_{1}^{*}})}_{x} - ε {(\frac{\partial F_{1}^{*} (v^{*})}{\partial v_{2}^{*}})}_{y} \\ + {(v_{1}^{*} + f_{1})}_{x} + {(v_{2}^{*})}_{y} \\ = & - ε u_{x x} - ε u_{y y} + {(v_{1}^{*})}_{x} + {(v_{2}^{*})}_{y} + f = 0 . \end{matrix}

(40)

Replacing the expressions of

v_{1}^{*}

and

v_{2}^{*}

into this last equation, we have

- ε u_{x x} - ε u_{y y} + {(\frac{\partial F_{1} (u_{x} - φ_{y}, u_{y} + φ_{x})}{\partial v_{1}})}_{x} + {(\frac{\partial F_{1} (u_{x} - φ_{y}, u_{y} + φ_{x})}{\partial v_{2}})}_{y} + f = 0,

so that

{(\frac{\partial F (u_{x} - φ_{y}, u_{y} + φ_{x})}{\partial v_{1}})}_{x} + {(\frac{\partial F (u_{x} - φ_{y}, u_{y} + φ_{x})}{\partial v_{2}})}_{y} + f = 0, in Ω .

(41)

Observe that if

\nabla^{2} φ = 0

then there exists

\hat{u}

such that u and

φ

are also such that

u_{x} - φ_{y} = {\hat{u}}_{x}

and

u_{y} + φ_{x} = {\hat{u}}_{y} .

The boundary conditions for

φ

must be such that

\hat{u} \in W_{0}^{1, 2} .

From this and equation (41) we obtain

δ J (\hat{u}) = 0 .

Summarizing, we may obtain a solution

\hat{u} \in W_{0}^{1, 2}

of equation

δ J (\hat{u}) = 0

by minimizing

J^{*}

A^{*}

Finally, observe that clearly

J^{*}

is convex in an appropriate large ball

B_{r} (0, 0)

for some appropriate

r > 0

8. Another primal dual formulation for a related model

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular boundary denoted by

\partial Ω .

Consider the functional

J : V \to R

where

\begin{matrix} J (u) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x \\ - {〈 u, f 〉}_{L^{2}}, \end{matrix}

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α > 0, β > 0, γ > 0

V = W_{0}^{1, 2} (Ω)

and

f \in L^{2} (Ω) .

Denoting

Y = Y^{*} = L^{2} (Ω)

, define now

J_{1}^{*} : V \times Y^{*} \to R

\begin{matrix} J_{1}^{*} (u, v_{0}^{*}) & = & - \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x - {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} \\ + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 v_{0}^{*} u - f)}^{2} d x + {〈 u, f 〉}_{L^{2}} \\ + \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x + β \int_{Ω} v_{0}^{*} d x, \end{matrix}

(43)

Define also

A^{+} = {u \in V : u f \geq 0, a . e . in Ω},

V_{2} = {{u \in V : ∥ u ∥}_{\infty} \leq K_{3}},

and

V_{1} = V_{2} \cap A^{+}

for some appropriate

K_{3} > 0

to be specified.

Moreover define

B^{*} = {v_{0}^{*} \in Y^{*} : ∥ v_{0}^{*} ∥_{\infty} \leq K}

for some appropriate

K > 0

to be specified.

Observe that, denoting

φ = - γ \nabla^{2} u + 2 v_{0}^{*} u - f

we have

\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*})}{\partial {(v_{0}^{*})}^{2}} = \frac{1}{α} + 4 K_{1} u^{2}

\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*})}{\partial u^{2}} = γ \nabla^{2} - 2 v_{0}^{*} + K_{1} {(- γ \nabla^{2} + 2 v_{0}^{*})}^{2}

and

\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*})}{\partial u \partial v_{0}^{*}} = K_{1} (2 φ + 2 (- γ \nabla^{2} u + 2 v_{0}^{*} u)) - 2 u

so that

\begin{matrix} det {δ^{2} J_{1}^{*} (u, v_{0}^{*})} \\ = & \frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*})}{\partial {(v_{0}^{*})}^{2}} \frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*})}{\partial u^{2}} - {(\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*})}{\partial u \partial v_{0}^{*}})}^{2} \\ = & \frac{K_{1} {(- γ \nabla^{2} + 2 v_{0}^{*})}^{2}}{α} - \frac{γ \nabla^{2} + 2 v_{0}^{*} + 4 α u^{2}}{α} \\ - 4 K_{1}^{2} φ^{2} - 8 K_{1} φ (- γ \nabla^{2} + 2 v_{0}^{*}) u + 8 K_{1} φ u \\ + 4 K_{1} (- γ \nabla^{2} u + 2 v_{0}^{*} u) u . \end{matrix}

(44)

Observe now that a critical point

φ = 0

and

(- γ \nabla^{2} u + 2 v_{0}^{*} u) u = f u \geq 0

Ω

Therefore, for an appropriate large

K_{1} > 0

, also at a critical point, we have

\begin{matrix} det {δ^{2} J_{1}^{*} (u, v_{0}^{*})} \\ = & 4 K_{1} f u - \frac{δ^{2} J (u)}{α} + K_{1} \frac{{(- γ \nabla^{2} + 2 v_{0}^{*})}^{2}}{α} > 0 . \end{matrix}

(45)

Remark 8.1.

From this last equation we may observe that

J_{1}^{*}

has a large region of convexity about any critical point

(u_{0}, {\hat{v}}_{0}^{*})

, that is, there exists a large

r > 0

such that

J_{1}^{*}

is convex on

B_{r} (u_{0}, {\hat{v}}_{0}^{*}) .

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

Theorem 8.2.

Assume

K_{1} ≫ max {1, K, K_{3}}

and suppose

(u_{0}, {\hat{v}}_{0}^{*}) \in V_{1} \times B^{*}

is such that

δ J_{1}^{*} (u_{0}, {\hat{v}}_{0}^{*}) = 0 .

Under such hypotheses, there exists

r > 0

such that

J_{1}^{*}

is convex in

E^{*} = B_{r} (u_{0}, {\hat{v}}_{0}^{*}) \cap (V_{1} \times B^{*})

δ J (u_{0}) = 0,

and

- J (u_{0}) = J_{1} (u_{0}, {\hat{v}}_{0}^{*}) = inf_{(u, v_{0}^{*}) \in E^{*}} J_{1}^{*} (u, v_{0}^{*}) .

9. A third primal dual formulation for a related model

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular boundary denoted by

\partial Ω .

Consider the functional

J : V \to R

where

\begin{matrix} J (u) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x \\ - {〈 u, f 〉}_{L^{2}}, \end{matrix}

(46)

α > 0, β > 0, γ > 0

V = W_{0}^{1, 2} (Ω)

and

f \in L^{2} (Ω) .

Denoting

Y = Y^{*} = L^{2} (Ω)

, define now

J_{1}^{*} : V \times Y^{*} \times Y^{*} \to R

\begin{matrix} J_{1}^{*} (u, v_{0}^{*}, v_{1}^{*}) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{1}{2} \int_{Ω} K u^{2} d x \\ - {〈 u, v_{1}^{*} 〉}_{L^{2}} + \frac{1}{2} \int_{Ω} \frac{{(v_{1}^{*})}^{2}}{(- 2 v_{0}^{*} + K)} d x \\ + \frac{1}{2 (α + ε)} \int_{Ω} {(v_{0}^{*} - α (u^{2} - β))}^{2} d x + {〈 u, f 〉}_{L^{2}} \\ - \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x - β \int_{Ω} v_{0}^{*} d x, \end{matrix}

(47)

where

ε > 0

is a small real constant.

Define also

A^{+} = {u \in V : u f \geq 0, a . e . in Ω},

V_{2} = {{u \in V : ∥ u ∥}_{\infty} \leq K_{3}},

and

V_{1} = V_{2} \cap A^{+}

for some appropriate

K_{3} > 0

to be specified.

Moreover define

B^{*} = {v_{0}^{*} \in Y^{*} : ∥ v_{0}^{*} ∥_{\infty} \leq K_{4}}

and

D^{*} = {v_{1}^{*} \in Y^{*} : ∥ v_{1}^{*} ∥ \leq K_{5}},

for some appropriate real constants

K_{4}, K_{5} > 0

to be specified.

Remark 9.1.

Define now

H_{1} (u, v_{0}^{*}) = - γ \nabla^{2} + 2 v_{0}^{*} + 4 α u^{2}

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

M_{1}

define

B_{1}^{*} = {v_{0}^{*} \in B^{*} : 2 v_{0}^{*} + M_{1} \geq ε_{1}},

for some small parameter

ε_{1} > 0 .

Moreover, define

E^{*} = {u \in V_{1} : \sqrt{4 α} | u | \geq \sqrt{| M_{1} + γ \nabla^{2} |} .

Since for

(u, v_{0}^{*}) \in V_{1} \times B_{1}^{*}

we have

u f \geq 0, in Ω

, so that for

u_{1}, u_{2} \in V_{1}

we have

sign (u_{1}) = sign (u_{2}) in Ω,

we may infer that

E^{*}

is a convex set.

Moreover if

(u, v_{0}^{*}) \in E^{*} \times B_{1}^{*}

, then

\sqrt{4 α} | u | \geq \sqrt{| M_{1} + γ \nabla^{2} |}

so that

4 α u^{2} \geq M_{1} + γ \nabla^{2}

and

2 v_{0}^{*} + M_{1} \geq ε_{1}

so that

H_{1} (u, v_{0}^{*}) = - γ \nabla^{2} + 2 v_{0}^{*} + 4 α u^{2} \geq ε_{1} .

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

Observe that, defining

φ = v_{0}^{*} - α (u^{2} - β)

we may obtain

\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*}, v_{1}^{*})}{\partial u^{2}} = - γ \nabla^{2} + K + \frac{α}{α + ε} 4 u^{2} - 2 φ \frac{α}{α + ε}

\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*}, v_{1}^{*})}{\partial {(v_{1}^{*})}^{2}} = \frac{1}{- 2 v_{0}^{*} + K}

and

\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*}, v_{1}^{*})}{\partial u \partial v_{1}^{*}} = - 1

so that

\begin{matrix} det \{\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*}, v_{1}^{*})}{\partial u \partial v_{1}^{*}}\} \\ = & \frac{\partial^{2} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})}{\partial {(v_{1}^{*})}^{2}} \frac{\partial^{2} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})}{\partial u^{2}} - {(\frac{\partial^{2} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})}{\partial u \partial v_{1}^{*}})}^{2} \\ = & \frac{- γ \nabla^{2} + 2 v_{0}^{*} + 4 \frac{α^{2}}{α + ε} u^{2} - 2 \frac{α}{α + ε} φ}{- 2 v_{0}^{*} + K} \\ \equiv & H (u, v_{0}^{*}) . \end{matrix}

(48)

However, at a critical point, we have

φ = 0

so that, for a fixed

v_{0}^{*} \in B^{*}

we define the non-active but convex restriction

{(C_{1})}_{v_{0}^{*}}^{*} = {u \in V_{1} : {(φ)}^{2} \leq ε},

for a small parameter

ε > 0 .

From such results, assuming

K ≫ max {K_{3}, K_{4}, K_{5}},

and

0 < ε ≪ ε_{1} ≪ 1

, we have that

H (u, v_{0}^{*}) > 0,

for

v_{0}^{*} \in B_{1}^{*}

and

u \in E^{*} \cap {(C_{1})}_{v_{0}^{*}}^{*} .

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

Theorem 9.2.

Suppose

(u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \in (E^{*} \cap {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*}) \times D^{*} \times B_{1}^{*}

is such that

δ J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = 0 .

Under such hypotheses, we have that

δ J (u_{0}) = 0

and

\begin{matrix} J (u_{0}) & = & inf_{u \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*}} J (u) \\ = & J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \\ = & inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} \{sup_{v_{0}^{*} \in B^{*}} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})\} \\ = & sup_{v_{0}^{*} \in B^{*}} \{inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})\} . \end{matrix}

(49)

Proof.

The proof that

δ J (u_{0}) = 0

and

J (u_{0}) = J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*})

may be easily made similarly as in the previous sections.

Moreover, observe that for

K > 0

sufficiently large, we have

\frac{\partial^{2} J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, v_{0}^{*})}{\partial {(v_{0}^{*})}^{2}} < 0, \forall v_{0}^{*} \in B^{*}

so that this and the other hypotheses, we have also

J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} J_{1}^{*} (u, v_{1}^{*}, {\hat{v}}_{0}^{*})

and

J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = sup_{v_{0}^{*} \in B^{*}} J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, v_{0}^{*}) .

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

\begin{matrix} J (u_{0}) & = & J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \\ = & inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} \{sup_{v_{0}^{*} \in B^{*}} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})\} \\ = & sup_{v_{0}^{*} \in B^{*}} \{inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})\} . \end{matrix}

(50)

Moreover, observe that

\begin{matrix} J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) & = & inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} J_{1}^{*} (u, v_{1}^{*}, {\hat{v}}_{0}^{*}) \\ \leq & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{K}{2} \int_{Ω} u^{2} d x \\ + {〈 u^{2}, {\hat{v}}_{0}^{*} 〉}_{L^{2}} - \frac{K}{2} \int_{Ω} u^{2} d x \\ - \frac{1}{2 α} \int_{Ω} {({\hat{v}}_{0}^{*})}^{2} d x - β \int_{Ω} {\hat{v}}_{0}^{*} d x \\ + \frac{1}{2 (α + ε)} \int_{Ω} {({\hat{v}}_{0}^{*} - α (u^{2} - β))}^{2} d x - {〈 u, f 〉}_{L^{2}} \\ \leq & sup_{v_{0}^{*} \in Y^{*}} \{\frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + 〈 u^{2}, v_{0}^{*} 〉 \\ - \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x - β \int_{Ω} v_{0}^{*} d x \\ + \frac{1}{2 (α + ε)} \int_{Ω} {(v_{0}^{*} - α (u^{2} - β))}^{2} d x - {〈 u, f 〉}_{L^{2}}\} \\ = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x \\ - {〈 u, f 〉}_{L^{2}}, \forall u \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} . \end{matrix}

(51)

Summarizing, we have got

J (u_{0}) = J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \leq inf_{u \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*}} J (u) .

From such results, we may infer that

\begin{matrix} J (u_{0}) & = & inf_{u \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*}} J (u) \\ = & J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \\ = & inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} \{sup_{v_{0}^{*} \in B^{*}} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})\} \\ = & sup_{v_{0}^{*} \in B^{*}} \{inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})\} . \end{matrix}

(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

Ω \subset R^{3}

with a regular (Lipschitzian) boundary denoted by

\partial Ω = Γ_{0} \cup Γ_{t}

where

Γ_{0} \cap Γ_{t} = \emptyset .

Consider also the problem of minimizing the functional

\hat{J} : U \times B \to R

where

\hat{J} (u, t) = \frac{1}{2} {〈 u_{i}, f_{i} 〉}_{L^{2} (Ω)} + \frac{1}{2} {〈 u_{i}, {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})},

subject to

\{\begin{matrix} {(H_{i j k l} (t) e_{k l} (u))}_{, j} + f_{i} = 0 in Ω, \\ H_{i j k l} (t) e_{k l} (u) n_{j} - {\hat{f}}_{i} = 0, on Γ_{t}, \forall i \in {1, 2, 3} . \end{matrix}

(53)

Here

n = (n_{1}, n_{2}, n_{3})

denotes the outward normal to

\partial Ω

and

U = {u = (u_{1}, u_{2}, u_{3}) \in W^{1, 2} (Ω; R^{3}) : u = (0, 0, 0) = 0 on Γ_{0}},

B = \{t : Ω \to [0, 1] measurable : \int_{Ω} t (x) d x = t_{1} | Ω |\},

where

0 < t_{1} < 1

and

| Ω |

denotes the Lebesgue measure of

Ω .

Moreover

u = (u_{1}, u_{2}, u_{3}) \in W^{1, 2} (Ω; R^{3})

is the field of displacements relating the cartesian system

(0, x_{1}, x_{2}, x_{3})

, resulting from the action of the external loads

f \in L^{2} (Ω; R^{3})

and

\hat{f} \in L^{2} (Γ_{t}; R^{3}) .

We also define the stress tensor

{σ_{i j}} \in Y^{*} = Y = L^{2} (Ω; R^{3 \times 3}),

σ_{i j} (u) = H_{i j k l} (t) e_{k l} (u),

and the strain tensor

e : U \to L^{2} (Ω; R^{3 \times 3})

e_{i j} (u) = \frac{1}{2} (u_{i, j} + u_{j, i}), \forall i, j \in {1, 2, 3} .

Finally,

{H_{i j k l} (t)} = {t H_{i j k l}^{0} + (1 - t) H_{i j k l}^{1}},

where

H^{0}

corresponds to a strong material and

H^{1}

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 10.1.

Consider the statements and assumptions indicated in the last section, in particular those refereing to Ω and the functional

\hat{J} : U \times B \to R .

Define

J_{1} : U \times B \to R

J_{1} (u, t) = - G (e (u), t) + {〈 u_{i}, f_{i} 〉}_{L^{2} (Ω)} + {〈 u_{i}, {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})},

where

G (e (u), t) = \frac{1}{2} \int_{Ω} H_{i j k l} (t) e_{i j} (u) e_{k l} (u) d x,

and where

d x = d x_{1} d x_{2} d x_{3} .

Define also

J^{*} : U \to R

\begin{matrix} J^{*} (u) & = & inf_{t \in B} {J_{1} (u, t)} \\ = & inf_{t \in B} {- G (e (u), t) + {〈 u_{i}, f_{i} 〉}_{L^{2} (Ω)} + {〈 u_{i}, {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})}} . \end{matrix}

(54)

Assume there exists

c_{0}, c_{1} > 0

such that

H_{i j k l}^{0} z_{i j} z_{k l} > c_{0} z_{i j} z_{i j}

and

H_{i j k l}^{1} z_{i j} z_{k l} > c_{1} z_{i j} z_{i j}, \forall z = {z_{i j}} \in R^{3 \times 3}, such that z \neq 0 .

Finally, define

J : U \times B \to R \cup {+ \infty}

J (u, t) = \hat{J} (u, t) + I n d (u, t),

where

I n d (u, t) = \{\begin{matrix} 0, & if (u, t) \in A^{*}, \\ + \infty, & otherwise, \end{matrix}

(55)

where

A^{*} = A_{1} \cap A_{2},

A_{1} = {(u, t) \in U \times B : {(σ_{i j} (u))}_{, j} + f_{i} = 0, in Ω, \forall i \in {1, 2, 3}}

and

A_{2} = {(u, t) \in U \times B : σ_{i j} (u) n_{j} - {\hat{f}}_{i} = 0, on Γ_{t}, \forall i \in {1, 2, 3}} .

Under such hypotheses, there exists

(u_{0}, t_{0}) \in U \times B

such that

\begin{matrix} J (u_{0}, t_{0}) & = & inf_{(u, t) \in U \times B} J (u, t) \\ = & sup_{\hat{u} \in U} J^{*} (\hat{u}) \\ = & J^{*} (u_{0}) \\ = & \hat{J} (u_{0}, t_{0}) \\ = & inf_{(t, σ) \in B \times C^{*}} G^{*} (σ, t) \\ = & G^{*} (σ (u_{0}), t_{0}), \end{matrix}

(56)

where

\begin{matrix} G^{*} (σ, t) & = & sup_{v \in Y} {{〈 v_{i j}, σ_{i j} 〉}_{L^{2} (Ω)} - G (v, t)} \\ = & \frac{1}{2} \int_{Ω} {\bar{H}}_{i j k l} (t) σ_{i j} σ_{k l} d x, \end{matrix}

(57)

{{\bar{H}}_{i j k l} (t)} = {H_{i j k l} (t)}^{- 1}

and

C^{*} = C_{1} \cap C_{2},

where

C_{1} = {σ \in Y^{*} : σ_{i j, j} + f_{i} = 0, in Ω, \forall i \in {1, 2, 3}}

and

C_{2} = {σ \in Y^{*} : σ_{i j} n_{j} - {\hat{f}}_{i} = 0, on Γ_{t}, \forall i \in {1, 2, 3}} .

Proof.

Observe that

\begin{matrix} inf_{(u, t) \in U \times B} J (u, t) & = & inf_{t \in B} \{inf_{u \in U} J (u, t)\} \\ = & inf_{t \in B} \{sup_{\hat{u} \in U} \{inf_{u \in U} \{\frac{1}{2} \int_{Ω} H_{i j k l} (t) e_{i j} (u) e_{k l} (u) d x \\ + {〈 {\hat{u}}_{i}, {(H_{i j k l} (t) e_{k l} (u))}_{, j} + f_{i} 〉}_{L^{2} (Ω)} \\ - {〈 {\hat{u}}_{i}, H_{i j k l} (t) e_{k l} (u) n_{j} - {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})}\}\}\} \\ = & inf_{t \in B} \{sup_{\hat{u} \in U} \{inf_{u \in U} \{\frac{1}{2} \int_{Ω} H_{i j k l} (t) e_{i j} (u) e_{k l} (u) d x \\ - \int_{Ω} H_{i j k l} (t) e_{i j} (\hat{u}) e_{k l} (u) d x \\ + {〈 {\hat{u}}_{i}, f_{i} 〉}_{L^{2} (Ω)} + {〈 {\hat{u}}_{i}, {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})}\}\}\} \\ = & inf_{t \in B} \{sup_{\hat{u} \in U} \{- \int_{Ω} H_{i j k l} (t) e_{i j} (\hat{u}) e_{k l} (\hat{u}) d x \\ {〈 {\hat{u}}_{i}, f_{i} 〉}_{L^{2} (Ω)} + {〈 {\hat{u}}_{i}, {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})}\}\} \\ = & inf_{t \in B} \{inf_{σ \in C^{*}} G^{*} (σ, t)\} . \end{matrix}

(58)

Also, from this and the min-max theorem, there exist

(u_{0}, t_{0}) \in U \times B

such that

\begin{matrix} inf_{(u, t) \in U \times B} J (u, t) & = & inf_{t \in B} \{sup_{\hat{u} \in U} J_{1} (u, t)\} \\ = & sup_{u \in U} \{inf_{t \in B} J_{1} (u, t)\} \\ = & J_{1} (u_{0}, t_{0}) \\ = & inf_{t \in B} J_{1} (u_{0}, t) \\ = & J^{*} (u_{0}) . \end{matrix}

(59)

Finally, from the extremal necessary condition

\frac{\partial J_{1} (u_{0}, t_{0})}{\partial u} = 0

we obtain

{(H_{i j k l} (t_{0}) e_{k l} (u_{0}))}_{, j} + f_{i} = 0 in Ω,

and

H_{i j k l} (t_{0}) e_{k l} (u_{0}) n_{j} - {\hat{f}}_{i} = 0 on Γ_{t}, \forall i \in {1, 2, 3},

so that

G (e (u_{0})) = \frac{1}{2} {〈 {(u_{0})}_{i}, f_{i} 〉}_{L^{2} (Ω)} + \frac{1}{2} {〈 {(u_{0})}_{i}, {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})} .

Hence

(u_{0}, t_{0}) \in A^{*}

so that

I n d (u_{0}, t_{0}) = 0

and

σ (u_{0}) \in C^{*} .

Moreover

\begin{matrix} J^{*} (u_{0}) & = & - G (e (u_{0})) + {〈 {(u_{0})}_{i}, f_{i} 〉}_{L^{2} (Ω)} + {〈 {(u_{0})}_{i}, {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})} \\ = & G (e (u_{0})) \\ = & G (e (u_{0})) + I n d (u_{0}, t_{0}) \\ = & J (u_{0}, t_{0}) \\ = & G^{*} (σ (u_{0}), t_{0}) . \end{matrix}

(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 $t_{1} = 0.5 in Ω$ and $n = 1$ .
Calculate $u_{n} \in U$ such that

$J_{1} (u_{n}, t_{n}) = sup_{u \in U} J_{1} (u, t_{n}) .$
Calculate $t_{n + 1} \in B$ such that

$J_{1} (u_{n}, t_{n + 1}) = inf_{t \in B} J_{1} (u_{n}, t) .$
If $∥ t_{n + 1} - t_{n} ∥_{\infty} < 10^{- 4}$ or $n > 100$ then stop, else set $n : = n + 1$ 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= $210 * 10^{5}$ ; (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 $(b 14 > 10^{- 3.5})$ and $(k 3 < 5)$

k3=k3+1;

b12=1.0;

k=0;

while $(b 12 > 10^{- 4.0})$ 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

$E 1 = L o {(i, j)}^{2} * (E a - E b);$

ex=du1(i,j);

ey=dv2(i,j);

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

$S x = E 1 * (e x + w * e y) / (1 - w^{2});$

$S y = E 1 * (w * e x + e y) / (1 - w^{2});$

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, $[]$ , $[]$ ,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;

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;

S=0;

for i=1:m9-1

for j=1:m8-1

E 1 = L o {(i, j)}^{3} * E a + (1 - L o {(i, j)}^{3}) * E b;

ex=du1(i,j);

ey=dv2(i,j);

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

S x = E 1 * (e x + w * e y) / (1 - w^{2});

S y = E 1 * (w * e x + e y) / (1 - w^{2});

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

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

end;

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

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

For a two dimensional beam of dimensions $1 m \times 0.5 m$ and $t_{1} = 0.63$ we have obtained the following results:

Case A: For the optimal shape for a clamped beam at left (cantilever) and load $P = - 4 10^{6} N j$ at $(x, y) = (1, 0.25)$ , please Figure 5.
Case B :For the optimal shape for a simply supported beam at $(0, 0)$ and $(1, 0)$ and load $P = - 4 10^{6} N j$ at $(x, y) = (1 / 3, 0.5)$ , 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

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω

. Let

J : V \to R

be a functional where

J (u) = G (\nabla u_{1}, \dots, \nabla u_{N}) - {〈 u, f 〉}_{L^{2}},

where

V = W_{0}^{1, 2} (Ω; R^{N})

and

f = (f_{1}, \dots,