On Duality Principles and Concerned Convex Dual Formulations Applied to a Non-Linear Plate Theory and Related Models

1. Introduction

This article develops a duality principle applicable to a large class of models in the calculus of variations. Specifically in this text, we present applications to the non-linear Kirchhoff-Love plate model.

We emphasize the results on duality theory here addressed and developed are inspired mainly in the approaches of J.J.Telega, W.R. Bielski and co-workers presented in the articles [1,2,3,4]. Other main reference is the article by Toland,[5].

Moreover, details on the Sobolev spaces involved may be found in [6].

Similar results and models are addressed in [7,8,9,10,11].

Basic results on convex analysis are addressed in [12]. Other similar results and approaches may be found in [13,14,15].

Now we start to describe the primal variational formulation for the plate model in question.

Let $\mathsf{\Omega }\subset {\mathbb{R}}^2$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \mathsf{\Omega }.$

We assume such a $\mathsf{\Omega }$ set represents the middle surface of a thin plate with a constant thickness $h>0$.

Moreover, we suppose such a plate is subject to a external load $(P_{\alpha },P)\in L^2(\mathsf{\Omega };{\mathbb{R}}^3)$ resulting a field of displacements denoted by

$$(u_{\alpha },w)=(u_1,u_2,w)\in W_0^{1,2}(\mathsf{\Omega };{\mathbb{R}}^2)\times W_0^{2,2}\left(\mathsf{\Omega }\right)=V.$$

Both the load and displacements fields refers to a cartesian system $(0,x_1,x_2,x_3)$ and related canonical basis in ${\mathbb{R}}^3.$

Finally, we denote $Y_1=Y_1^{*}=L^2(\mathsf{\Omega };{\mathbb{R}}^4)$ and $Y_2=Y_2^{*}=L^2(\mathsf{\Omega };{\mathbb{R}}^2).$

We also emphasize the boundary conditions in question refer to a clamped plate.

The strain tensors are defined by

$${\gamma }_{\alpha \beta }\left(u\right)={\displaystyle \frac{u_{\alpha ,\beta }+u_{\beta ,\alpha }}{2}}+{\displaystyle \frac{1}{2}}{w}_{,\alpha }{w}_{,\beta },$$

and

$${\kappa }_{\alpha \beta }\left(w\right)=-w_{,\alpha \beta }.$$

The plate total energy functional is defined by

$$\begin{array}{ccc}\hfill J\left(u\right)& =& {\displaystyle \frac{1}{2}}{\int }_{\mathsf{\Omega }}{H}_{\alpha \beta \lambda \mu }{\gamma }_{\alpha \beta }\left(u\right){\gamma }_{\lambda \mu }\left(u\right)dx\hfill \\ & & +{\displaystyle \frac{1}{2}}{\int }_{\mathsf{\Omega }}{h}_{\alpha \beta \lambda \mu }{\kappa }_{\alpha \beta }\left(u\right){\kappa }_{\lambda \mu }\left(u\right)dx\hfill \\ & & -{\langle w,P\rangle }_{L^2}-{\langle u_{\alpha },P_{\alpha }\rangle }_{L^2}.\hfill \end{array}$$

(1)

Here $\left\{H_{\alpha \beta \lambda \mu }\right\}$ is a fourth order positive definite symmetric constant tensor.

Moreover

$$\left\{h_{\alpha \beta \lambda \mu }\right\}={\displaystyle \frac{h^2}{12}}\left\{H_{\alpha \beta \lambda \mu }\right\}$$

and we denote

$$\left\{{\overline{H}}_{\alpha \beta \lambda \mu }\right\}={\left\{H_{\alpha \beta \lambda \mu }\right\}}^{-1}$$

and

$$\left\{{\overline{h}}_{\alpha \beta \lambda \mu }\right\}={\left\{h_{\alpha \beta \lambda \mu }\right\}}^{-1}$$

in an appropriate tensor sense.

2. The Main Duality Principle and Related Convex Dual Approximate Formulation

We start by defining the approximate functional $J_1:V\to \mathbb{R}$ by

$$J_1\left(u\right)=J\left(u\right)+\sum _{\alpha =1}^2{\displaystyle \frac{{\epsilon }_1}{2}}{\int }_{\mathsf{\Omega }}{\left(u_{\alpha }\right)}^{2}dx,$$

and considering an appropriate real constant $K>0$, the functionals $F_1:V\to \mathbb{R}$, $F_2:V\times Y_1^{*}\to \mathbb{R}$, $F_3:V\to \mathbb{R}$ and $F_4:V\to \mathbb{R}$, by

$$\begin{array}{ccc}\hfill F_1\left(u\right)& =& {\displaystyle \frac{1}{2}}{\int }_{\mathsf{\Omega }}{h}_{\alpha \beta \lambda \mu }{\kappa }_{\alpha \beta }\left(u\right){\kappa }_{\lambda \mu }\left(u\right)dx+\sum _{\alpha =1}^2{\displaystyle \frac{K}{2}}{\int }_{\mathsf{\Omega }}{w}_{,\alpha }^{2}dx\hfill \\ & & +\sum _{\alpha =1}^2\epsilon {\int }_{\mathsf{\Omega }}{w}_{,\alpha }^{2}dx-{\langle w,P\rangle }_{L^2}\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill F_2(u,N)& =& {\displaystyle \frac{1}{2}}{\int }_{\mathsf{\Omega }}{N}_{\alpha \beta }{w}_{,\alpha }{w}_{,\beta }dx\hfill \\ & & -\sum _{\alpha =1}^2{\displaystyle \frac{K}{2}}{\int }_{\mathsf{\Omega }}{w}_{,\alpha }^{2}dx\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill F_3\left(u\right)& =& \sum _{\alpha =1}^2{\displaystyle \frac{{\epsilon }_1}{2}}{\int }_{\mathsf{\Omega }}{u}_{\alpha }^{2}dx\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill F_4\left(u\right)& =& \sum _{\alpha =1}^2\epsilon {\int }_{\mathsf{\Omega }}{w}_{,\alpha }^{2}dx.\hfill \end{array}$$

(5)

Moreover, we define the polar functionals $F_1^{*}:{\left[Y_2^{*}\right]}^2\to \mathbb{R}$, $F_2^{*}:{\left[Y_2^{*}\right]}^2\times Y_1^{*}\to \mathbb{R},$ $F_3^{*}:Y_1^{*}\to \mathbb{R}$ and $F_4^{*}:{\left[Y_2^{*}\right]}^2\to \mathbb{R}$, by

$$\begin{array}{ccc}\hfill F_1^{*}(R,L)& =& \underset{u\in V}{sup}\{{\langle w_{,\alpha },R_{\alpha }-L_{\alpha }\rangle }_{L^2}-F_1\left(u\right)\},\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill F_2^{*}(Q,L)& =& \underset{v\in Y_2}{inf}\left\{{\langle v_{\alpha },Q_{\alpha }+L_{\alpha }\rangle }_{L^2}-{\displaystyle \frac{1}{2}}{\int }_{\mathsf{\Omega }}{N}_{\alpha \beta }{v}_{\alpha }{v}_{\beta }dx\right.\hfill \\ & & \left.+\sum _{\alpha =1}^2{\displaystyle \frac{K}{2}}{\int }_{\mathsf{\Omega }}{v}_{\alpha }^{2}dx+{\displaystyle \frac{1}{2}}{\int }_{\mathsf{\Omega }}{\overline{H}}_{\alpha \beta \lambda \mu }{N}_{\alpha \beta }{N}_{\lambda \mu }dx\right\}\hfill \\ & =& {\displaystyle \frac{1}{2}}{\int }_{\mathsf{\Omega }}\overline{N_{\alpha \beta }^{(-K)}}(Q_{\alpha }+L_{\alpha })(Q_{\beta }+L_{\beta })dx\hfill \\ & & +{\displaystyle \frac{1}{2}}{\int }_{\mathsf{\Omega }}{\overline{H}}_{\alpha \beta \lambda \mu }{N}_{\alpha \beta }{N}_{\lambda \mu }dx,\hfill \end{array}$$

(7)

if $N=\left\{N_{\alpha \beta }\right\}\in B^{*}$, where

$$B^{*}=\{N\in Y_1^{*}:\parallel N_{\alpha \beta }{\parallel }_{\infty }\le K/8,\forall \alpha ,\beta \in \{1,2\}\},$$

$$\left\{N_{\alpha \beta }^{(-K)}\right\}=\{N_{\alpha \beta }-K{\delta }_{\alpha \beta }\},$$

and

$$\left\{\overline{N_{\alpha \beta }^{(-K)}}\right\}={\left\{N_{\alpha \beta }^{(-K)}\right\}}^{-1}.$$

Furthermore,

$$\begin{array}{ccc}\hfill F_3^{*}\left(N\right)& =& \underset{u\in V}{sup}\{{\langle u_{\alpha },N_{\alpha \beta ,\beta }+P_{\alpha }\rangle }_{L_2}-F_3\left(u\right)\}\hfill \\ & =& \sum _{\alpha =1}^2{\displaystyle \frac{1}{2{\epsilon }_1}}{\int }_{\mathsf{\Omega }}{(N_{\alpha \beta ,\beta }+P_{\alpha })}^{2}dx,\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill F_4^{*}(R,Q)& =& \underset{(v_1,v_2)\in Y_2^{*}}{sup}\left\{{\langle {\left(v_1\right)}_{\alpha },R_{\alpha }\rangle }_{L^2}-{\displaystyle \frac{\epsilon }{2}}{\int }_{\mathsf{\Omega }}{\left(v_1\right)}_{\alpha }^{2}dx\right.\hfill \\ & & \left.+{\langle {\left(v_2\right)}_{\alpha },Q_{\alpha }\rangle }_{L^2}-{\displaystyle \frac{\epsilon }{2}}{\int }_{\mathsf{\Omega }}{\left(v_2\right)}_{\alpha }^{2}dx\right\}\hfill \\ & =& \sum _{\alpha =1}^2\left({\displaystyle \frac{1}{2\epsilon }}{\int }_{\mathsf{\Omega }}{R}_{\alpha }^{2}dx+{\displaystyle \frac{1}{2\epsilon }}{\int }_{\mathsf{\Omega }}{Q}_{\alpha }^{2}dx\right).\hfill \end{array}$$

(9)

At this point, denoting

$$D^{*}=\{Q=\left\{Q_{\alpha }\right\}\in Y_2^{*}:\parallel Q_{\alpha }\parallel \le 5,\forall \alpha \in \{1,2\}\},$$

we define $J_1^{*}:{\left(D^{*}\right)}^2\times B^{*}\times Y_2^{*}\to \mathbb{R}$ by

$$J_1^{*}(R,Q,N,L)=-F_1^{*}(R,L)-F_2^{*}(Q,L,N)-F_3^{*}\left(N\right)+F_4^{*}(R,Q),$$

and $J_2^{*}:{\left(D^{*}\right)}^2\times B^{*}\to \mathbb{R}$ by

$$J_2^{*}(R,Q,N)={\mathrm{sta}}_{L\in Y_2^{*}}{J}_1^{*}(R,Q,N,L)=J_1^{*}(R,Q,N,\widehat{L}(R,Q,N)),$$

where $\widehat{L}=\widehat{L}(R,Q,N)\in Y_2^{*}$ is the only solution of the linear equation in L

$${\displaystyle \frac{\partial J_1^{*}(R,Q,N,\widehat{L})}{\partial L}}=\mathbf{0}.$$

Moreover, we define $J_3^{*}:{\left(D^{*}\right)}^2\times B^{*}\to \mathbb{R}$, by

$$\begin{array}{ccc}& & J_3^{*}(R,Q,N)\hfill \\ & =& J_2^{*}(R,Q,N)\hfill \\ & & -\sum _{\alpha =1}^2\sum _{\beta =1}^2{\displaystyle \frac{K_1}{2}}{∥{\overline{H}}_{\alpha \beta \lambda \mu }{N}_{\lambda \mu }-{\displaystyle \frac{{(N_{\alpha \rho ,\rho }+P_{\alpha })}_{,\beta }+{(N_{\beta \rho ,\rho }+P_{\alpha })}_{,\alpha }}{2{\epsilon }_1}}-{\displaystyle \frac{1}{2}}{\tilde{v}}_{\alpha }{\tilde{v}}_{\beta }∥}_{0,2}^2,\hfill \end{array}$$

(10)

where

$${\tilde{v}}_{\alpha }=\overline{N_{\alpha \beta }^{(-K)}}(Q_{\beta }+L_{\beta }(R,Q,N)),\forall \alpha \in \{1,2\}.$$

Here, we assume

$$K_1\gg max\{1,K,max\{{\overline{h}}_{\alpha \beta \lambda \mu },\alpha ,\beta ,\lambda ,\mu \in \{1,2\}\}\},$$

$$0<\epsilon ,{\epsilon }_1\ll 1$$

and

$${\displaystyle \frac{1}{\epsilon }}\gg max\left\{K_1,{\displaystyle \frac{1}{{\epsilon }_1}}\right\}.$$

Observe that

$${\displaystyle \frac{{\partial }^2{J}_3^{*}(R,Q,N)}{\partial Q_{\alpha }^2}}=\mathcal{O}\left({\displaystyle \frac{1}{\epsilon }}\right)>\mathbf{0},$$

$${\displaystyle \frac{{\partial }^2{J}_3^{*}(R,Q,N)}{\partial R_{\alpha }^2}}=\mathcal{O}\left({\displaystyle \frac{1}{\epsilon }}\right)>\mathbf{0},$$

$\forall \alpha \in \{1,2\}.$

Thus, considering also the remaining mixed variations in $Q_{\alpha }$ and $R_{\alpha }$, we may infer that

$$det\left\{{\displaystyle \frac{{\partial }^2{J}_3^{*}(R,Q,N)}{\partial Q_{\alpha }\partial R_{\beta }}}\right\}>\mathbf{0},$$

in ${\left(D^{*}\right)}^2\times B^{*}$.

Moreover, by direct computation, clearly

$${\displaystyle \frac{{\partial }^2{J}_3^{*}(R,Q,N)}{\partial {\left(N_{\alpha \beta }\right)}^2}}<\mathbf{0},\forall \alpha ,\beta \in \{1,2\}$$

and considering the remaining mixed variations of $J_3^{*}$ in $N_{11}$, $N_{22}$, $N_{12}$ and $N_{21}$ and the concerned remaining minor determinants, we may infer that

$$det\left\{{\displaystyle \frac{{\partial }^2{J}_3^{*}(R,Q,N)}{\partial N_{\alpha \beta }\partial N_{\lambda \mu }}}\right\}>\mathbf{0},$$

in ${\left(D^{*}\right)}^2\times B^{*}$.

From such results, we may also infer that $J_3^{*}$ is convex in $(R,Q)$ and concave in N in ${\left(D^{*}\right)}^2\times B^{*}$.

Let $(\widehat{R},\widehat{Q},\widehat{N},\widehat{L})\in {\left(D^{*}\right)}^2\times B^{*}\times Y_2^{*}$ be such that

$$\delta J_1^{*}(\widehat{R},\widehat{Q},\widehat{N},\widehat{L})=\mathbf{0}.$$

Let $u_0=({\left(u_0\right)}_{\alpha },w_0)\in V$ be such that

$${\left(u_0\right)}_{\alpha }={\displaystyle \frac{{\widehat{N}}_{\alpha \beta ,\beta }+P_{\alpha }}{{\epsilon }_1}},$$

and

$${\left(w_0\right)}_{,\alpha }={\displaystyle \frac{{\widehat{Q}}_{\alpha }}{\epsilon }},$$

$\forall \alpha \in \{1,2\}.$ From standard results in Duality Theory and the Legendre Transform properties, we may obtain

$$\delta J_1\left(u_0\right)=\mathbf{0},$$

$$\delta J_2^{*}(\widehat{R},\widehat{Q},\widehat{N})=\mathbf{0},$$

$$\delta J_3^{*}(\widehat{R},\widehat{Q},\widehat{N})=\mathbf{0}$$

and

$$\begin{array}{ccc}\hfill J_1\left(u_0\right)& =& J_1^{*}(\widehat{R},\widehat{Q},\widehat{N},\widehat{L})\hfill \\ & =& J_2^{*}(\widehat{R},\widehat{Q},\widehat{N})\hfill \\ & =& J_3^{*}(\widehat{R},\widehat{Q},\widehat{N}).\hfill \end{array}$$

(11)

From such results and the Min-Max Theorem, we have

$$J_3^{*}(\widehat{R},\widehat{Q},\widehat{N})=\underset{(R,Q)\in {\left(D^{*}\right)}^2}{inf}\left\{\underset{N\in B^{*}}{sup}{J}_3^{*}(R,Q,N)\right\}.$$

Joining the pieces, we have got

$$\begin{array}{ccc}\hfill J_1\left(u_0\right)& =& J_1^{*}(\widehat{R},\widehat{Q},\widehat{N},\widehat{L})\hfill \\ & =& J_2^{*}(\widehat{R},\widehat{Q},\widehat{N})\hfill \\ & =& \underset{(R,Q)\in {\left(D^{*}\right)}^2}{inf}\left\{\underset{N\in B^{*}}{sup}{J}_3^{*}(R,Q,N)\right\}\hfill \\ & =& J_3^{*}(\widehat{R},\widehat{Q},\widehat{N}).\hfill \end{array}$$

(12)

Remark 2.1.

Defining $J_5^{*}:{\left(D^{*}\right)}^2\to \mathbb{R}$ by

$$J_5^{*}(R,Q)=\underset{N\in B^{*}}{sup}{J}_3^{*}(R,Q,N),$$

we have that such a functional $J_5^{*}$ is convex in ${\left(D^{*}\right)}^2$ as the supremum in $N\in B^{*}$ of a family of convex functionals in $(R,Q).$

In such a case, we have also obtained

$$\begin{array}{ccc}\hfill J_1\left(u_0\right)& =& J_1^{*}(\widehat{R},\widehat{Q},\widehat{N},\widehat{L})\hfill \\ & =& J_2^{*}(\widehat{R},\widehat{Q},\widehat{N})\hfill \\ & =& \underset{(R,Q)\in {\left(D^{*}\right)}^2}{inf}\left\{\underset{N\in B^{*}}{sup}{J}_3^{*}(R,Q,N)\right\}\hfill \\ & =& J_3^{*}(\widehat{R},\widehat{Q},\widehat{N})\hfill \\ & =& \underset{(R,Q)\in {\left(D^{*}\right)}^2}{inf}{J}_5^{*}(R,Q)\hfill \\ & =& J_5^{*}(\widehat{R},\widehat{Q}).\hfill \end{array}$$

(13)

3. One More Duality Principle and Related Convex Dual Formulation

In this section we develop another new duality principle with a related convex dual functional applied to a Ginzburg-Landau type equation.

Let $\mathsf{\Omega }\subset {\mathbb{R}}^3$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \mathsf{\Omega }.$

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

$$\begin{array}{ccc}\hfill J\left(u\right)& =& {\displaystyle \frac{\gamma }{2}}{\int }_{\mathsf{\Omega }}\nabla u·\nabla udx\hfill \\ & & +{\displaystyle \frac{\alpha }{2}}{\int }_{\mathsf{\Omega }}{(u^2-\beta )}^{2}dx-{\langle u,f\rangle }_{L^2},\hfill \end{array}$$

(14)

where $\gamma >0$, $\alpha >0,$ $\beta >0$ and $f\in L^2\left(\mathsf{\Omega }\right).$

Here $u\in V=W_0^{1,2}\left(\mathsf{\Omega }\right)$ and we denote $Y=Y^{*}=L^2\left(\mathsf{\Omega }\right).$

Define the functionals $F_1:V\times Y^{*}\to \mathbb{R}$, $F_2:V\times Y^{*}\to \mathbb{R}$ and $F_3:V\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill F_1(u,v_0^{*})& =& {\displaystyle \frac{\gamma }{4}}{\int }_{\mathsf{\Omega }}\nabla u·\nabla udx+{\displaystyle \frac{1}{2}}{\langle u^2,v_0^{*}\rangle }_{L^2}\hfill \\ & & +{\displaystyle \frac{K}{2}}{\int }_{\mathsf{\Omega }}{u}^{2}dx,\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill F_2(u,v_0^{*})& =& {\displaystyle \frac{\gamma }{4}}{\int }_{\mathsf{\Omega }}\nabla u·\nabla udx+{\displaystyle \frac{1}{2}}{\langle u^2,v_0^{*}\rangle }_{L^2}\hfill \\ & & +{\displaystyle \frac{K}{2}}{\int }_{\mathsf{\Omega }}{u}^{2}dx-{\langle u,f\rangle }_{L^2}\hfill \\ & & -{\displaystyle \frac{1}{2\alpha }}{\int }_{\mathsf{\Omega }}{\left(v_0^{*}\right)}^{2}dx-\beta {\int }_{\mathsf{\Omega }}{v}_0^{*}dx,\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill F_3\left(u\right)& =& K{\int }_{\mathsf{\Omega }}{u}^{2}dx,\hfill \end{array}$$

(17)

where $K>0$ is an appropriate real constant.

Define also the polar functionals $F_1^{*}:{\left[Y^{*}\right]}^3\to \mathbb{R}$, $F_2^{*}:{\left[Y^{*}\right]}^3\to \mathbb{R}$ and $F_3^{*}:Y^{*}\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill F_1^{*}(v_1^{*},v_0^{*},z^{*})& =& \underset{u\in V}{sup}\{{\langle u,v_1^{*}+z^{*}/2\rangle }_{L^2}-F_1(u,v_0^{*})\}\hfill \\ & =& {\displaystyle \frac{1}{2}}{\int }_{\mathsf{\Omega }}{\displaystyle \frac{{(v_1^{*}+z^{*}/2)}^2}{((-\gamma {\nabla }^2+2v_0^{*})/2+K)}}dx\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill F_2^{*}(v_1^{*},v_0^{*},z^{*})& =& \underset{u\in V}{sup}\{{\langle u,-v_1^{*}+z^{*}/2\rangle }_{L^2}-F_2(u,v_0^{*})\}\hfill \\ & =& {\displaystyle \frac{1}{2}}{\int }_{\mathsf{\Omega }}{\displaystyle \frac{{(-v_1^{*}+z^{*}/2+f)}^2}{((-\gamma {\nabla }^2+2v_0^{*})/2+K)}}dx\hfill \\ & & +{\displaystyle \frac{1}{2\alpha }}{\int }_{\mathsf{\Omega }}{\left(v_0^{*}\right)}^{2}dx+\beta {\int }_{\mathsf{\Omega }}{v}_0^{*}dx,\hfill \end{array}$$

(19)

if $v_0^{*}\in B^{*}$, where

$$B^{*}=\{v_0^{*}\in Y^{*}:\parallel 2v_0^{*}\parallel \le K/4\},$$

$$\begin{array}{ccc}\hfill F_3^{*}\left(z^{*}\right)& =& \underset{w\in L^2}{sup}\{{\langle w,z^{*}\rangle }_{L^2}-F_3\left(w\right)\}\hfill \\ & =& {\displaystyle \frac{1}{4K}}{\int }_{\mathsf{\Omega }}{\left(z^{*}\right)}^{2}dx.\hfill \end{array}$$

(20)

Moreover, define

$$D^{*}=\{v_1^{*}\in Y^{*}:\parallel 2v_1^{*}-f{\parallel }_{\infty }\le 5\},$$

$$D_1^{+}=\{z^{*}\in Y^{*}:z^{*}f\ge 0,\mathrm{in}\mathsf{\Omega }\},$$

and

$$D_1^{*}=\{z^{*}\in D^{+}:\parallel z^{*}{\parallel }_{\infty }\le (5/2)K\}.$$

Assuming $K_1\gg max\{\gamma ,\alpha ,\beta ,1/\alpha ,1,K\}$ and

$$K\gg \alpha ,$$

define $J^{*}:D^{*}\times B^{*}\times D_1^{*}\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill J^{*}(v_1^{*},v_0^{*},z^{*})& =& -F_1^{*}(v_1^{*},v_0^{*},z^{*})-F_2^{*}(v_1^{*},v_0^{*},z^{*})+F_3^{*}\left(z^{*}\right).\hfill \end{array}$$

(21)

From the variation of $J^{*}$ in $v_1^{*}$, we have

$$-{\displaystyle \frac{v_1^{*}+z^{*}/2}{((-\gamma {\nabla }^2+2v_0^{*})/2+K)}}+{\displaystyle \frac{-v_1^{*}+z^{*}/2+f}{((-\gamma {\nabla }^2+2v_0^{*})/2+K)}}=0,\mathrm{in}\mathsf{\Omega },$$

so that

$$-2v_1^{*}+f=0,\mathrm{in}\mathsf{\Omega }.$$

Moreover, denoting

$$u_0={\displaystyle \frac{v_1^{*}+z^{*}/2}{((-\gamma {\nabla }^2+2v_0^{*})/2+K)}},$$

from the variation of $J^{*}$ in $z^{*}$, we obtain

$$u_0={\displaystyle \frac{z^{*}}{2K}}.$$

From such results, we may also obtain

$$\delta J\left(u_0\right)=-\gamma {\nabla }^2{u}_0+2\alpha (u_0^2-\beta )u_0-f=\mathbf{0},$$

so that

$$-\gamma {\nabla }^2\left({\displaystyle \frac{z^{*}}{2K}}\right)+2\alpha \left({\left({\displaystyle \frac{v_1^{*}+z^{*}/2}{(-\gamma {\nabla }^2+2v_0^{*})/2+K}}\right)}^2-\beta \right)\left({\displaystyle \frac{z^{*}}{2K}}\right)-f=\mathbf{0}$$

in $\mathsf{\Omega }.$

With such results in mind, we define also the exactly penalized functional $J_1^{*}:D^{*}\times B^{*}\times D_1^{*}\to \mathbb{R}$, where

$$\begin{array}{ccc}& & J_1^{*}(v_1^{*},v_0^{*},z^{*})\hfill \\ & =& J^{*}(v_1^{*},v_0^{*},z^{*})-{\displaystyle \frac{K_1}{2}}{\parallel 2v_1^{*}-f\parallel }_{0,2}^2\hfill \\ & & +30{\displaystyle \frac{K}{2}}{∥-\gamma {\nabla }^2\left({\displaystyle \frac{z^{*}}{2K}}\right)+2\alpha \left({\left({\displaystyle \frac{v_1^{*}+z^{*}/2}{(-\gamma {\nabla }^2+2v_0^{*})/2+K}}\right)}^2-\beta \right)\left({\displaystyle \frac{z^{*}}{2K}}\right)-f∥}_{0,2}^2\hfill \end{array}$$

(22)

Clearly, we have

$${\displaystyle \frac{{\partial }^2{J}_1^{*}(v_1^{*},v_2^{*},v_0^{*})}{\partial {\left(v_1^{*}\right)}^2}}=-\mathcal{O}\left(K_1\right)<\mathbf{0},$$

and

$${\displaystyle \frac{{\partial }^2{J}_1^{*}(v_1^{*},v_2^{*},v_0^{*})}{\partial {\left(v_0^{*}\right)}^2}}=-\mathcal{O}(1/\alpha )<\mathbf{0},$$

so that considering also the mixed variations of $J_1^{*}$ in $v_1^{*}$ and $v_0^{*}$, we may infer that

$$det\left\{{\displaystyle \frac{{\partial }^2{J}_1^{*}(v_1^{*},v_0^{*},z^{*})}{\partial v_1^{*}\partial v_0^{*}}}\right\}>\mathbf{0},\mathrm{in}{D}^{*}\times B^{*}\times D_1^{*}.$$

Hence, $J_1^{*}$ is concave in $(v_1^{*},v_0^{*})$ in $D^{*}\times B^{*}\times D_1.$

Furthermore,

$${\displaystyle \frac{{\partial }^2{J}_2^{*}(v_1^{*},v_0^{*},z^{*})}{\partial {\left(z^{*}\right)}^2}}=\mathcal{O}(30/\left(4K\right))>\mathbf{0},$$

in $D^{*}\times B^{*}\times D_1^{*}$, so that $J_1^{*}$ is convex in $z^{*}$ in $D^{*}\times B^{*}\times D_1^{*}$.

Let $({\widehat{v}}_1^{*},{\widehat{v}}_0^{*},{\widehat{z}}^{*})\in D^{*}\times B^{*}\times D_1^{*}$ be such that

$$\delta J_1^{*}({\widehat{v}}_1^{*},{\widehat{v}}_0^{*},{\widehat{z}}^{*})=\mathbf{0}.$$

From this, the last previous results and the Min-Max theorem we may infer that

$$J_1^{*}({\widehat{v}}_1^{*},{\widehat{v}}_0^{*},{\widehat{z}}^{*})=\underset{z^{*}\in D_1^{*}}{inf}\left\{\underset{(v_1^{*},v_0^{*})\in D^{*}\times B^{*}}{sup}{J}_1^{*}(v_1^{*},v_0^{*},z^{*})\right\}.$$

Let $u_0\in V$ be such that

$$u_0={\displaystyle \frac{{\widehat{z}}^{*}}{2K}}.$$

From fundamentals of duality theory and the Legendre Transform properties, we may obtain

$$\delta J\left(u_0\right)=\mathbf{0},$$

and

$$J\left(u_0\right)=J_1^{*}({\widehat{v}}_1^{*},{\widehat{v}}_0^{*},{\widehat{z}}^{*}).$$

Joining the pieces, we have got

$$\begin{array}{ccc}\hfill J\left(u_0\right)& =& J_1^{*}({\widehat{v}}_1^{*},{\widehat{v}}_0^{*},{\widehat{z}}^{*})\hfill \\ & =& \underset{z^{*}\in D_1^{*}}{inf}\left\{\underset{(v_1^{*},v_0^{*})\in D^{*}\times B^{*}}{sup}{J}_1^{*}(v_1^{*},v_0^{*},z^{*})\right\}.\hfill \end{array}$$

(23)

Remark 3.1.

Defining $J_3^{*}:D_1^{*}\to \mathbb{R}$ by

$$J_3^{*}\left(z^{*}\right)=\underset{(v_1^{*},v_0^{*})\in D^{*}\times B^{*}}{sup}{J}_1^{*}(v_1^{*},v_0^{*},z^{*}),$$

we have that $J_3^{*}$ is convex in $D_1^{*}$ as a supremum of a family of convex functionals in $z^{*}$.

In such case, we have

$$\begin{array}{ccc}\hfill J\left(u_0\right)& =& J_1^{*}({\widehat{v}}_1^{*},{\widehat{v}}_0^{*},{\widehat{z}}^{*})\hfill \\ & =& \underset{z^{*}\in D_1^{*}}{inf}\left\{\underset{(v_1^{*},v_0^{*})\in D^{*}\times B^{*}}{sup}{J}_1^{*}(v_1^{*},v_0^{*},z^{*})\right\}.\hfill \\ & =& \underset{z^{*}\in D_1^{*}}{inf}{J}_3^{*}\left(z^{*}\right)\hfill \\ & =& J_3^{*}\left({\widehat{z}}^{*}\right).\hfill \end{array}$$

(24)

The objective of this section is complete.

4. An Approximate Procedure for Improving the Convexity Conditions for an Originally Non-Convex Primal Formulation

In this section we obtain an approximate procedure for improving the convexity conditions for an originally non-convex variational formulation.

In this new version we present some corrections and improvements concerning the previous one.

Let $\mathsf{\Omega }\subset {\mathbb{R}}^3$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \mathsf{\Omega }.$

Let $V=W_0^{1,2}\left(\mathsf{\Omega }\right)$ and consider a continuously twice Fréchet differentiable functional $J:V\to \mathbb{R}$ such that

$$-50I_d\le {\displaystyle \frac{{\partial }^{2}J\left(u\right)}{\partial u^2}}\le 50I_d,$$

in $V_1,$ where the set $V_1$ will be specified in the next lines.

Let $K=5000002\pi$ and $K_3=1.0$.

Define

$$V_1=\{u\in V:0\le u\le K_3\}.$$

and the functional $J_2:V_1\to \mathbb{R}$ by

$$J_2\left(u\right)=-{\displaystyle \frac{{10}^{16}}{K^3}}{\int }_{\mathsf{\Omega }}cos\left({\displaystyle \frac{K^4}{u/{10}^{10}+K}}\right)dx.$$

Define also the functional $J_1:V_1\to \mathbb{R}$ by

$$J_1\left(u\right)=J\left(u\right)+J_2\left(u\right).$$

Observe that, with a help of the software MAT-LAB, we may obtain

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial J_1\left(u\right)}{\partial u}}& =& {\displaystyle \frac{\partial J\left(u\right)}{\partial u}}+{\displaystyle \frac{\partial J_2\left(u\right)}{\partial u}}\hfill \\ & =& {\displaystyle \frac{\partial J\left(u\right)}{\partial u}}+\mathcal{O}\left(0.3\right).\hfill \end{array}$$

(25)

Moreover,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{{\partial }^2{J}_1\left(u\right)}{\partial u^2}}& =& {\displaystyle \frac{{\partial }^{2}J\left(u\right)}{\partial u^2}}+{\displaystyle \frac{{\partial }^2{J}_2\left(u\right)}{\partial u^2}}\hfill \\ & & \hfill \\ & =& {\displaystyle \frac{{\partial }^{2}J\left(u\right)}{\partial u^2}}+\mathcal{O}\left(116\right)\hfill \\ & >& \mathbf{0},\mathrm{in}{V}_1.\hfill \end{array}$$

(26)

Thus, such an approximate functional $J_1$ is convex in $V_1$ and its first variation results, for a large class of appropriate numerical parameters, in a very close approximation for the first variation of the original functional $J.$

The functions

$${\displaystyle \frac{\partial J_2\left(x\right)}{\partial x}}$$

and

$${\displaystyle \frac{{\partial }^2{J}_2\left(x\right)}{\partial x^2}}$$

on the interval $[0,K_3=1]$ stands for

$${\displaystyle \frac{\partial J_2\left(x\right)}{\partial x}}={\displaystyle \frac{{10}^{16}}{K^3}}sin\left({\displaystyle \frac{K^4}{x/{10}^{10}+K}}\right)\left({\displaystyle \frac{-K^4}{{(x/{10}^{10}+K)}^2}}\right)\left({\displaystyle \frac{1}{{10}^{10}}}\right),$$

$$\begin{array}{ccc}\hfill {\displaystyle \frac{{\partial }^2{J}_2\left(x\right)}{\partial x^2}}& =& {\displaystyle \frac{{10}^{16}}{K^3}}cos\left({\displaystyle \frac{K^4}{x/{10}^{10}+K}}\right){\left({\displaystyle \frac{-K^4}{{(x/{10}^{10}+K)}^2}}\right)}^2{\left({\displaystyle \frac{1}{{10}^{10}}}\right)}^2\hfill \\ & & +2{\displaystyle \frac{{10}^{16}}{K^3}}sin\left({\displaystyle \frac{K^4}{x/{10}^{10}+K}}\right)\left({\displaystyle \frac{K^4}{{(x/{10}^{10}+K)}^3}}\right)\left({\displaystyle \frac{1}{{10}^{10}}}\right),\hfill \end{array}$$

(27)

respectively.

For their graphs, please see Figure 1 and Figure 2, repectively.

Remark 4.1.

The reason the graphs of $J_2^{\prime }\left(x\right)$ and $J_2^{″}\left(x\right)$ are straight lines is because ${\displaystyle \frac{x}{{10}^{10}}}$ varies very little on the interval $[0,1].$

The objective of this section is complete.

5. Conclusion

In this article, we have developed duality principles and related convex dual variational formulations for originally non-convex primal ones.

We highlight the results here obtained are applicable to a large class of models in the calculus of variations, including other plate and shell non-linear theories, models in superconductivity, phase transition and micro-magnetism, among many others.

In a near future research we intend to apply such results to some of these mentioned related models.