\documentclass[twoside]{article} \usepackage{amssymb,amsmath,epic,eepic} \pagestyle{myheadings} \markboth{\hfil Optimal control and homogenization \hfil EJDE--2002/27} {EJDE--2002/27\hfil Hakima Zoubairi \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2002}(2002), No. 27, pp. 1--32. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Optimal control and homogenization in a mixture of fluids separated by a rapidly oscillating interface % \thanks{ {\em Mathematics Subject Classifications:} 35B27, 35B37, 49J20, 76D07. \hfil\break\indent {\em Key words:} Optimal control, homogenization, Stokes equations. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Submitted January 23, 2002. Published March 5, 2002.} } \date{} % \author{Hakima Zoubairi} \maketitle \begin{abstract} We study the limiting behaviour of the solution to optimal control problems in a mathematical mixture of two homogeneous viscous fluids. These fluids are separated by a rapidly oscillating periodic interface with constant amplitude. We show that the limit of the optimal control is the optimal control for the limiting problem \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode`@=11 \@addtoreset{equation}{section} \catcode`@=12 \allowdisplaybreaks \section{Introduction} The aim of this paper is to study the optimal control problem in a mixture of fluids. More precisely, we consider a mixture of two viscous, homogeneous and incompressible fluids occupying sub-domains of a bounded domain $\Omega \subset \mathbb{R}^{n+1}$ ($n=1$ or 2). These fluids are separated by a given interface whose form is determined using a rapidly oscillating function of period $\varepsilon>0$ and constant amplitude $h_1>0$. We assume that the velocity and pressure of both fluids satisfy the Stokes equations. On the interface, we assume that the velocity is continuous and that the normal forces that the fluids exert each other are equal in magnitude and opposite in direction (hence, surface tension effects are neglected). We associate an optimal control problem to these equations and our aim is to study the limiting behaviour of the solutions when the oscillating period $\varepsilon $ tends to $0$. To do so, we use some homogenization tools (see Bensoussan-Lions-Papanicolaou \cite{blp} and Sanchez-Palencia \cite{sp}) and Murat's compactness result \cite{mu}. This work is based on the mathematical framework of Baffico \& Conca \cite{bc3} for the Stokes problem, of Brizzi \cite{br} for the transmission problem, and of Baffico \& Conca \cite{bc1,bc2} for the transmission problem in elasticity. The plan of this paper is as follows. In Section 2, we present the domain with the rapidly oscillating interface, the Stokes problem posed in this domain and the definition of the associated optimal control problem. In Section 3, we introduce the adjoint problem and present related convergence results. In Section 4, we prove the results announced in the previous section. In Section 5, we sketch the proof of the convergence results concerning the optimal control. In Section 6, we present the case $\Omega \subset \mathbb{R}^2$. \section{Setting of the problem} For $n=1$ or $2$, let $Y=]0,1[^n$ and $\widetilde{\Omega}=]0,L_i[^n$ with $L_i>0$, $i=1,\dots, n$. Let $h: \bar {Y} \to \mathbb{R}, h\geq 0$ be a smooth function such that \begin{enumerate} \item[i)] $h |_{\partial Y}= h_1 $ where $ h_1= \max \{ h(y) : y \in \bar {Y}\}$ and $h_1>0$. \item[ii)] Exists $y_0 \in Y$ such that $h(y_0)=0$ and $\nabla_y h(y_0)=0$. \end{enumerate} For $z_o \in \mathbb{R}^+$, define $\Omega \subset \mathbb{R}^{n+1}$ by $\Omega= \widetilde{\Omega}\times ]-z_0,z_0[$ and $\Gamma$ the boundary $\Omega$ by $\Gamma= \widetilde{\Omega}\times \{-z_0 \}\cup \partial \widetilde{\Omega}\times ]-z_0,z_0[ \cup \widetilde{\Omega}\times \{z_0 \}$. To define the reference cell, we introduce the sub-domains: \begin{gather*} \Omega_1^1=\{ (y,z) \in Y\times \mathbb{R} : h(y) < z z\},\quad O(z)= \{ y \in Y : h(y) < z\},\\ \gamma (z)= \{ y \in Y : h(y) = z\}. \end{gather*} Let $\varepsilon >0$ be a small positive parameter. We extend $h$ by $Y$-periodicity to $\mathbb{R}^n$, we restrict this function to $\widetilde{\Omega}$ (this function is still denoted by $h$). Let $$ h^\varepsilon(x) = h( {x \over \varepsilon}) \quad x \in \widetilde{\Omega}. $$ Now we introduce \begin{gather*} \Omega_1^\varepsilon=\{ (x,z) \in \widetilde{\Omega}\times \mathbb{R} : h^\varepsilon(x) < z0$, $\mu_1 \not= \mu_2$, and $\chi_{\Omega_i^\varepsilon}$ correspond to the characteristic functions of $\Omega_i^\varepsilon$ $(i=1,2)$. We denote by $ \vec {v}=(\underline{v},v_{n+1})$, a vector of $\mathbb{R}^{n+1}$. Throughout this paper, $C$ denotes various real positive constants independent of $\varepsilon$. We also denote by $|\cdot |$ the $n$-dimensional Lebesgue measure and by $(\underline{e}_k)_{1\leq k\leq n}$ the canonical basis of $ \mathbb{R}^n$ ($y_k$ is the $k$-th component of $y \in \mathbb{R}^n$ in this basis). We define the optimal control as follows. Let $ \mathcal{U}_{ad}\subset L^2(\Omega)^{n+1}$ be a closed non empty convex subset. Let $N>0$ be a given constant. For $\vec{\theta} \in \mathcal{U}_{ad}$ the state equation is given by the Stokes problem \begin{equation}\begin{gathered} -\mathop{\rm div}(2 \mu^\varepsilon e(\vec {u}^\varepsilon)) =\vec {f}^\varepsilon-\nabla p^\varepsilon +\vec{\theta} \quad\mbox{in } \Omega \\ \mathop{\rm div} \vec{ u}^\varepsilon =0 \quad\mbox{in } \Omega\\ \vec{ u}^\varepsilon =0 \quad\mbox{on }\partial\Omega. \end{gathered}\label{2.2.1} \end{equation} where $(\vec{ u}^\varepsilon, p^\varepsilon)$ are respectively the velocity and the pressure of the fluid, $\vec{\theta}$ is the control and $\vec {f}^\varepsilon$ is a density of external forces defined by $\vec {f}^\varepsilon=\vec {f}^1\chi_{\Omega^\varepsilon_1} +\vec {f}^2\chi_{\Omega^\varepsilon_2}$ with $\vec {f}^i \in L^2(\Omega)^{n+1} (i=1,2)$. The rate-of-strain tensor $e(\vec {u}^\varepsilon)$ is $$ e(\vec {u}^\varepsilon)= \frac 12 \big( \nabla \vec{ u}^\varepsilon + ^t \nabla \vec{ u}^\varepsilon \big). $$ The cost function is \begin{equation} J_\varepsilon (\vec{\theta})=\frac 12\int_\Omega e(\vec {u}^\varepsilon) : e(\vec {u}^\varepsilon) dx + {N\over 2}\int_\Omega |\vec{\theta}|^2 dx. \label{2.2.2} \end{equation} The optimal control $\vec{\theta}_\star^\varepsilon$ is the function in $\mathcal{U}_{ad} $ which minimizes $ J_\varepsilon (\vec{\theta} )$ for $ \vec{\theta}\in \mathcal{U}_{ad}$, in other words \begin{equation} J_\varepsilon (\vec{\theta}_\star^\varepsilon ) =\min_{\vec{\theta} \in \mathcal{U}_{ad}} J_\varepsilon (\vec{\theta} ). \label{2.2.3} \end{equation} This problem is standard and admits an unique optimal solution $\vec{\theta}_\star^\varepsilon \in \mathcal{U}_{ad}$ (see Lions \cite{li}). Our aim is to study the limiting behaviour of $\vec{\theta}_\star^\varepsilon $ as $\varepsilon \to 0$. In particular, it can be shown that (for a subsequence) $$ \vec{\theta}_\star^\varepsilon \rightharpoonup \vec{\theta}_\star\quad \mbox{weakly in } L^2(\Omega)^{n+1}. %\label{2.2.4} $$ Our objective is to characterize $ \vec{\theta}_\star$ as the optimal control of a similar problem with limiting tensors $\mathcal{A}$ and $\mathcal{B}$ and to identify these tensors. The homogenization of the problem (\ref{2.2.1}) is thanks to Baffico \& Conca \cite{bc3}. About the optimal control, Saint Jean Paulin \& Zoubairi \cite{sz1} studied the problem of a mixture of two fluids periodically distributed one in the other. Also this type of problem has been studied by Kesavan \& Vanninathan \cite{kv} in the periodic case for a problem which the state equation is a second order elliptic problem with rapidly oscillating coefficients and by Kesavan \& Saint Jean Paulin \cite{ks1} and \cite{ks2} in the general case (with H-convergence). In this paper, we adapt these methods to the Stokes problem following the technique used by Baffico \& Conca \cite{bc2} and \cite{bc3}, by Kesavan \& Saint Jean Paulin \cite{ks1} and \cite{ks2}, and by Saint Jean Paulin \& Zoubairi \cite{sz1}. \paragraph{Definition} % 2.2.1 We define $ L_0^2 (\Omega)$ by $$ L_0^2 (\Omega)= \big \{ g \in L^2 (\Omega) : \int_\Omega g(y) dy=0 \big \}. $$ The problem (\ref{2.2.1})--(\ref{2.2.3}) can be reduced to a system of equations by introducing the adjoint state $(\vec {v}^\varepsilon,p'^\varepsilon)\in H^1 (\Omega)^{n+1} \times L_0 ^2 (\Omega)$. Thus we get \begin{equation} \begin{gathered} -\mathop{\rm div}(2 \mu^\varepsilon e(\vec {u}^\varepsilon)) =\vec {f}^\varepsilon-\nabla p^\varepsilon +\vec{\theta} \quad\mbox{in } \Omega \\ \mathop{\rm div}(2 \mu^\varepsilon e(\vec {v}^\varepsilon) -e(\vec{ u}^\varepsilon)) =-\nabla p'^\varepsilon \quad\mbox{in } \Omega \\ \mathop{\rm div} \vec{ u}^\varepsilon =\mathop{\rm div} \vec{v}^\varepsilon= 0 \quad\mbox{in } \Omega\\ \vec{ u}^\varepsilon= \vec{v}^\varepsilon = 0 \quad\mbox{on } \partial\Omega, \end{gathered} \label{2.2.5} \end{equation} the optimal control $\vec{\theta}_\star^\varepsilon$ is characterized by the variationnal inequality $$\vec{\theta}_\star^\varepsilon \in \mathcal{U}_{ad} \mbox{ and } \int_ \Omega \ (\vec{v}^\varepsilon +N\vec{\theta}_\star^\varepsilon) .(\vec{\theta} - \vec{\theta}_\star^\varepsilon) dx\geq 0 \forall \vec{\theta} \in \mathcal{U}_{ad}. %\label{2.2.6} $$ \section{Convergence results} \subsection*{The homogenized adjoint problem} Let $\mu=\mu(y,z)$ (where $ y \in Y$ and $z\in ]0,h_1[$), be the variable viscosity given by $$ \mu(y,z)= \mu_1 \chi_{O(z)}(y) + \mu_2 \chi_{Y^\star (z)}(y). $$ Let us introduce some functions as solution of the Stokes problem defined on $Y$. These functions, introduced by Baffico \& Conca \cite{bc3}, are associated to the state equation. Let $1\leq k,l\leq n$, and let $(\underline {\chi}^{kl}, r_1^{kl})$ be the solution of \begin{equation} \begin{gathered} -\mathop{\rm div}_y (2 \mu e_y(-\underline {\chi}^{kl}+\underline {P}^{kl} )) =-\nabla_y r_1^{kl} \quad\mbox{in } Y \\ \mathop{\rm div}_y \underline {\chi}^{kl} = 0 \quad\mbox{in }Y \\ \underline {\chi}^{kl}, r_1^{kl}\quad Y\mbox{-periodic, } \end{gathered} \label{3.1.1} \end{equation} where $ \underline {P}^{kl}=\frac 12 \big(y_k \underline {e}_l + y_l\underline {e}_k\big) $. We define $M^{kl}$ the $n\times n$ matrix by $M^{kl}= e_y(\underline {P}^{kl})$. We notice that $[M^{kl}]_{ij}=\frac 12 (\delta_{ik}\delta_{jl}+\delta_{il} \delta_{jk})$ for all $1\leq i,j,k,l\leq n$. We also define the $n+1 \times n+1$ matrix $$ [E^{kl}]_{ij}=\frac 12 (\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}) \quad \forall 1\leq i,j,k,l\leq n+1. $$ For each $z\in ]0,h_1[$, problem (\ref{3.1.1}) admits an unique solution in $\big(H^1_\sharp(Y)^n / \mathbb{R} \big)\times L^2_0(Y)$ (see Sanchez-Palencia \cite{sp} or Baffico \& Conca \cite{co}). Now, we consider the periodic problem \begin{equation} \begin{gathered} -\mathop{\rm div}_y (\mu \nabla_y(-\varphi^k+2y_k )) =0 \quad\mbox{in } Y \\ \varphi^k \quad Y\mbox{-periodic. } \end{gathered} \label{3.1.2} \end{equation} For each $z\in ]0,h_1[$ fixed, problem (\ref{3.1.2}) has an unique solution in $H^1(Y)$ up to an additive constant. Let $\mathcal{A} (z)$ be the fourth-order tensor whose coefficients are defined by \begin{equation} a_{ijkl}= \begin{cases} 2\mu_1 [E^{kl}]_{ij} & h_10$ such that for all $\xi$, $n+1 \times n+1$ symmetric matrix, $$ \mathcal{A}(z)\xi : \xi \geq \alpha \xi : \xi \quad \forall z \in ]-z_0, z_0[. $$ \end{proposition} Now we give a result concerning some symmetry and ellipticity properties of the tensor $\mathcal{B}$. \begin{proposition} \label{prop3.2} The coefficients of $\mathcal{B}$ (voir $(\ref{3.1.7})$) are such that: \\ a) $b_{ijkl}(z)=b_{klij}(z)=b_{ijlk}(z)$ for $1\leq i,j,k,l\leq n+1$, for all $z \in ]-z_0, z_0[$ \\ b) There exists $\beta >0$ such that for all $\xi$, the $n+1 \times n+1$ symmetric matrix, $$ \mathcal{B}(z)\xi : \xi \geq \beta \xi : \xi \quad \forall z \in ]-z_0, z_0[. $$ \end{proposition} \paragraph{Proof.} Throughout this proof, we adopt the convention of summation over repeated indices. To prove a), we first study the coefficients of tensor $\mathcal{B}$ with indexes $1\leq i,j,k,l\leq n$. The symmetry of these coefficients is evident when $z \in ] h_1,z_0[$ and $z \in ]-z_0, h_1[$. Let study the case where $z \in ]0, h_1[.$ In this case (cf (\ref{3.1.8})) \begin{equation} b_{ijkl}=\widetilde{b_{ijkl}} = \int_Y \big(2\mu \big[e_y (\underline {\psi}^{kl})\big]_{ij} + \big[e_y ( -\underline {\chi}^{kl}+\underline {P}^{kl} )\big]_{ij}\big) dy. \label{3.1.9} \end{equation} Following the ideas in \cite{ks1,sz1,sz3}, we transform the above expression to obtain a symmetric form. Let $(\underline {Y}^{kl}, r_3^{kl})$ be the solution of \begin{equation} \begin{gathered} -\mathop{\rm div}_y (2 \mu e_y(-\underline {Y}^{kl}+\underline {P}^{kl} )) =-\nabla_y r_3^{kl} \quad\mbox{in } Y \\ \mathop{\rm div}_y \underline {Y}^{kl} = 0 \quad\mbox{in }Y \\ \underline {Y}^{kl}, r_1^{kl}\quad Y\mbox{-periodic, } \end{gathered} \label{3.1.10} \end{equation} Introducing $\underline {Y}^{kl}$ the solution of the previous local problem, the coefficients (\ref{3.1.9}) can be rewritten as \begin{equation} \widetilde{b_{ijkl}} = \int_Y \big[e_y ( -\underline {Y}^{kl}+\underline {P}^{kl} )\big]_{ij} dy + \int_Y \big(2\mu\big[e_y (\underline {\psi}^{kl})\big]_{ij} - \big[e_y ( \underline {\chi}^{kl}-\underline {Y}^{kl} )\big]_{ij}\big) dy . \label{3.1.11} \end{equation} The first term of the second integral of the right-hand side of this equation is evaluated as follows (using the fact that $[e_y (\underline {\psi}^{kl})\big]_{ij}=[e_y (\underline {\psi}^{kl})\big]_{ji}$) \begin{align*} \int_Y 2\mu \big[e_y (\underline {\psi}^{kl})\big]_{ij} dy =&\int_Y 2\mu \big[e_y (\underline {\psi}^{kl})\big]_{\beta m} \delta_{\beta i}\delta_{mj} dy\\ =&\int_Y \mu \big[e_y (\underline {\psi}^{kl})\big]_{\beta m} (\delta_{\beta i}\delta_{mj}+\delta_{\beta j}\delta_{mi}) dy\\ =&\int_Y 2\mu \big[e_y (\underline {\psi}^{kl})\big]_{\beta m} \big[e_y (\underline {P}^{ij})\big]_{\beta m} dy. \end{align*} Using successively (\ref{3.1.1}),(\ref{3.1.5}) and (\ref{3.1.10}), we have \begin{align*} \int_Y 2\mu \big[e_y (\underline {\psi}^{kl})\big]_{\beta m} \big[e_y (\underline {P}^{ij})\big]_{\beta m} dy =&\int_Y 2\mu \big[e_y (\underline {\psi}^{kl})\big]_{\beta m} \big[e_y (\underline {\chi}^{ij})\big]_{\beta m} dy\\ =&\int_Y \big[e_y (\underline {\chi}^{kl}- \underline{P}^{kl})\big]_{\beta m} \big[e_y (\underline {\chi}^{ij})\big]_{\beta m} dy \\ =&\int_Y \big[e_y (\underline {\chi}^{kl}- \underline{Y}^{kl})\big]_{\beta m} \big[e_y (\underline {\chi}^{ij})\big]_{\beta m} dy. \end{align*} Moreover using (\ref{3.1.10}), we can rewrite the last integral as \begin{align*} \int_Y &\big[e_y (\underline {\chi}^{kl}- \underline{Y}^{kl})\big]_{\beta m} \big[e_y (\underline {\chi}^{ij})\big]_{\beta m} dy\\ =& \int_Y \big[e_y (\underline {\chi}^{kl}- \underline{Y}^{kl})\big]_{\beta m} \big[e_y (\underline {\chi}^{ij}-\underline {Y}^{ij})\big]_{\beta m} dy\\ &+\int_Y \big[e_y (\underline {\chi}^{kl}- \underline{Y}^{kl})\big]_{\beta m} \big[e_y (\underline {Y}^{ij})\big]_{\beta m} dy\\ =& \int_Y \big[e_y (\underline {\chi}^{kl}- \underline{Y}^{kl})\big]_{\beta m} \big[e_y (\underline {\chi}^{ij}-\underline {Y}^{ij})\big]_{\beta m} dy\\ &+\int_Y \big[e_y (\underline {\chi}^{kl}- \underline{Y}^{kl})\big]_{\beta m} \big[e_y (\underline {P}^{ij})\big]_{\beta m} dy\\ =& \int_Y \big[e_y (\underline {\chi}^{kl}- \underline{Y}^{kl})\big]_{\beta m} \big[e_y (\underline {\chi}^{ij}-\underline {Y}^{ij})\big]_{\beta m} dy +\int_Y \big[e_y (\underline {\chi}^{kl}- \underline{Y}^{kl})\big]_{ij} dy \end{align*} Thus the second integral of the right-hand side of (\ref{3.1.11}) can be rewritten as \begin{multline*} \int_Y \big(2\mu \big[e_y (\underline {\psi}^{kl})\big]_{ij} - \big[e_y ( \underline {\chi}^{kl}-\underline {Y}^{kl} )\big]_{ij}\big) dy \big]\\ =\int_Y \big[e_y (\underline {\chi}^{kl}- \underline{Y}^{kl})\big]_{\beta m} \big[e_y (\underline {\chi}^{ij}-\underline {Y}^{ij})\big]_{\beta m}dy. \end{multline*} Let us now consider the first integral in (\ref{3.1.11}). Multiplying the first equation of (\ref{3.1.10}) by $Y^{ij}$ and integrating by parts we have $$ \int_Y \big[e_y (-\underline {Y}^{kl}+ \underline{P}^{kl})\big]_{\beta m} \big[e_y (\underline {Y}^{ij})\big]_{\beta m} dy=0. $$ Using the fact that $\big[e_y (\underline{P}^{ij})\big]_{\beta m}=\frac 12 \big(\delta_{\beta i}\delta_{mj}+\delta_{\beta j}\delta_{mi} \big)$, we obtain \begin{align*} \int_Y &\big[e_y (-\underline {Y}^{kl}+ \underline{P}^{kl})\big]_{\beta m} \big[e_y (-\underline {Y}^{ij}+\underline{P}^{ij})\big]_{\beta m} dy\\ = & \int_Y \big[e_y (-\underline {Y}^{kl}+ \underline{P}^{kl})\big]_{\beta m} \big[e_y (\underline{P}^{ij})\big]_{\beta m} dy \\ = & \int_Y \big[e_y (-\underline {Y}^{kl}+ \underline{P}^{kl})\big]_{ij} dy. \end{align*} Then using definition (\ref{3.1.11}), we derive \begin{equation} \widetilde{b_{ijkl}}= \int_Y e_y (-\underline {Y}^{kl} + \underline{P}^{kl}) : e_y (-\underline {Y}^{ij}+\underline{P}^{ij}) dy + \int_Y e_y (\underline {\chi}^{kl} - \underline{Y}^{kl}) : e_y (\underline {\chi}^{ij}-\underline {Y}^{ij}) dy. \label{3.1.12} \end{equation} It is immediate from the above form that the coefficients of $\mathcal{B}$ satisfy $\widetilde{b_{ijkl}}=\widetilde{b_{klij}}$. On the other hand, since $e_y (\underline{P}^{kl})=e_y (\underline{P}^{lk})$ then by uniqueness of problem (\ref{3.1.1}), we have $ \chi^{kl}=\chi^{lk}$ (up to an additive constant) and then $b_{ijkl}=b_{ijlk}$. We now study the coefficients $b_{ijkl}$ with $i=k=n+1$ and $1\leq j,l\leq n$. From the definition of $\mathcal{B}$ (cf (\ref{3.1.7})), these coefficients are symmetric when $z \in ]h_1, z_0[$ and $z \in ]-z_0, h_1[$. To prove the symmetry when $z \in ]0, h_1[$, we proceed as we did before. These coefficients are as follows \begin{equation} \widetilde{b_{n+1 j n+1 l}}= \frac 14 \int_Y \big({ {\partial(- \varphi^l + 2y_l)} \over {\partial y_j}} + 2\mu { {\partial \psi^l}\over {\partial y_j}} \big) dy. \label{3.1.13} \end{equation} Let $\tau^k$ be the solution of \begin{equation} \begin{gathered}-\Delta_y (-\tau^k+2y_k ) =0 \quad\mbox{in } Y \\ \tau^k\quad Y\mbox{-p\'eriodic. } \end{gathered} \label{3.1.14} \end{equation} The expression (\ref{3.1.13}) can be rewritten as follows (using $\tau^l$): \begin{equation} \widetilde{b_{n+1 j n+1 l}}=\frac 14 \int_Y { {\partial(- \tau^l + 2y_l)} \over {\partial y_j}} dy + \frac 14 \int_Y \big(2\mu { {\partial \psi^l}\over {\partial y_j}} - { {\partial( \varphi^l -\tau^l)} \over{\partial y_j}}\big) dy. \label{3.1.15} \end{equation} Using exactly the same technique used above, we obtain (using (\ref{3.1.2}), (\ref{3.1.6}) and (\ref{3.1.14})) $$ \int_Y 2\mu { {\partial \psi^l}\over {\partial y_j}}= \frac 12 \int_Y { {\partial( \varphi^l -\tau^l)} \over{\partial y_k}} { {\partial( \varphi^j -\tau^j)} \over{\partial y_k}} dy + \int_Y { {\partial( \varphi^l -\tau^l)} \over{\partial y_j}} dy. $$ Therefore, $$ \int_Y \big( 2\mu { {\partial \psi^l}\over {\partial y_j}} - { {\partial( \varphi^l -\tau^l)} \over{\partial y_j}}\big) dy= \frac 12 \int_Y { {\partial( \varphi^l -\tau^l)} \over{\partial y_k}} { {\partial( \varphi^j -\tau^j)} \over{\partial y_k}} dy. $$ Concerning the first integral in (\ref{3.1.15}), we consider $\tau^l$ the solution of (\ref{3.1.14}): multiplying the first equation by $\tau^j$ and integrating by parts, we have $$\int_Y { {\partial(- \tau^l + 2y_l)} \over {\partial y_k}}{ {\partial\tau^j} \over{\partial y_k}} dy=0, $$ so we derive $$\int_Y { {\partial(- \tau^l + 2y_l)} \over {\partial y_k}}{ {\partial(-\tau^j+ 2y_j)} \over{\partial y_k}} dy =2\int_Y { {\partial(- \tau^l + 2y_l)} \over{\partial y_j}} dy. $$ Finally, we obtain the following expression \begin{equation} \widetilde{b_{n+1 j n+1 l}}=\frac 18 \int_Y \nabla(- \tau^l + 2y_l) . \nabla(-\tau^j+ 2y_j) dy + \frac 18 \int_Y \nabla ( \varphi^l -\tau^l) . \nabla ( \varphi^j -\tau^j) dy. \label{3.1.16} \end{equation} From the form of (\ref{3.1.16}), it is evident that $\widetilde{b_{n+1 j n+1 l}}=\widetilde{b_{n+1 l n+1 j}}$. By construction of $\mathcal{B}$, we also have $\widetilde{b_{n+1 l n+1 j}}=\widetilde{b_{n+1 l j n+1 }}$. For the other nonzero terms, the same method can be used to obtain \begin{gather*} \widetilde{b_{i n+1 k n+1 }}=\widetilde{b_{ k n+1 i n+1 }} =\widetilde{b_{i n+1 n+1 k}}\\ \widetilde{b_{i n+1 n+1 l}}=\widetilde{b_{ i n+1 l n+1 }} =\widetilde{b_{ n+1 l i n+1}}\\ \widetilde{b_{ n+1 j k n+1}}=\widetilde{b_{ n+1 j n+1 k }} =\widetilde{b_{k n+1 n+1 j}}. \end{gather*} To prove part b), we first notice that the coerciveness of $\mathcal{B}$ when $z \in ]h_1,z_0[$ and $z \in ]-z_0, h_1[$ is evident. When $z \in ]0, h_1[$, from the form of $\widetilde{b_{ijkl}} 1\leq i,j,k,l\leq n$ (see (\ref{3.1.12})) and the form of $\widetilde{b_{n+1 j n+1 l}} 1\leq j,l\leq n $ (see (\ref{3.1.16}), we have that $\mathcal{B}$ is elliptic. \hfill$\Box$\smallskip Now we introduce the homogenized problem. Let $(\vec {u},p)$ and $(\vec {v},p')$ be in $\big(H^1 (\Omega)^{n+1} \times L_0 ^2 (\Omega)\big)^2$ and be the solution of \begin{equation} \begin{gathered} -\mathop{\rm div}\big(\mathcal{A} e(\vec {u})\big) =\vec {f}-\nabla p +\vec{\theta} \quad\mbox{in } \Omega \\ \mathop{\rm div}\big(\mathcal{A} e(\vec {v}) -\mathcal{B} e(\vec {u})\big) =-\nabla p' \quad\mbox{in } \Omega \\ \mathop{\rm div} \vec{ u} =\mathop{\rm div} \vec{ v}= 0 \quad\mbox{in } \Omega\\ \vec{ u}= \vec{ v} = 0 \quad\mbox{on } \partial\Omega, \end{gathered} \label{3.1.17} \end{equation} where $\vec {f}$ is the weak limit of $\vec {f^\varepsilon}$ in $L^2(\Omega)^{n+1}$ and we precise it later, (\ref{4.1.17}). \paragraph{Remark 3.3} Since Propositions \ref{prop3.1} and \ref{prop3.2} hold, problem (\ref{3.1.17}) admits an unique solution. Now we state the main result of this paper, and we will prove it in the next section. \begin{theorem} \label{thm3.4} Under some regularity hypotheses concerning the solutions of (\ref{3.1.1}), (\ref{3.1.2}), (\ref{3.1.5}), and (\ref{3.1.6}) (detailed in Section 4) the solutions $(\vec {u}^\varepsilon,p^\varepsilon)$ and $(\vec {v}^\varepsilon,p'^\varepsilon)$ of (\ref{2.2.5}) are such that $\vec {u}^\varepsilon \rightharpoonup \vec {u}$ weakly in $H^1 (\Omega)^{n+1}$, $\vec {v}^\varepsilon \rightharpoonup \vec {v}$ weakly in $H^1 (\Omega)^{n+1}$, $p^\varepsilon \rightharpoonup p$ weakly in $L_0^2 (\Omega)$, $p'^\varepsilon \rightharpoonup p'$ weakly in $L_0^2 (\Omega)$, where $(\vec {u},p)$ and $(\vec {v},p')$ are the unique solutions of (\ref{3.1.17}). \end{theorem} \section{Proof of the convergence result} \paragraph{A priori estimates} Let \begin{gather} \xi^\varepsilon =2 \mu^\varepsilon e(\vec {u}^\varepsilon), \label{4.1.1}\\ q^\varepsilon = 2 \mu^\varepsilon e(\vec {v}^\varepsilon) -e(\vec {u}^\varepsilon). \label{4.1.2} \end{gather} \begin{proposition} \label{prop4.1} The sequences $(\vec {u}^\varepsilon,p^\varepsilon)$, $(\vec {v}^\varepsilon,p'^\varepsilon)$, $\xi^\varepsilon$ and $q^\varepsilon$ are such that (up to subsequences) \begin{gather*} \vec {u}^\varepsilon \rightharpoonup \vec {u} \mbox{ weakly in } H^1 (\Omega)^{n+1} \qquad \vec {v}^\varepsilon \rightharpoonup \vec {v} \mbox{ weakly in } H^1 (\Omega)^{n+1} \\ p^\varepsilon \rightharpoonup p \mbox{ weakly in }L_0^2 (\Omega) \qquad p'^\varepsilon \rightharpoonup p' \mbox{ weakly in }L_0^2 (\Omega) \\ \xi^\varepsilon \rightharpoonup \xi \mbox{ weakly in }L^2 (\Omega)^{n+1 \times n+1} \qquad q^\varepsilon \rightharpoonup q \mbox{ weakly in }L^2 (\Omega)^{n+1 \times n+1}. \end{gather*} % \label{4.1.3} \end{proposition} \paragraph{Proof.} Using $\vec {u}^\varepsilon$ as a test function in the first equation of (\ref{2.2.5}), we can easily see that there exists a constant $C>0$ independent of $\varepsilon$ such that \begin{equation} \Vert \vec {u}^\varepsilon \Vert_{H^1 (\Omega)^{n+1}} \leq C; \label{4.1.4} \end{equation} therefore, we have for a subsequence (still denoted by $\varepsilon$) \begin{equation} \vec {u}^\varepsilon \rightharpoonup \vec {u} \mbox{ weakly in }H^1 (\Omega)^{n+1}. \label{4.1.5} \end{equation} Similarly, multiplying the second equation of (\ref{2.2.5}) by $\vec {v}^\varepsilon$, integrating by parts and using (\ref{4.1.4}), we obtain $$\Vert \vec {v}^\varepsilon \Vert_{H^1 (\Omega)^{n+1}} \leq C,$$ so we have (for a subsequence) \begin{equation} \vec {v}^\varepsilon \rightharpoonup \vec {v} \mbox{ weakly in }H^1 (\Omega)^{n+1}. \label{4.1.7} \end{equation} Now since $\Vert \mbox{div }(2 \mu^\varepsilon e(\vec {u}^\varepsilon))\Vert_{H^{-1} (\Omega)^{n+1}}$ is bounded, we have $\Vert \nabla p^\varepsilon \Vert_{H^{-1} (\Omega)^{n+1}}\leq C$, %\label{4.1.8} this implies (see Temam \cite{te}) $|p^\varepsilon |_{L_0^2 (\Omega)}\leq C$, %\label{4.1.9} we derive (for a subsequence), $$ p^\varepsilon \rightharpoonup p \mbox{ weakly in }L_0^2 (\Omega). %\label{4.1.10} $$ Also by similar arguments, we get $$ p'^\varepsilon \rightharpoonup p'\mbox{ weakly in }L_0^2 (\Omega). %\label{4.1.11} $$ The boundedness of $\Vert \xi^\varepsilon \Vert_{L^2 (\Omega)^{(n+1)^2 }}$ provides from (\ref{4.1.4}) and we derive by extraction of subsequences $$ \xi^\varepsilon \rightharpoonup \xi \mbox{ weakly in }L^2 (\Omega)^{(n+1) ^2}. %\label{4.1.12} $$ Similarly, the boundedness of $\Vert q^\varepsilon \Vert_{L^2 (\Omega)^{(n+1) ^2}}$ provides from (\ref{4.1.4}) and (4.1.6), so we can extract a subsequence such that \begin{equation} q^\varepsilon \rightharpoonup q \mbox{ weakly in }L^2 (\Omega)^{(n+1) ^2}. \quad \label{4.1.13} \end{equation} \hfill$\Box$\smallskip Since $(\vec {u}^\varepsilon,p^\varepsilon)$ and $(\vec {v}^\varepsilon,p'^\varepsilon)$ are solutions of (\ref{2.2.5}) and since Proposition \ref{prop4.1} holds, we obtain that $(\vec {u},p), (\vec {v},p'), \xi$ and $q$ satisfy in the distribution sense \begin{equation} \begin{gathered} -\mathop{\rm div}\big(\xi \big) =\vec {f}-\nabla p +\vec{\theta} \quad\mbox{in } \Omega \\ \mathop{\rm div}\big(q\big) =-\nabla p' \mbox{ in } \Omega \\ \mathop{\rm div} \vec{ u} =\mathop{\rm div} \vec{ v}= 0 \quad\mbox{in } \Omega\\ \vec{ u}= \vec{ v} = 0 \quad\mbox{on } \partial\Omega, \end{gathered} \label{4.1.14} \end{equation} where $\vec {f}$ is the weak limit of $\vec {f^\varepsilon}$ in $L^2(\Omega)^{n+1}$. This limit can be identified explicitly (cf \cite{bc3} or \cite{br}). Indeed, the characteristic functions $ \chi_{\Omega_1^\varepsilon}$ ($i=1,2$) are such that \begin{equation} \chi_{\Omega_i^\varepsilon}\rightharpoonup \rho \mbox{ and } \chi_{\Omega_2^\varepsilon} \rightharpoonup (1- \rho) \mbox{ weakly } \star \quad\mbox{in } L^{\infty}(\Omega), \label{4.1.15} \end{equation} where \begin{equation} \rho (x,z)= \begin{cases} 1 &\mbox{in } \Omega_1 \\ \frac { |O(z) |} { |Y |} &\mbox{in } \Omega_m \\ 0 &\mbox{in } \Omega_2. \end{cases} \label{4.1.16} \end{equation} Then we have \begin{equation} \vec {f^\varepsilon} \rightharpoonup \vec {f}=\vec {f^1} \rho + \vec {f^2} (1- \rho) \mbox{ weakly in }L^2(\Omega)^{n+1}. \label{4.1.17} \end{equation} \begin{proposition}[Baffico \& Conca \cite{bc3}] \label{prop4.2} Under the hypotheses (\ref{4.2.1}), (\ref{4.2.4}) and (\ref{4.3.1}) (introduced in the next subsections), $ \xi = \mathcal{A} e(\vec {u})$, where $\mathcal{A}$ is defined by (\ref{3.1.4}). \end{proposition} To prove Theorem \ref{thm3.4}, we have to show that $q, \vec {u}$ and $ \vec {v}$ are related by \begin{equation} q= \mathcal{A} e(\vec {v})-\mathcal{B} e(\vec {u}). \label{4.1.18} \end{equation} Using the same method as Baffico \& Conca\cite{bc3}, we show that the identification of $q$ is carried out in $\Omega_1, \Omega_m \mbox{ and } \Omega_2$ independently. In $\Omega_1$ and $\Omega_2$, this identification will pose no particular problem. In $\Omega_m$, following the ideas of Baffico \& Conca (\cite{bc3}), there is three steps: we first identify the components $[q]_{ij}$ of $q$ for $1\leq i,j \leq n$, and then $[q]_{n+1 j}$ for $1\leq j \leq n $ and finally we identify $[q]_{n+1 n+1}$. To do so, we use some suitable test functions and the energy method ( cf Bensoussan, Lions \& Papanicolaou \cite{blp} or Sanchez-Palencia \cite{sp}). \subsection*{Identification of $[q]_{ij} 1\leq i,j \leq n $ in $\Omega_m$} In what follows, we construct at first the test functions which allow us the identification of $[q]_{ij}$, then we introduce the regularity conditions that these functions must satisfy and finally we establish the identification. Let $\underline{w}^{kl}=-\underline{\chi}^{kl}+ \underline{P}^{kl} $ and $\sigma^{kl}=2\mu e_y (\underline{w}^{kl})$ where $(\underline{\chi}^{kl}, r_1^{kl})$ be the solution of (\ref{3.1.1}). We assume that $(\underline{\chi}^{kl}, r_1^{kl})$, as function of $(y,z) \in Y\times ]0, h_1[$, satisfies the regularity hypothesis \begin{equation} \begin{aligned} &a) \quad \underline{\chi}^{kl} \in L_{ \mbox{loc}}^2(0,h_1, H_{\sharp}^1(Y)^n) \cap (L_{ \mbox{loc}}^2(]0,h_1[ \times \mathbb{R}^n))^n \\ &b)\quad \frac{\partial}{\partial z} \big( (\underline{\chi}^{kl})_i \big) \in L_{ \mbox{loc}}^2(0,h_1, L_{\sharp}^2(Y)^n) \cap L_{ \mbox{loc}}^2(]0,h_1[ \times \mathbb{R}) \end{aligned} \quad 1\leq i,j \leq n \label{4.2.1} \end{equation} We define the following functions by extension by $Y$-periodicity to $\mathbb{R}^{n+1}$ and by restriction to $\Omega_m$: \begin{equation} \begin{gathered} \underline{w}^{\varepsilon, kl}(x,z)= \varepsilon \underline{w}^{kl}( { x \over \varepsilon}, z) \\ r_1^{\varepsilon, kl}(x,z)=r_1^{kl}({ x \over \varepsilon}, z) \\ \sigma^{\varepsilon, kl}(x,z)= \underline{\sigma}^{kl}({ x \over \varepsilon}, z) \end{gathered} \label{4.2.2} \end{equation} It is easy to check that \begin{equation} \begin{gathered} \mathop{\rm div}_x (\sigma^{\varepsilon, kl})= - \nabla_x r_1^{\varepsilon, kl} \quad\mbox{in } \Omega_m \\ \mathop{\rm div}_x (\underline{w}^{\varepsilon, kl})=\mathop{\rm div}_x (\underline{P}^{kl})=\delta_{kl} \quad\mbox{in } \Omega_m. \end{gathered}\label{4.2.3} \end{equation} We also need the Murat's compactness result \cite{mu}. \begin{lemma} \label{lm4.3} If the sequence $(g_n)_n$ belongs to a bounded subset of $W^{-1,p}(\Omega)$ for some $ p>2$, and $(g_n)_n \geq 0$ in the following sense i.e., for all $\phi \in {\cal D}(\Omega)$ such that $\phi\geq 0$ then for all $n>0 \langle g_n, \phi\rangle \geq 0$. Then $(g_n)_n$ belongs to a compact subset of $H^{-1}(\Omega)$. \end{lemma} If we suppose that $r_1^{\varepsilon, kl}$ satisfy \begin{equation} \begin{aligned} &a)\quad r_1^{\varepsilon, kl} \in L_{ \mbox{loc}}^p(\Omega_m) \mbox{ for some }p>2 , \mbox{ locally bounded} \\ &b)\quad \frac{\partial}{\partial z} \big( r_1^{\varepsilon, kl} \big) \geq 0 \mbox{ in distribution sense,} \end{aligned} \label{4.2.4} \end{equation} Then using Lemma \ref{lm4.3} and hypothesis (\ref{4.2.1}), we have the following result. \begin{proposition}[Baffico \& Conca \cite{bc3}]\label{prop4.4 } If (\ref{4.2.1}) and (\ref{4.2.4}) hold. Then for all $\Omega' \subset\subset \Omega_m$, we have the following convergence \begin{equation} \begin{aligned} &\mbox{ a) $\underline{w}^{\varepsilon, kl}\rightharpoonup \underline{P}^{kl}$ weakly in $H^1(\Omega')^n$ } \\ &\mbox{ b) $\frac{\partial}{\partial z} \big( (\underline{w}^{\varepsilon,kl})_i \big) \to 0$ strongly in $L^2(\Omega')^n$, $1\leq i\leq n$} \\ &\mbox{ c) $r_1^{\varepsilon, kl} \to 0$ weakly in $ L^2(\Omega')^n$} \\ &\mbox{ d) $\frac{\partial} {\partial z} \big( r_1^{\varepsilon, kl} \big) \to 0$ strongly in $H^{-1}(\Omega')^n$, $1\leq i\leq n$} \\ &\mbox{ e) $\sigma^{\varepsilon, kl} \rightharpoonup \sigma^{kl}= m _ Y (2\mu e_y (\underline{w}^{kl})$ weakly in $L^2(\Omega')^{n\times n}$.} \\ \end{aligned} \label{4.2.5} \end{equation} \end{proposition} In the same way, we assume that the solution $(\underline {\psi}^{kl}, r_2^{kl})$ of (\ref{3.1.5}) satisfies the following convergence hypothesis: \begin{equation} \begin{aligned} &a)\quad \underline {\psi}^{kl} \in L_{ \mbox{loc}}^2(0,h_1, H_{\sharp}^1(Y)^n) \cap (L_{ \mbox{loc}}^2(]0,h_1[ \times \mathbb{R}^n))^n \\ &b)\quad \frac{\partial}{\partial z} \big( (\underline {\psi}^{kl})_i \big) \in L_{ \mbox{loc}}^2(0,h_1, L_{\sharp}^2(Y)^n) \cap L_{ \mbox{loc}}^2(]0,h_1[ \times \mathbb{R} ) \end{aligned} \quad 1\leq i,j \leq n \label{4.2.6} \end{equation} We define \begin{equation} \underline {\psi}^{\varepsilon ,kl}(x,z)= \varepsilon \underline {\psi}^{kl}({ x \over \varepsilon}, z),\quad\mbox{and}\quad r_2^{\varepsilon, kl}(x,z)=r_2^{kl}({ x \over \varepsilon}, z) \label{4.2.7} \end{equation} so we obtain \begin{equation} \begin{gathered} -\mathop{\rm div}_x \big(2 \mu^\varepsilon e_x(\underline {\psi}^{\varepsilon ,kl})+ e_x(\underline{w}^{\varepsilon, kl})\big) =-\nabla_x r_2^{\varepsilon, kl} \quad\mbox{in } \Omega_m \\ \mathop{\rm div}_x \underline {\psi}^{\varepsilon ,kl} = 0 \quad\mbox{in } \Omega_m \end{gathered} \label{4.2.8} \end{equation} If we suppose that $r_2^{\varepsilon, kl}$ satisfy \begin{equation} \begin{aligned} &a)\quad r_2^{\varepsilon, kl} \in L_{ \mbox{loc}}^p(\Omega_m) \mbox{ for some }p>2 , \mbox{ locally bounded} \\ &b)\quad \frac{\partial}{\partial z} \big( r_2^{\varepsilon, kl} \big) \geq 0 \mbox{ in distribution sense,} \end{aligned} \label{4.2.9} \end{equation} then using Lemma \ref{lm4.3}, we derive $$ \frac{\partial}{\partial z} \big( r_2^{\varepsilon, kl} \big) \to 0 \mbox{ strongly in } H^{-1}(\Omega')^n. 1\leq i\leq n %\label{4.2.10} $$ We have the following result concerning these functions \begin{proposition} \label{prop4.5 } If (\ref{4.2.1}) and (\ref{4.2.4}) hold. Then for all $\Omega' \subset\subset \Omega_m$, we have \begin{equation} \begin{aligned} &a)\quad \underline{\psi}^{\varepsilon, kl}\rightharpoonup 0 \mbox{ weakly in } H^1(\Omega')^n \\ &b)\quad \frac{\partial}{\partial z} \big( (\underline{\psi}^{\varepsilon, kl})_i \big) \to 0 \mbox{ strongly in } L^2(\Omega')^n ,\quad 1\leq i\leq n \\ &c)\quad r_2^{\varepsilon, kl} \rightharpoonup 0 \mbox{ weakly in } L^2(\Omega')^n. \end{aligned} \label{4.2.11} \end{equation} \end{proposition} To prove this proposition, we use well-known results concerning the convergence of periodic functions.\smallskip We shall prove now the principal result of this section \begin{proposition} \label{prop4.6} If (\ref{4.2.1}), (\ref{4.2.4}), (\ref{4.2.6}), and (\ref{4.2.9}) hold, then $[q^\varepsilon]_{kl} \rightharpoonup [q]_{kl}$ weakly in $L^2(\Omega_m)$ (up to a subsequence) for all $1\leq k,l \leq n$ where \begin{equation*} \begin{aligned}{} [q]_{kl}= {1 \over {|Y |}} \sum_{i,j=1}^n \big \{ \int_Y 2 \mu [e_y(-\underline {\chi}^{ij}+\underline {P}^{ij} )]_{kl} dy\big \} \big[ e(\vec {v})\big]_{ij} \\ - {1 \over {|Y |}} \sum_{i,j=1}^n \big \{ \int_Y \big( 2 \mu [e_y(-\underline {\psi}^{ij} )]_{kl} + [e_y(-\underline {\chi}^{ij}+\underline {P}^{ij} )]_{kl} \big) dy \big \} \big[ e(\vec {u})\big]_{ij} \end{aligned} %\label{4.2.12} \end{equation*} \end{proposition} \paragraph{Proof.} Let $\phi \in {\cal D} (\Omega_m)$ and $\vec {w}^{\varepsilon, kl}=( \underline{w}^{\varepsilon, kl}, 0)$. Multiplying the second equation of (\ref{2.2.5}) by $\phi \vec {w}^{\varepsilon, kl}$, integrating by parts and using (\ref{4.1.2}), we obtain \begin{align*} - \int_{\Omega_m} &\nabla p'^\varepsilon . (\phi \vec {w}^{\varepsilon, kl}) dxdz\\ =& - \int_{\Omega_m} (q^\varepsilon \nabla \phi). \vec {w}^{\varepsilon, kl} dxdz +\int_{\Omega_m} e(\vec{u}^\varepsilon) : \nabla \vec {w}^{\varepsilon, kl}\phi dxdz \\ & -\int_{\Omega_m} 2 \mu^\varepsilon e(\vec {v}^\varepsilon) : \nabla \vec {w}^{\varepsilon, kl}\phi dxdz. \end{align*} %\label{4.2.13} Developing the second and third integral of the right-hand side of the above equation, \begin{equation} \begin{aligned} - \int_{\Omega_m}& \nabla p'^\varepsilon . \phi \vec {w}^{\varepsilon, kl}dxdz\\ =& - \int_{\Omega_m} (q^\varepsilon \nabla \phi). \vec {w}^{\varepsilon, kl}dxdz +\int_{\Omega_m} e_x(\underline{u}^\varepsilon) : e_x(\underline{w}^{\varepsilon, kl})\phi dxdz \\ &-\int_{\Omega_m} 2 \mu^\varepsilon e_x(\underline {v}^\varepsilon) : e_x( \underline {w}^{\varepsilon, kl})\phi dxdz - \int_{\Omega_m} \sum_{j=1}^n [q^\varepsilon]_{n+1 j} \frac{\partial}{\partial z} \big( (\underline{w}^{\varepsilon, kl})_j \big) \phi dxdz. \end{aligned} \label{4.2.14} \end{equation} Let $\vec {\psi}^{\varepsilon, kl}=( \underline{\psi}^{\varepsilon, kl}, 0)$. Multiplying the first equation of (\ref{2.2.5}) by $\phi \vec {\psi}^{\varepsilon, kl}$, integrating by parts and using definition (\ref{4.1.1}), we obtain (after algebraic developments) \begin{align*} \int_{\Omega_m} &(\vec {f}^\varepsilon-\nabla p^\varepsilon +\vec{\theta}) . \phi \vec {\psi}^{\varepsilon, kl}dxdz\\ =& \int_{\Omega_m} (\xi^\varepsilon \nabla \phi). \vec {\psi}^{\varepsilon, kl} +\int_{\Omega_m} 2 \mu^\varepsilon e_x(\underline {u }^\varepsilon) : e_x( \underline {\psi}^{\varepsilon, kl})\phi dxdz\\ &+ \int_{\Omega_m} \sum_{j=1}^n [\xi^\varepsilon]_{n+1 j} \frac{\partial}{\partial z} \big( (\underline{\psi}^{\varepsilon, kl})_j \big) \phi dxdz. % \label{4.2.15} \end{align*} Integrating by parts now the second integral of the right-hand side, we have \begin{align*} \int_{\Omega_m}&(\vec {f}^\varepsilon-\nabla p^\varepsilon +\vec{\theta}) . \phi \vec {\psi}^{\varepsilon, kl}dxdz \\ =& \int_{\Omega_m} (\xi^\varepsilon \nabla \phi). \vec {\psi}^{\varepsilon, kl} -\int_{\Omega_m} \mathop{\rm div}_x \big( 2 \mu^\varepsilon e_x( \underline {\psi}^{\varepsilon, kl})\big). (\underline {u }^\varepsilon \phi) dxdz \\ & - \int_{\Omega_m} (2 \mu^\varepsilon e_x( \underline {\psi}^{\varepsilon, kl}) \nabla_x \phi). \underline {u }^\varepsilon dxdz + \int_{\Omega_m} \sum_{j=1}^n [\xi^\varepsilon]_{n+1 j} \frac{\partial}{\partial z} \big( (\underline{\psi}^{\varepsilon, kl})_j \big) \phi dxdz. %\label{4.2.16} \end{align*} Using the equation that $\underline{\psi}^{\varepsilon, kl}$ satisfy (see (\ref{4.2.8})), \begin{align*} \int_{\Omega_m} &(\vec {f}^\varepsilon-\nabla p^\varepsilon +\vec{\theta}) . \phi \vec {\psi}^{\varepsilon, kl}dxdz \\ =& \int_{\Omega_m} (\xi^\varepsilon \nabla \phi). \vec {\psi}^{\varepsilon, kl} -\int_{\Omega_m} \underline {u }^\varepsilon \mathop{\rm div}_x \big(e_x( \underline {w}^{\varepsilon, kl})\big)\phi dxdz +\int_{\Omega_m} \nabla_x r_2^{\varepsilon, kl}. (\phi \underline {u }^\varepsilon) dxdz \\ & - \int_{\Omega_m} \big(2 \mu^\varepsilon e_x( \underline {\psi}^{\varepsilon, kl}) \nabla_x \phi \big). \underline {u }^\varepsilon dxdz + \int_{\Omega_m} \sum_{j=1}^n [\xi^\varepsilon]_{n+1 j} \frac{\partial}{\partial z} \big( (\underline{\psi}^{\varepsilon, kl})_j \big) \phi dxdz. %\label{4.2.17} \end{align*} Integrating again by parts the second integral, \begin{align*} \int_{\Omega_m} &(\vec {f}^\varepsilon+\vec{\theta}-\nabla p^\varepsilon) . \phi \vec {\psi}^{\varepsilon, kl}dxdz\\ =& \int_{\Omega_m} (\xi^\varepsilon \nabla \phi). \vec {\psi}^{\varepsilon, kl} -\int_{\Omega_m} e_x(\underline {u }^\varepsilon): e_x( \underline {w}^{\varepsilon, kl})\phi dxdz \\ &+\int_{\Omega_m} \nabla_x r_2^{\varepsilon, kl}.(\phi \underline {u }^\varepsilon) dxdz - \int_{\Omega_m} \big(2 \mu^\varepsilon e_x( \underline {\psi}^{\varepsilon, kl}) \nabla_x \phi \big). \underline {u }^\varepsilon dxdz \\ &- \int_{\Omega_m} \big( e_x( \underline {w}^{\varepsilon, kl}) \nabla_x \phi\big).\underline {u }^\varepsilon dxdz + \int_{\Omega_m} \sum_{j=1}^n [\xi^\varepsilon]_{n+1 j} \frac{\partial}{\partial z} \big( (\underline{\psi}^{\varepsilon, kl})_j \big) \phi dxdz. %\label{4.2.18} \end{align*} Adding (\ref{4.2.14}) and the above equation, we obtain \begin{equation} \begin{aligned} \int_{\Omega_m} &(\vec {f}^\varepsilon+\vec{\theta}) . \phi \vec {\psi}^{\varepsilon, kl}dxdz - \int_{\Omega_m} \nabla p^\varepsilon . \phi \vec {\psi}^{\varepsilon, kl}dxdz - \int_{\Omega_m} \nabla p'^\varepsilon . \phi \vec {w}^{\varepsilon, kl}dxdz \\ =&- \int_{\Omega_m} (q^\varepsilon \nabla \phi). \vec {w}^{\varepsilon, kl}dxdz -\int_{\Omega_m} 2 \mu^\varepsilon e_x(\underline {v}^\varepsilon) : e_x( \underline {w}^{\varepsilon, kl})\phi dxdz \\ &- \int_{\Omega_m} \sum_{j=1}^n [q^\varepsilon]_{n+1 j} \frac{\partial}{\partial z} \big( (\underline{w}^{\varepsilon, kl})_j \big) \phi dxdz + \int_{\Omega_m} (\xi^\varepsilon \nabla \phi). \vec {\psi}^{\varepsilon, kl} \\ & +\int_{\Omega_m} \nabla_x r_2^{\varepsilon, kl}. (\phi \underline {u }^\varepsilon) dxdz - \int_{\Omega_m} (b^{\varepsilon, kl}\nabla_x \phi). \underline {u }^\varepsilon dxdz \\ &+ \int_{\Omega_m} \sum_{j=1}^n [\xi^\varepsilon]_{n+1 j} \frac{\partial}{\partial z} \big( (\underline{\psi}^{\varepsilon, kl})_j \big) \phi dxdz. \end{aligned} \label{4.2.19} \end{equation} where \begin{equation} b^{\varepsilon, kl}=2 \mu^\varepsilon e_x( \underline {\psi}^{\varepsilon, kl}) + e_x( \underline {w}^{\varepsilon, kl}). \label{4.2.20} \end{equation} We obtain easily that (using Problem (\ref{3.1.5})) $$ \mathop{\rm div}_x (b^{\varepsilon, kl})=-\nabla_x r_2^{\varepsilon, kl} \quad\mbox{in } \Omega_m. $$ We now pass to the limit in (\ref{4.2.19}) as $\varepsilon$ tends to 0. In order to do so, we need some preliminaries results. By Definition (\ref{4.2.20}) and classical arguments concerning the convergence of periodic functions, we conclude that for all $\Omega' \subset\subset \Omega_m$, \begin{equation} \begin{gathered} b^{\varepsilon, kl} \rightharpoonup b^{kl}= \mbox{m}_Y (2 \mu e_y( \underline {\psi}^{kl}) + e_y( \underline {w}^{kl})) \mbox{ weakly in } L^2(\Omega')^{n\times n} \\ \mbox{and}\quad \mathop{\rm div}_x (b^{kl})=0 \quad\mbox{in } \Omega_m. \end{gathered} \label{4.2.21} \end{equation} By the convergence (\ref{4.2.5}) a), we have $$\vec {w}^{\varepsilon, kl} \to \vec {P}^{kl}=(\underline {P}^{kl},0) \mbox{ strongly in } L^2(\Omega')^{n+1}. %\label{4.2.22} $$ Also by (\ref{4.2.11}) $a)$, we get $$\vec {\psi}^{\varepsilon, kl} \to 0 \mbox{ strongly in } L^2(\Omega')^{n+1}. %\label{4.2.23} $$ Now passing to the limit in (\ref{4.2.19}) taking into account the precedent convergence results, we obtain \begin{align*} - \int_{\Omega_m} &\nabla p' . \phi \vec {P}^{kl}dxdz\\ =&-\int_{\Omega_m}(q \nabla \phi). \vec {P}^{kl}dxdz -\int_{\Omega_m} \sigma^{kl}: e_x (\underline {v})\phi dxdz - \int_{\Omega_m} (b^{kl}\nabla_x \phi). \underline {u } dxdz \end{align*} %\label{4.2.24} Integrating by parts the right-hand side of the above expression, using the second equation of (4.1.14) and the expression (\ref{4.2.21}), we obtain \begin{equation*} 0=\int_{\Omega_m}q : e(\vec {P}^{kl})\phi dxdz -\int_{\Omega_m} \sigma^{kl}: e_x (\underline {v}) \phi dxdz + \int_{\Omega_m} e_x(\underline {u }) : b^{kl} \phi dxdz %\label{4.2.25} \end{equation*} Since $\big[ e(\vec {P}^{kl})\big]_{ij} = \big[M^{kl}\big]_{ij}$, then we obtain in the distribution sense $$ [q]_{kl} = \sum_{i,j=1}^n \big[ \sigma^{kl}\big]_{ij} \big[ e_x (\underline {v})\big]_{ij} - \sum_{i,j=1}^n \big[ b^{kl}\big]_{ij} \big[e_x (\underline {u })\big]_{ij}. %\label{4.2.26} $$ Now since (\ref{4.2.5}) e), (\ref{4.2.21}) hold and since $\big[ e_x (\underline {v})\big]_{ij}=\big[ e(\vec {v})\big]_{ij}$ and $\big[ e_x (\underline {u})\big]_{ij}=\big[ e(\vec {u})\big]_{ij}$ for all $1\leq i,j \leq n$, we get \begin{equation} \begin{aligned}{} [q]_{kl}= & {1 \over {|Y |}} \sum_{i,j=1}^n \big \{ \int_Y 2 \mu [e_y(-\underline {\chi}^{kl}+\underline {P}^{kl} )]_{ij} dy \big \}\big[ e(\vec {v})\big]_{ij} \\ & - {1 \over {|Y |}} \sum_{i,j=1}^n \big \{ \int_Y \big( 2 \mu [e_y(-\underline {\psi}^{kl} )]_{ij} + [e_y(-\underline {\chi}^{kl}+\underline {P}^{kl} )]_{ij} \big) dy \big \} \big[ e(\vec {u})\big]_{ij}. \end{aligned}\label{4.2.27} \end{equation} Also since the following symmetry property holds (see Proposition 3.1) $$ \int_Y 2 \mu [e_y(-\underline {\chi}^{kl}+\underline {P}^{kl} )]_{ij} dy = \int_Y 2 \mu [e_y(-\underline {\chi}^{ij}+\underline {P}^{ij} )]_{kl} dy, %\label{4.2.28} $$ and the following's one holds too (see Proposition 3.2) \begin{multline} \int_Y \big( 2 \mu \big[ e_y(-\underline {\psi}^{kl} ) + e_y(-\underline {\chi}^{kl}+\underline {P}^{kl} ) \big]_{ij} \big)dy\\ =\int_Y \big( 2 \mu \big[ e_y(-\underline {\psi}^{ij} ) + e_y(-\underline {\chi}^{ij}+\underline {P}^{ij} ) \big]_{kl} \big)dy, \label{4.2.29} \end{multline} we conclude that \begin{align*}{} [q]_{kl}= &{1 \over {|Y |}} \sum_{i,j=1}^n \big \{ \int_Y 2 \mu [e_y(-\underline {\chi}^{ij}+\underline {P}^{ij} )]_{kl} dy\big \} \big[ e(\vec {v})\big]_{ij} \\ &- {1 \over {|Y |}} \sum_{i,j=1}^n \big \{ \int_Y \big( 2 \mu [e_y(-\underline {\psi}^{ij} )]_{kl} + [e_y(-\underline {\chi}^{ij}+\underline {P}^{ij} )]_{kl} \big) dy \big \} \big[ e(\vec {u})\big]_{ij} %\label{4.2.30} \end{align*} This completes the proof. \hfill$\Box$\smallskip \subsection*{Identification of $[q]_{n+1 j}$, $1\leq j \leq n$ in $\Omega_m$} Let $\varphi^k$ be the solution of Problem (\ref{3.1.2}). We assume that $\varphi^k= \varphi^k (y,z)$ satisfy the regularity hypotheses \begin{equation} \begin{aligned} &\mbox{a)\quad $\varphi^k \in L_{ \mbox{loc}}^2(0,h_1, H_{\sharp}^1(Y)) \cap L_{ \mbox{loc}}^2(]0,h_1[ \times \mathbb{R}^n)$} \\ &\mbox{b)\quad ${{\partial \varphi^k} \over {\partial z}} \in L_{ \mbox{loc}}^2(0,h_1, L_{\sharp}^2(Y)) \cap L_{ \mbox{loc}}^2(]0,h_1[ \times \mathbb{R}^n )$} \end{aligned} \label{4.3.1} \end{equation} Let us define $\zeta^k= -\varphi^k +2 y_k \mbox{ and } \underline{\eta}^k = \mu \nabla_y \zeta^k$. We also define the following functions by $Y$-periodicity: \begin{equation} \zeta^{\varepsilon ,k}(x,z)=\varepsilon \zeta^k({x \over \varepsilon}, z), \quad \underline{\eta}^{\varepsilon ,k}(x,z)=\underline{\eta}^k ({x \over \varepsilon}, z) \label{4.3.2} \end{equation} It is easy to see that $-\mathop{\rm div}_x \underline{\eta}^{\varepsilon ,k}= 0$ in $\Omega_m$. %\label{4.3.3} We introduce a supplementary hypothesis concerning $\underline{\eta}^{\varepsilon ,k}$: \begin{equation} \begin{aligned} &\mbox{a)}\quad \{ \big(\underline{\eta}^{\varepsilon ,k} \big)_j \} _{\varepsilon>0} \subset L_{ \mbox{loc}}^p(\Omega_m) \mbox{ for some $p>2$, locally bounded} \\ &\mbox{b)}\quad \frac{\partial}{\partial z} \big(\underline{\eta}^{\varepsilon,k} \big)_j \geq 0 \mbox{ in the distribution sense.} \end{aligned} \label{4.3.4} \end{equation} Then we have the following result. \begin{proposition}[Baffico \& Conca \cite{bc3}] \label{prop4.7} Assume (\ref{4.3.1}) and (\ref{4.3.4}). Then for all $ \Omega' \subset\subset \Omega_m$, we have \begin{equation}\begin{aligned} &\mbox{ a) $\zeta^{\varepsilon ,k}\rightharpoonup 2y_k$ weakly in $H^1(\Omega')$} \\ &\mbox{ b) $\frac{\partial}{\partial z} \big( \zeta^{\varepsilon ,k} \big) \to 0$ strongly in $L^2(\Omega')$} \\ &\mbox{ c) $\underline{\eta}^{\varepsilon ,k} \rightharpoonup \underline{\eta}^{k}= m _ Y (\underline{\eta}^k)$ weakly in $L^2(\Omega')^n$} \\ &\mbox{ d) $\frac{\partial}{\partial z} \big(\underline{\eta}^{\varepsilon ,k} \big)_j \to \frac{\partial}{\partial z} \big(\underline{\eta}^{k} \big)_j$ strongly in $H^{-1}(\Omega')$, $1\leq j\leq n$.} \end{aligned}\label{4.3.5} \end{equation} \end{proposition} In view of (4.3.5) c) and (4.3.3), we get $-\mathop{\rm div}_x \underline{\eta}^{k}= 0$ in $\Omega_m$. %\label{4.3.6} Similarly we assume that $ \psi^k$, the solution of (\ref{3.1.6}), satisfies \begin{equation} \begin{aligned} &\mbox{a)}\quad \psi^k \in L_{ \mbox{loc}}^2(0,h_1, H_{\sharp}^1(Y)) \cap L_{ \mbox{loc}}^2(]0,h_1[ \times \mathbb{R}^n) \\ &\mbox{b)} \quad \frac{\partial \psi^k} {\partial z} \in L_{ \mbox{loc}}^2(0,h_1, L_{\sharp}^2(Y)) \cap L_{ \mbox{loc}}^2(]0,h_1[ \times \mathbb{R}^n ) \end{aligned} \label{4.3.7} \end{equation} Let \begin{equation} \psi^{\varepsilon ,k}(x,z)=\psi^k ({x \over \varepsilon}, z) \quad\mbox{and}\quad %\label{4.3.8} \underline{d}^{\varepsilon ,k}=2\mu^\varepsilon \nabla_x \psi^{\varepsilon ,k} + \nabla_x \zeta^{\varepsilon ,k}, \label{4.3.9} \end{equation} then using (\ref{3.1.6}), (\ref{4.3.2}) and (\ref{4.3.9}), we get \begin{equation} -\mathop{\rm div}_x \underline{d}^{\varepsilon ,k}= 0 \quad\mbox{in } \Omega_m. \label{4.3.10} \end{equation} We assume that $\underline{d}^{\varepsilon ,k}$ satisfies the regularity conditions: \begin{equation} \begin{aligned} &a) \quad \big\{ \big(\underline{d}^{\varepsilon ,k} \big)_j \big \}_{\varepsilon>0} \subset L_{ \mbox{loc}}^p(\Omega_m) \mbox{ for some } p>2, \mbox{ locally bounded} \\ &b) \quad \frac{\partial}{\partial z} \big(\underline{d}^{\varepsilon ,k} \big)_j \geq 0 \mbox{ in the distribution sense.} \end{aligned} \label{4.3.11} \end{equation} We have the following result. \begin{proposition} \label{prop4.8} Assume hypotheses (4.3.7) and (\ref{4.3.11}) hold. Then for all $\Omega' \subset\subset \Omega_m$, we have \begin{equation} \begin{aligned} &a)\quad \psi^{\varepsilon ,k}\rightharpoonup 0 \mbox{ weakly in } H^1(\Omega') \\ &b)\quad \frac{\partial}{\partial z} \big( \psi^{\varepsilon ,k} \big) \to 0 \mbox{ strongly in } L^2(\Omega') \\ &c)\quad \underline{d}^{\varepsilon ,k} \rightharpoonup \underline{d}^{k}= m _ Y (2\mu \nabla_y \psi^{k}+ \nabla_y \zeta^{k}) \mbox{ weakly in } L^2(\Omega')^n \\ & d) \quad \frac{\partial}{\partial z} \big(\underline{d}^{\varepsilon ,k} \big)_j \to \frac{\partial}{\partial z} \big(\underline{d}^{k} \big)_j \mbox{ strongly in } H^{-1}(\Omega'),\; 1\leq j\leq n. \\ \end{aligned} \label{4.3.12} \end{equation} \end{proposition} \begin{remark} \label{rmk4.9} \rm From (\ref{4.3.10}) and (\ref{4.3.12}) c), we have $-\mathop{\rm div}_x \underline{d}^{k}= 0$ in $\Omega_m$. %\label{4.3.13} \end{remark} \paragraph{Proof of Proposition \ref{prop4.8}} Using classical arguments concerning convergence of periodic functions, we can prove the three first assertions. For the last one, we use the compactness Lemma \ref{lm4.3} (the hypotheses of this lemma hold since we suppose that (\ref{4.3.11}) is satisfied). \hfill$\Box$\smallskip \begin{proposition} \label{prop4.10} If (\ref{4.3.1}), (\ref{4.3.4}), (\ref{4.3.7}) and (\ref{4.3.11}) hold, then up to a subsequence, we have $[q^\varepsilon]_{n+1 k} \rightharpoonup [q]_{n+1 k} \mbox{ weakly in } L^2(\Omega_m)$ $ \forall 1\leq j \leq n $, where \begin{align*}{} [q]_{n+1 k}=& {1 \over {|Y |}} \sum_{i=1}^n \big \{ \int_Y \mu {{\partial (-\varphi^i+2y_i)} \over {\partial y_k}} dy\big \} \big[ e(\vec {v})\big]_{n+1 i} \\ &- {1 \over {2|Y |}} \sum_{i=1}^n \big \{ \int_Y \big( 2 \mu {{\partial \psi^i} \over {\partial y_k}} + {{\partial (-\varphi^i+2y_i)} \over {\partial y_k}} dy \big \} \big[ e(\vec {u})\big]_{n+1 i}. %\label{4.3.14} \end{align*} \end{proposition} \paragraph{Proof.} Let $\phi \in {\cal D} (\Omega_m)$ and $\vec {\zeta}^{\varepsilon, k}=(\underline 0 , \zeta^{\varepsilon, k})$. Multiplying the second equation of (\ref{2.2.5}) by $\phi \vec {\zeta}^{\varepsilon, k}$, integrating by parts and using (\ref{4.1.2}), we obtain \begin{align*} - \int_{\Omega_m} \nabla p'^\varepsilon . \vec {\zeta}^{\varepsilon, k} \phi dxdz =& - \int_{\Omega_m} (q^\varepsilon \nabla \phi). \vec {\zeta}^{\varepsilon, k} dxdz +\int_{\Omega_m} e(\vec{u}^\varepsilon) : \nabla \vec {\zeta}^{\varepsilon, k} \phi dxdz \\ & -\int_{\Omega_m} 2 \mu^\varepsilon e(\vec {v}^\varepsilon) : \nabla \vec {\zeta}^{\varepsilon, k} \phi dxdz. %\label{4.3.15} \end{align*} After some elementary computations on the second and third integral of the right-hand side of the above equation, \begin{equation} \begin{aligned} - \int_{\Omega_m} &\nabla p'^\varepsilon . \vec {\zeta}^{\varepsilon, k} \phi dxdz \\ =& - \int_{\Omega_m} (q^\varepsilon \nabla \phi). \vec {\zeta}^{\varepsilon,k}dxdz -\int_{\Omega_m} \mu^\varepsilon \nabla_x v_{n+1}^\varepsilon. \nabla_x \zeta^{\varepsilon, k} \phi dxdz \\ &-\int_{\Omega_m}\sum_{i=1}^n \mu^\varepsilon {{\partial v_i^\varepsilon} \over {\partial z}} {{\partial \zeta^{\varepsilon, k}} \over {\partial x_i}}\phi dxdz + \frac 12 \int_{\Omega_m} \nabla_x u_{n+1}^\varepsilon. \nabla_x \zeta^{\varepsilon, k} \phi dxdz \\ &- \int_{\Omega_m} \sum_{j=1}^n [q^\varepsilon]_{n+1 n+1} {{\partial \zeta^{\varepsilon, k}} \over {\partial z}} \phi dxdz + \frac 12 \int_{\Omega_m} \sum_{i=1}^n \mu^\varepsilon {{\partial u_i^\varepsilon} \over {\partial z}} {{\partial \zeta^{\varepsilon, k}} \over {\partial x_i}}\phi dxdz . \end{aligned} \label{4.3.16} \end{equation} Let $\vec {\psi}^{\varepsilon, k}=(\underline 0, \psi^{\varepsilon, k})$. Multiplying the first equation of (\ref{2.2.5}) by $\phi \vec {\psi}^{\varepsilon, k}$, integrating by parts and using Definition (\ref{4.1.1}), we get $$ \int_{\Omega_m} (\vec {f}^\varepsilon-\nabla p^\varepsilon +\vec{\theta}) . \vec {\psi}^{\varepsilon, k}\phi dxdz = \int_{\Omega_m} (\xi^\varepsilon \nabla \phi). \vec {\psi}^{\varepsilon, k} +\int_{\Omega_m} 2 \mu^\varepsilon e(\vec {u }^\varepsilon) : \nabla \vec {\psi}^{\varepsilon, k}\phi dxdz %\label{4.3.17} $$ Rewriting the second integral of the right-hand side differently, the above expression becomes \begin{equation}\begin{aligned} \int_{\Omega_m} &(\vec {f}^\varepsilon-\nabla p^\varepsilon +\vec{\theta}) . \vec {\psi}^{\varepsilon, k}\phi dxdz\\ =& \int_{\Omega_m} (\xi^\varepsilon \nabla \phi). \vec {\psi}^{\varepsilon, k} +\int_{\Omega_m} \mu^\varepsilon \nabla_x u_{n+1}^\varepsilon. \nabla_x \psi^{\varepsilon, k} \phi dxdz \\ &+ \int_{\Omega_m} \sum_{i=1}^n (\mu^\varepsilon {{\partial u_i^\varepsilon} \over {\partial z}}) {{\partial \psi^{\varepsilon, k}} \over {\partial x_i}} \phi dxdz + \int_{\Omega_m} \sum_{i=1}^n [\xi^\varepsilon]_{n+1 n+1} {{\partial \psi^{\varepsilon, k}} \over {\partial z}} \phi dxdz . \end{aligned}\label{4.3.18} \end{equation} Integrating by parts the second integral of the right-hand side of the above equation and using (\ref{4.3.9}) and(4.3.10), we derive \begin{align*} \int_{\Omega_m} & \mu^\varepsilon \nabla_x u_{n+1}^\varepsilon. \nabla_x \psi^{\varepsilon, k} \phi dxdz \\ =& -\frac 12 \int_{\Omega_m}\hskip-3mm u_{n+1}^\varepsilon \mathop{\rm div}_x ( 2\mu^\varepsilon \nabla_x \psi^{\varepsilon, k})\phi dxdz -\int_{\Omega_m} \mu^\varepsilon u_{n+1}^\varepsilon \nabla_x \phi. \nabla_x \psi^{\varepsilon, k} dxdz \\ =&\frac 12 \int_{\Omega_m} u_{n+1}^\varepsilon \mathop{\rm div}_x ( \nabla_x \zeta^{\varepsilon, k}) \phi dxdz -\int_{\Omega_m} \mu^\varepsilon u_{n+1}^\varepsilon \nabla_x \phi. \nabla_x \psi^{\varepsilon, k} dxdz %\label{4.3.19} \end{align*} Now, integrating by parts the first integral of the right-hand side of the above equation and using (\ref{4.3.9}), we get \begin{align*} \int_{\Omega_m} & \mu^\varepsilon \nabla_x u_{n+1}^\varepsilon. \nabla_x \psi^{\varepsilon, k} \phi dxdz \\ =& -\frac 12 \int_{\Omega_m} \nabla_x u_{n+1}^\varepsilon . \nabla_x \zeta^{\varepsilon, k} \phi dxdz -\frac 12 \int_{\Omega_m} u_{n+1}^\varepsilon (\underline{d}^{\varepsilon , k}.\nabla_x \phi) dxdz %\label{4.3.20} \end{align*} Therefore the expression (\ref{4.3.18}) becomes \begin{align*} \int_{\Omega_m} &(\vec {f}^\varepsilon-\nabla p^\varepsilon +\vec{\theta}) . \vec {\psi}^{\varepsilon, k} \phi dxdz \\ = & \int_{\Omega_m} (\xi^\varepsilon \nabla \phi). \vec {\psi}^{\varepsilon, k} -\frac 12 \int_{\Omega_m} \nabla_x u_{n+1}^\varepsilon . \nabla_x \zeta^{\varepsilon, k} \phi dxdz \\ &-\frac 12 \int_{\Omega_m} u_{n+1}^\varepsilon (\underline{d}^{\varepsilon , k}.\nabla_x \phi) dxdz + \int_{\Omega_m} \sum_{i=1}^n (\mu^\varepsilon {{\partial u_i^\varepsilon} \over {\partial z}}) {{\partial \psi^{\varepsilon, k}} \over {\partial x_i}} \phi dxdz \\ &+ \int_{\Omega_m} \sum_{i=1}^n [\xi^\varepsilon]_{n+1 n+1} {{\partial \psi^{\varepsilon, k}} \over {\partial z}} \phi dxdz . \end{align*} Adding (\ref{4.3.16}) and the above equation, we have (using Definition (\ref{4.3.9})) \begin{equation} \begin{aligned} \int_{\Omega_m} &(\vec {f}^\varepsilon-\nabla p^\varepsilon +\vec{\theta}) . \vec {\psi}^{\varepsilon, k}\phi dxdz - \int_{\Omega_m} \nabla p'^\varepsilon . \vec {\zeta}^{\varepsilon, k}\phi dxdz \\ =& - \int_{\Omega_m} (q^\varepsilon \nabla \phi). \vec {\zeta}^{\varepsilon, k}dxdz -\int_{\Omega_m} \underline{\eta}^{\varepsilon, k}.\nabla_x v_{n+1}^\varepsilon \phi dxdz \\ & -\int_{\Omega_m}\sum_{i=1}^n \mu^\varepsilon {{\partial q_i^\varepsilon} \over {\partial z}} {{\partial \zeta^{\varepsilon, k}} \over {\partial x_i}}\phi dxdz - \int_{\Omega_m} \sum_{j=1}^n [q^\varepsilon]_{n+1 n+1} {{\partial \zeta^{\varepsilon, k}} \over {\partial z}} \phi dxdz\\ &+ \frac 12 \int_{\Omega_m} \sum_{i=1}^n {{\partial u_i^\varepsilon} \over {\partial z}} (\underline{d}^{\varepsilon, k})_i \phi dxdz +\int_{\Omega_m} (\xi^\varepsilon \nabla \phi). \vec {\psi}^{\varepsilon, k}\\ &-\frac 12 \int_{\Omega_m} u_{n+1}^\varepsilon (\underline{d} ^{\varepsilon , k}.\nabla_x \phi) dxdz + \int_{\Omega_m} \sum_{i=1}^n [\xi^\varepsilon]_{n+1 n+1} {{\partial \psi^{\varepsilon, k}} \over {\partial z}} \phi dxdz . \end{aligned} \label{4.3.22} \end{equation} We now pass to the limit in the above equation as $\varepsilon \to 0$. To do so, we need some preliminary results. From convergence (4.3.5) $a)$, we have $$ \vec{\zeta}^{\varepsilon, k} \to \vec{\zeta}^{k}=(\underline 0, 2y_k) \mbox{ weakly in } L^2(\Omega_m)^{n+1}. %\label{4.3.23} $$ and from (4.3.12) a), we get $\vec{\psi}^{\varepsilon, k} \to 0 \mbox{ weakly in } L^2(\Omega_m)^{n+1}$. %\label{4.3.24} Now passing to the limit in (4.3.22) taking into account the precedent convergence results, we obtain \begin{align*} \int_{\Omega_m} \nabla p' . \vec {\zeta}^{k} \phi dxdz = &- \int_{\Omega_m} (q \nabla \phi). \vec {\zeta}^{k}dxdz -\int_{\Omega_m} \underline{\eta}^{k}.\nabla_x v_{n+1} \phi dxdz \\ &-\int_{\Omega_m}\sum_{i=1}^n (\underline{\eta}^{k})_i {{\partial v_i} \over {\partial z}} \phi dxdz -\frac 12 \int_{\Omega_m} u_{n+1} (\underline{d}^{k}.\nabla_x \phi) dxdz \\ &+ \frac 12 \int_{\Omega_m} \sum_{i=1}^n (\underline{d}^{k})_i {{\partial u_i} \over {\partial z}} \phi dxdz. \end{align*} %\label{4.3.25} Integrating by parts the above expression, we derive \begin{align*} 0=& \int_{\Omega_m} q : e(\vec {\zeta}^{k}) \phi dxdz -\int_{\Omega_m}\sum_{i=1}^n 2 (\underline{\eta}^{k})_i \big[e(\vec {v})\big]_{n+1 i} \phi dxdz\\ &+\int_{\Omega_m}\sum_{i=1}^n (\underline{d}^{k})_i \big[e(\vec {u})\big]_{n+1 i} \phi dxdz. %\label{4.3.26} \end{align*} Using the fact that $q : e(\vec {\zeta}^{k})=2 [q]_{n+1 k}$, we obtain, in the distribution sense, $$ [q]_{n+1 k} = \sum_{i=1}^n (\underline{\eta}^{k})_i \big[ e (\vec {v})\big]_{n+1 i} -\frac 12 \sum_{i=1}^n ( \underline{d}^{k})_i \big[e (\vec {u})\big]_{n+1 i}. %\label{4.3.27} $$ Also by Definition (\ref{4.3.5}) c) and (\ref{4.3.12}) c) of $\underline{\eta}^{k}$ and of $\underline{d}^{k}$, we have \begin{equation} \begin{aligned} [q]_{n+1 k}= {1 \over {|Y |}} \sum_{i=1}^n \big \{ \int_Y \mu {{\partial (-\varphi^k+2y_k)} \over {\partial y_i}} dy\big \} \big[ e(\vec {v})\big]_{n+1 i} \\ - {1 \over {2|Y |}} \sum_{i=1}^n \big \{ \int_Y \big( 2 \mu {{\partial \psi^k} \over {\partial y_i}} + {{\partial (-\varphi^k+2y_k)} \over {\partial y_i}}\big) dy \big \} \big[ e(\vec {u})\big]_{n+1 i}. \end{aligned} \label{4.3.28} \end{equation} Hence, since Proposition \ref{prop3.1} holds, \begin{equation} \int_Y \mu {{\partial (-\varphi^k+2y_k)} \over {\partial y_i}} dy = \int_Y \mu {{\partial (-\varphi^i+2y_i)} \over {\partial y_k}} dy, \label{4.3.29} \end{equation} and since Proposition \ref{prop3.2} holds, \begin{equation} \int_Y \big( 2 \mu {{\partial \psi^k} \over {\partial y_i}} + {{\partial (-\varphi^k+2y_k)} \over {\partial y_i}}\big) dy =\int_Y \big( 2 \mu {{\partial \psi^i} \over {\partial y_k}} + {{\partial (-\varphi^i+2y_i)} \over {\partial y_k}}\big) dy . \label{4.3.30} \end{equation} Finally using (\ref{4.3.29}) and (\ref{4.3.30}) in (\ref{4.3.28}) we obtain the announced result, i.e., \begin{align*} [q]_{n+1 k}=& {1 \over {|Y |}} \sum_{i=1}^n \big \{ \int_Y \mu {{\partial (-\varphi^i+2y_i)} \over {\partial y_k}} dy\big \} \big[ e(\vec {v})\big]_{n+1 i} \\ &- {1 \over {2|Y |}} \sum_{i=1}^n \big \{ \int_Y \big( 2 \mu {{\partial \psi^i} \over {\partial y_k}} + {{\partial (-\varphi^i+2y_i)} \over {\partial y_k}} dy \big \} \big[ e(\vec {u})\big]_{n+1 i}. %\label{4.3.31} \end{align*} Since the matrix $q^\varepsilon$ is symmetric, this implies that $q$ is symmetric. Hence $[q]_{n+1 k}=[q]_{ k n+1}$ which completes the proof. \hfill$\Box$ \subsection*{Identification of $[q]_{n+1 n+1}$ in $\Omega_m$} The following proposition gives a result concerning the identification of the last component of $q$. \begin{proposition} \label{prop4.11} $[q^\varepsilon ]_{n+1 n+1} \to [q ]_{n+1 n+1}$ weakly in $L^2(\Omega_m) $ (up to a subsequence), where $$ [q ]_{n+1 n+1}= \big \{ {1 \over {|Y |}} \int_Y 2\mu dy\big \} \big[ e(\vec {v})\big]_{n+1 n+1} -\big[ e(\vec {u})\big]_{n+1 n+1}. %\label{4.4.1} $$ \end{proposition} \paragraph{Proof.} To prove this result, we proceed as in \cite{bc1,bc2}; We use the method of Brizzi \cite{br}. From (\ref{4.1.2}) we have $ [q^\varepsilon ]_{n+1 n+1}= 2 \mu^\varepsilon {{\partial v_{n+1}^\varepsilon} \over {\partial z}} -{{\partial u_{n+1}^\varepsilon} \over {\partial z}}$ and $\mu^\varepsilon= \mu_1 \chi_{\Omega_1^\varepsilon \cap \Omega_m} + \mu_2 \chi_{\Omega_2^\varepsilon \cap \Omega_m}$, hence \begin{multline*} [q^\varepsilon ]_{n+1 n+1}\\ = 2 \mu_1 P_1 \big({{\partial (v_{n+1}^\varepsilon)_1} \over {\partial z}}\big) +2 \mu_2 P_2 \big({{\partial (v_{n+1}^\varepsilon)_2} \over {\partial z}}\big) - P_1 \big({{\partial (u_{n+1}^\varepsilon)_1} \over {\partial z}}\big) -P_2 \big({{\partial (u_{n+1}^\varepsilon)_2} \over {\partial z}}\big), %\label{4.4.2} \end{multline*} where $ (u_{n+1}^\varepsilon)_i= u_{n+1}^\varepsilon |_{\Omega_i^\varepsilon}, (v_{n+1}^\varepsilon)_i= v_{n+1}^\varepsilon |_{\Omega_i^\varepsilon} (i=1,2)$ and $P_i$ represent the extension by 0 in $\Omega \setminus \Omega_i^\varepsilon (i=1,2)$. It is easy to see that $P_i \big({{\partial (u_{n+1}^\varepsilon)_i}\over {\partial z}}\big)$ and $P_i \big({{\partial (v_{n+1}^\varepsilon)_i} \over {\partial z}}\big)$ are bounded in $L^2 (\Omega_m)$. Therefore (up to a subsequence), there exists $\nu_i$ and $\gamma_i \in L^2 (\Omega_m)$ such that \begin{gather} P_i \big({{\partial (u_{n+1}^\varepsilon)_i} \over {\partial z}}\big) \rightharpoonup \nu_i \mbox{ weakly in } L^2(\Omega_m), \label{4.4.3} \\ P_i \big({{\partial (v_{n+1}^\varepsilon)_i} \over {\partial z}}\big) \rightharpoonup \gamma_i \mbox{ weakly in } L^2(\Omega_m). \label{4.4.4} \end{gather} We now proceed to identify these limits. Concerning $\nu_i$, we have \begin{equation} \nu_1 = \rho (x,z) {{\partial u_{n+1}} \over {\partial z}}, \quad \nu_2 = (1-\rho (x,z)) {{\partial u_{n+1}} \over {\partial z}} \label{4.4.5} \end{equation} where $\rho$ is defined by (\ref{4.1.16}) (see Baffico \& Conca \cite{bc3}). In the same way, we can find explicitly $\gamma_i$: Let $\phi \in {\cal D}(\Omega)$, then \begin{multline} _ {H^{-1}(\Omega')}\langle {{\partial } \over {\partial z}} (\chi_{\Omega_1^\varepsilon \cap \Omega_m}) , \phi v_{n+1}^\varepsilon \rangle_{H_0^{1}(\Omega')} \\ = -\int_{\Omega_m}\chi_{\Omega_1^\varepsilon \cap \Omega_m} v_{n+1}^\varepsilon {{\partial \phi} \over {\partial z}} dxdz -\int_{\Omega_m}P_i \big({{\partial (v_{n+1}^\varepsilon)_i} \over {\partial z}}\big) \phi dxdz. \label{4.4.6} \end{multline} Using Lemma \ref{lm4.3} for the sequence $ \big \{ {\partial \over {\partial z }} (\chi_{\Omega_1^\varepsilon \cap \Omega_m})\big \}_{\varepsilon >0}$, it is shown that this sequence satisfies the hypotheses required \cite{bc1,br}. Using (\ref{4.1.7}), we can pass to the limit in the left-hand side of (\ref{4.4.6}). For the right-hand side, from (\ref{4.1.5}), (\ref{4.1.15}) and (\ref{4.4.4}), in the limit, we have \begin{multline*} {H^{-1}(\Omega')}\langle {{\partial } \over {\partial z}} \big({{|O(z) |} \over { |Y |}}\big) , \phi v_{n+1} \rangle_{H_0^{1}(\Omega')} \\ = -\int_{\Omega_m}\big({{|O(z) |} \over { |Y |}}\big) v_{n+1} {{\partial \phi} \over {\partial z}} dxdz -\int_{\Omega_m}\gamma_1 \phi dxdz \end{multline*}\label{4.4.7} and developing the duality product in the last equation, we obtain in the distribution sense the identity \begin{equation} \gamma_1 = \rho (x,z) {{\partial v_{n+1}} \over {\partial z}}. \label{4.4.8} \end{equation} In the same way, passing to the limit in $\langle {{\partial } \over {\partial z}}(\chi_{\Omega_2^\varepsilon \cap \Omega_m}) , \phi v_{n+1}^\varepsilon\rangle $ and using again the compactness Lemma, we get \begin{equation} \gamma_2 = (1-\rho (x,z)) {{\partial v_{n+1}} \over {\partial z}}. \label{4.4.9} \end{equation} From (\ref{4.4.3}) and (\ref{4.4.4}), we have $[q ]_{n+1 n+1}= 2 \mu_1 \gamma_1 + 2\mu_2 \gamma_2 -\nu_1 -\nu_2$. Hence using (4.4.5), (\ref{4.4.8}) and (\ref{4.4.9}), we conclude $$ [q ]_{n+1 n+1}= 2\big(\mu_1 \rho (x,z) + \mu_2 \big(1-\rho (x,z)\big)\big) {{\partial v_{n+1}} \over {\partial z}}- {{\partial u_{n+1}} \over {\partial z}}, %\label{4.4.10} $$ which gives the announced result, that is $$ [q]_{n+1 n+1}= \big \{ {1 \over {|Y |}} \int_Y 2\mu dy\big \} \big[ e(\vec {v})\big]_{n+1 n+1} -\big[ e(\vec {u})\big]_{n+1 n+1}. %\label{4.4.11} $$ Hence Proposition \ref{prop4.11} is proved. \hfill$\Box$ \subsection*{Identification of $q$ in $\Omega_1$ and $\Omega_2$} \begin{proposition} \label{prop4.12} For $i=1,2$ $q^\varepsilon |_{\Omega_i} \rightharpoonup q |_{\Omega_i}$ weakly in $L^2(\Omega_m)^{n+1^2}$ (up to a subsequence), where $$ q|_{\Omega_i}= 2 \mu_i e(\vec {v}|_{\Omega_i}) -e(\vec {u}|_{\Omega_i}). %\label{4.5.1} $$ \end{proposition} \paragraph{Proof.} From (\ref{4.1.2}), we have $$ q^\varepsilon |_{\Omega_i}= 2 \mu_i e(\vec {v}^\varepsilon |_{\Omega_i}) -e(\vec {u}^\varepsilon |_{\Omega_i}). %\label{4.5.2} $$ From the convergence (\ref{4.1.5}), (\ref{4.1.7}), and (\ref{4.1.13}), we easily obtain the result of Proposition \ref{prop4.12}. \hfill$\Box$\smallskip \paragraph{Conclusion} From Propositions \ref{prop4.6}, \ref{prop4.10}, \ref{prop4.11}, and \ref{prop4.12}, from the definition of tensors $\mathcal{A}$ and $\mathcal{B}$, we conclude that $q$, $\vec{u}$, and $\vec{v} $ are related by (\ref{4.1.18}). Therefore, since $q$ satisfies (\ref{4.1.14}) by Proposition \ref{prop4.2}, we have that $(\vec{u}, p)$ and $(\vec{v}, p)$ are solutions of (\ref{3.1.16}). From the properties of the tensors $\mathcal{A}$ and $\mathcal{B}$, problem (\ref{3.1.16}) admits a unique solution and hence, the whole sequence $\vec{u}^\varepsilon$ and $\vec{v}^\varepsilon$ converge weakly to $\vec{u}$ and $\vec{v}$ respectively. This completes the proof of Theorem \ref{thm3.4}. \hfill$\Box$\smallskip \section{Optimal control} The following theorem gives a convergence result of the optimal control. \begin{theorem} \label{thm5.1} For $\theta$ fixed in $\mathcal{U}_{ad}$, we consider $(\vec{u}, p)$ as the solution of \begin{equation} \begin{gathered} -\mathop{\rm div}({\cal A } e(\vec{u}) ) =\vec{f}-\nabla p +\vec{\theta} \quad\mbox{in } \Omega \\ \mathop{\rm div}\vec{u}=0 \quad\mbox{in }\Omega \\ \vec{u} =0 \quad\mbox{on }\partial\Omega. \end{gathered} \label{5.1} \end{equation} The cost function is $$ J_0 (\vec{\theta})=\frac 12\int_\Omega {\cal B } e(\vec{u}): e(\vec{u}) dxdz + {N\over 2}\int_\Omega |\vec{\theta}|^2 dxdz. %\label{5.2} $$ Then the optimal control $\vec{\theta}_\star^\varepsilon$ of problem (\ref{2.2.1})--(\ref{2.2.3}) satisfies $$ \vec{\theta}_\star^\varepsilon \to \vec{\theta}_\star \mbox{ strongly in } L^2(\Omega)^{n+1} $$ and $\vec{\theta}_\star $ satisfies the optimality condition \begin{equation} J_0 (\vec{\theta}_\star ) =\min_{\vec{\theta} \in \mathcal{U}_{ad} } J_0 (\vec{\theta} ), \label{5.3} \end{equation} Furthermore there is convergence of the minimal cost, i.e., \begin{equation} \lim_{\varepsilon \to 0} J_\varepsilon (\vec{\theta}_\star^\varepsilon ) = J_0 (\vec{\theta}_\star ). \label{5.4} \end{equation} \end{theorem} \paragraph{Proof.} {\bf Step 1: A priori estimates} By the optimal control definition, we have for all $\vec{\theta}$ in $\mathcal{U}_{ad}$ $$ {N\over 2}\Vert \vec{\theta}_\star^\varepsilon \Vert_{L^2 (\Omega)^{n+1}} \leq J_\varepsilon (\vec{\theta}_\star^\varepsilon ) \leq J_\varepsilon (\vec{\theta}) \leq C, %\label{5.5} $$ so $\vec{\theta}_\star^\varepsilon $ is bounded in $L^2 (\Omega)^n $ and is such that (for a subsequence) \begin{equation} \vec{\theta}_\star^\varepsilon \rightharpoonup \vec{\theta}_\star \mbox{ weakly in } L^2(\Omega) ^{n+1}. \label{5.6} \end{equation} We will show later in the proof that the above weakly convergence is in fact a strong convergence (cf \ref{5.19}). Let $(\vec{u}_{\star}^{\varepsilon}, p_{\star}^{\varepsilon})$ and $ (\vec{v}_{\star}^{\varepsilon}, p^{'\varepsilon}_\star)$ the optimal state and the corresponding adjoint state respectively associated to $\vec{\theta}_\star^\varepsilon$. By the same arguments as those used in the proof of Theorem \ref{thm3.4} (the fact that $\vec{\theta}$ is replaced by $ \vec{\theta}_\star^\varepsilon$ poses no problem), we get \begin{equation} \begin{gathered} \vec{u}_{\star}^{\varepsilon} \rightharpoonup \vec{u}_{\star} \mbox{ weakly in } H^1(\Omega)^{n+1}, \quad p_{\star}^{\varepsilon} \rightharpoonup p_{\star} \mbox{ weakly in } L_0 ^2 (\Omega ), \\ \vec{v}_{\star}^{\varepsilon} \rightharpoonup \vec{v}_{\star} \mbox{ weakly in } H^1(\Omega)^{n+1}, \quad p^{'\varepsilon}_\star \rightharpoonup p'_{\star} \mbox{ weakly in } L_0 ^2 (\Omega ) \end{gathered} \label{5.7} \end{equation} where $(\vec{u}_{\star}, p_{\star}) $ and $(\vec{v}_{\star} ,p'_{\star})$ satisfy \begin{equation} \begin{gathered} -\mathop{\rm div}\big(\mathcal{A} e(\vec{u}_{\star})\big) =\vec {f}-\nabla p_{\star} +\vec{\theta}_\star \quad\mbox{in } \Omega \\ \mathop{\rm div}\big(\mathcal{A} e(\vec{v}_{\star}) -\mathcal{B} e(\vec{u}_{\star})\big) =-\nabla p'_{\star} \mbox{ dans } \Omega \\ \mathop{\rm div} \vec{u}_{\star} =\mathop{\rm div} \vec{v}_{\star}= 0 \quad\mbox{in } \Omega\\ \vec{u}_{\star}= \vec{v}_{\star} = 0 \quad\mbox{on } \partial\Omega, \end{gathered} \label{5.8} \end{equation} the optimal control $\vec{\theta}_\star$ is characterized by the variationnal inequality $$ \vec{\theta}_\star \in \mathcal{U}_{ad} \quad \mbox{and}\quad \int_\Omega (\vec{v}_{\star} + N \vec{\theta}_{\star}). (\vec{\theta} -\vec{\theta}_{\star}) \geq 0 \;\forall \vec{\theta} \in \mathcal{U}_{ad}. $$ {\bf Step 2: Energy convergence} Using integration by parts and the first equation of (\ref{2.2.5}), we have \begin{equation}\begin{aligned} \int_{\Omega} e(\vec{u}_{\star}^{\varepsilon}): e(\vec{u}_{\star}^{\varepsilon}) dxdz =& - \int_{\Omega} \mathop{\rm div}( e(\vec{u}_{\star}^{\varepsilon})) . \vec{u}_{\star}^{\varepsilon} dxdz \\ =& - \int_{\Omega} \mathop{\rm div}(2\mu^\varepsilon e(\vec{v}_{\star}^{\varepsilon})). \vec{u}_{\star}^{\varepsilon} dxdz - \int_{\Omega} \nabla p^{'\varepsilon}_\star . \vec{u}_{\star}^{\varepsilon} dxdz . \end{aligned} \label{5.9} \end{equation} Integrating now by parts the right-hand side of the above equation and using the second equation of (\ref{2.2.5}), \begin{equation} \begin{aligned} \int_{\Omega} e(\vec{u}_{\star}^{\varepsilon}): e(\vec{u}_{\star}^{\varepsilon}) dxdz =&\int_{\Omega} 2\mu^\varepsilon e(\vec{v}_{\star}^{\varepsilon}): e(\vec{u}_{\star}^{\varepsilon}) dxdz \\ =& \int_{\Omega} 2\mu^\varepsilon e(\vec{u}_{\star}^{\varepsilon}) : e(\vec{v}_{\star}^{\varepsilon}) dxdz \\ =& - \int_{\Omega} \mathop{\rm div}(2\mu^\varepsilon e(\vec{u}_{\star}^{\varepsilon})). \vec{v}_{\star}^{\varepsilon} dxdz \\ =& \int_{\Omega} (\vec {f}^\varepsilon-\nabla p_{\star}^\varepsilon +\vec{\theta}_{\star}^\varepsilon ). \vec{v}_{\star}^{\varepsilon} dxdz . \end{aligned} \label{5.10} \end{equation} Using (\ref{4.1.17}),(\ref{5.7}) and (\ref{5.8}), we have $$ \int_{\Omega} e(\vec{u}_{\star}^{\varepsilon}): e(\vec{u}_{\star}^{\varepsilon}) dxdz \to \int_{\Omega} (\vec {f}-\nabla p_{\star} +\vec{\theta}_{\star} ). \vec{v}_{\star} dxdz. %\label{5.11} $$ Using the first equation of (\ref{5.8}) and integrating by parts in the right-hand side of the above equation, we get (using the symmetry properties of $\mathcal{A}$), \begin{equation}\begin{aligned} \int_{\Omega} (\vec {f}-\nabla p_{\star} +\vec{\theta}_{\star} ) .\vec{v}_{\star} dxdz =& -\int_{\Omega} \mathop{\rm div} (\mathcal{A} e(\vec{u}_{\star})). \vec{v}_{\star} dxdz \\ =& -\int_{\Omega} \mathcal{A} e(\vec{u}_{\star}) : e(\vec{v}_{\star}) dxdz \\ =& -\int_{\Omega} e(\vec{u}_{\star}) : \mathcal{A} e(\vec{v}_{\star}) dxdz \\ =& -\int_{\Omega} \mathop{\rm div} (\mathcal{A} e(\vec{v}_{\star})). \vec{u}_{\star} dxdz. \end{aligned} \label{5.12} \end{equation} Using now the second equation of (\ref{5.9}) and integrating by parts in the last integral of (\ref{5.12}), \begin{equation} \begin{aligned} -\int_{\Omega} \mathop{\rm div} (\mathcal{A} e(\vec{v}_{\star})). \vec{u}_{\star} dxdz =&-\int_{\Omega} \mathop{\rm div} (\mathcal{B} e(\vec{u}_{\star})). \vec{u}_{\star} dxdz +\int_{\Omega} \nabla p'_{\star} . \vec{u}_{\star} dxdz \\ =& -\int_{\Omega} \mathop{\rm div} (\mathcal{B} e(\vec{u}_{\star})). \vec{u}_{\star} dxdz \\ =& \int_{\Omega} \mathcal{B} e(\vec{u}_{\star}) :e(\vec{u}_{\star}) dxdz . \end{aligned} \label{5.13} \end{equation} Finally, by (\ref{5.10})-(\ref{5.13}), we have the energy convergence \begin{equation} \int_{\Omega} e(\vec{u}_{\star}^{\varepsilon}): e(\vec{u}_{\star}^{\varepsilon}) dxdz \to \int_{\Omega} \mathcal{B} e(\vec{u}_{\star}) :e(\vec{u}_{\star}) dxdz . \label{5.14} \end{equation} {\bf Step 3} Taking into account the definition of optimal control (\ref{5.3}), we have $$ \forall \vec{\theta} \in \mathcal{U}_{ad} \quad J_\varepsilon (\vec{\theta}) \geq J_\varepsilon (\vec{\theta}_{\star} ^\varepsilon ). %\leqno (5.15) $$ Now passing to the limit in inequality (\ref{4.3.18}) and using (\ref{4.3.16}), we get \begin{equation} J_0(\vec{\theta})\geq { 1\over 2 } \int_{\Omega} \mathcal{B} e(\vec{u}_{\star}) :e(\vec{u}_{\star}) dxdz + { N\over 2 }\limsup_{\varepsilon \to 0} \int_{\Omega} |\vec{\theta}_{\star}^\varepsilon | ^2 dxdz. \label{5.16} \end{equation} Thus taking $ \vec{\theta}= \vec{\theta}_{\star} $ in the above equation, we have $$ \limsup_{\varepsilon \to 0} \int_{\Omega} |\vec{\theta}_{\star}^\varepsilon |^2 dxdz \leq \int_{\Omega} |\vec{\theta}_{\star} |^2 dxdz. %\leqno (5.17) $$ By (\ref{5.6}) and the above equation, we obtain \begin{equation} \lim_{\varepsilon \to 0} \int_{\Omega} |\vec{\theta}_{\star}^\varepsilon |^2 dxdz = \int_{\Omega} |\vec{\theta}_{\star} |^2 dxdz. \label{5.18} \end{equation} From (\ref{5.16}) and the above equation, we have (\ref{5.3}). Now from (\ref{5.14}) and the above equation, we get (\ref{5.4}). Finally from (\ref{5.6}) and (\ref{5.18}), we derive \begin{equation} \vec{\theta}_{\star}^\varepsilon \to \vec{\theta}_{\star} \quad\mbox{strongly in } L^2(\Omega)^{n+1}. \label{5.19} \end{equation} This completes the proof. \hfill$\Box$ \section{The case $\Omega \subset \mathbb{R}^2$} When $\Omega \subset \mathbb{R}^2$, it is possible to find explicit functions which are solutions of (\ref{3.1.1}), (\ref{3.1.2}), (\ref{3.1.5}), and (\ref{3.1.6}) and satisfies hypotheses (\ref{4.2.1}), (\ref{4.2.4}), (\ref{4.2.6}), (\ref{4.2.9}), (\ref{4.3.1}), (\ref{4.3.4}), (\ref{4.3.7}) and (\ref{4.3.11}). In this case $Y=]0, 1[$ is as in figure 2 (left) and Problems (\ref{3.1.1}) and (\ref{3.1.2}) become \begin{equation} \begin{gathered} -\frac{d}{dy} \big(2 \mu \frac{d}{dy}(-\chi+y )\big) = -{{d r_1}\over dy } \quad\mbox{in } ]0, 1[ \\ {{d \chi} \over dy } = 0 \quad\mbox{in } ]0, 1[\\ \chi (0)= \chi (1), r_1(0)=r_1(1)\end{gathered} \label{6.1} \end{equation} and \begin{equation} \begin{gathered} -\frac{d}{dy} \big(\mu \frac{d}{dy}(-\varphi+2y)\big) =0 \quad\mbox{in } ]0, 1[ \\ \varphi(0)= \varphi(1). \end{gathered} \label{6.2} \end{equation} We have the following result (see Baffico \& Conca \cite{bc3}): \begin{proposition} \label{prop6.1} For $z \in [0, h_1[$ fixed. Let $\chi =0$, $r_1= 2\mu $ and $$ \varphi(y)= \begin{cases} (2- {C \over \mu_2} ) y &\mbox{if } y \in ]0, a(z)[ \\ (2- {C \over \mu_1} ) y + a ({C \over \mu_1}-{C \over \mu_2}) &\mbox{if } y \in ]a(z), b(z)[ \\ (2- {C \over \mu_2} ) y + (a-b) ({C \over \mu_1}-{C \over \mu_2}) &\mbox{if } y \in ]b(z), 1[ \end{cases} $$ with $a=a(z)$, $b=b(z)$, and $$ C=C(z)= 2 \big( \frac 1{|Y|} \int_Y {1 \over \mu} dy \big) ^{-1}. $$ Then $(\chi, r_1)$ and $\varphi$ are the unique solution (up to an additive constant) of Problems (\ref{6.1}) and (\ref{6.2}). \hfill$\Box$ \end{proposition} \begin{remark} \label{rmk6.2} \rm We can calculate the value of $C(z)$ (see \cite{bc2,bc3}), we get $$ C(z)= {{ 2 \mu_1 \mu_2 } \over {(\mu_2 - \mu_1) (b(z) -a(z)) + \mu_1}}. $$ Studying the regularity of the solutions of Problems (6.1) and (6.2) and supposing that $\mu_2 < \mu_1$, hypotheses (\ref{4.2.1}), (\ref{4.2.4}), (\ref{4.3.1}) and (\ref{4.3.4}) are satisfied (cf \cite{bc3}). \end{remark} Let study now Problems (\ref{3.1.5}) and (\ref{3.1.6}), they becomes \begin{equation} \begin{gathered} -\frac{d}{dy} \big(2 \mu {d \lambda \over dy } +\frac{d}{dy}(-\chi+y )\big) = -{{d r_2} \over dy } \quad\mbox{in } ]0, 1[ \\ \frac{\lambda}{dy} = 0 \quad\mbox{in } ]0, 1[\\ \lambda (0)= \lambda (1), r_2(0)=r_2(1)\end{gathered} \label{6.3} \end{equation} and \begin{equation} \begin{gathered} -\frac{d}{dy} \big(\mu {d \psi \over dy }+\frac{d}{dy}(-\varphi+2y)\big) =0 \quad\mbox{in } ]0, 1[ \\ \psi(0)= \psi(1). \end{gathered} \label{6.4} \end{equation} \begin{proposition} \label{prop6.3} For $z$ fixed in $[0, h_1[$, Let $ \lambda =c_1$, $r_2= c_2 $, where $c_1$ and $c_2$ are constants, and let $$ \psi(y)= \begin{cases} { 1 \over 2 \mu_2}(K- {C \over \mu_2} ) y &\mbox{if } y \in ]0, a(z)[ \\ { 1 \over 2 \mu_1}(K- {C \over \mu_1} ) y + a \big( ({K \over 2\mu_2}-{K \over 2\mu_1})-({C \over 2\mu_2^2} -{C \over 2\mu_1^2})\big) & \mbox{if } y \in ]a(z), b(z)[ \\ { 1 \over 2 \mu_2} (K- {C \over \mu_1} ) y\\ + (a-b) \big( ({K \over 2\mu_2}-{K \over 2\mu_1})-({C \over 2\mu_2^2} -{C \over 2\mu_1^2})\big) &\mbox{if } y \in ]b(z), 1[ \end{cases} $$ with $$ K=K(z)= {C\over { 2\mu_1 \mu_2}}\big(2(\mu_1 +\mu_2)- C\big) $$ and the constant $C$ defined in Proposition \ref{prop6.1}. Then $(\lambda, r_2)$ and $\psi$ are the unique solutions (up to an additive constant) of Problems (\ref{6.3}) and (\ref{6.4}). \end{proposition} \begin{remark} \label{rmk6.4} \rm The hypotheses (\ref{4.2.6}), (\ref{4.2.9}), (\ref{4.3.7}) and (\ref{4.3.11}) are satisfied as shown next. \end{remark} From Proposition \ref{prop6.3}, it is easy to check that $\lambda = c_1$ verifies hypothesis (\ref{4.2.6}). It is no longer necessary to suppose hypothesis (\ref{4.2.9}), since we can pass to the limit in $\langle \frac{\partial}{\partial z} r_2^\varepsilon , u_{n+1}^\varepsilon\rangle$ using the explicit form of $r_2$ and $r_2^\varepsilon$. By the form of $\psi$ (see Proposition \ref{prop6.3}), and since the function $h$ satisfies i) and ii) (see Section 2), we show that $\psi=\psi(y,z) \in H_{\mbox{loc}}^1(Y\times [0, h_1[)$ (using the implicit function theorem). Therefore, Hypothesis (4.3.7) is fulfilled. Concerning Hypothesis (\ref{4.3.11}), since we can calculate explicitly $ { \partial \psi \over \partial y}$ and $ { \partial \varphi \over \partial y}$, the expression (\ref{4.3.9}) becomes $d^\varepsilon= K(z)$ where $K$ is defined in Proposition \ref{prop6.3}. Part a) of (\ref{4.3.11}) follows from regularity of $K(z)$ which follows from the regularity of $C(z)$. Concerning part b), we must show that $ { \partial d^\varepsilon \over \partial z} \geq 0$ in the distributions sense, i.e $$ \langle { \partial d^\varepsilon \over \partial z}, \phi\rangle \geq 0.\quad \forall \phi \in {\cal D}(\Omega), \phi \geq 0 $$ Indeed, from the form of $K$ and after elementary computations, we get $$ \langle { \partial d^\varepsilon \over \partial z}, \phi\rangle = - \langle C(z) \big({1 \over \mu_1} + {1 \over \mu_2}\big) - {C(z)^2 \over 2\mu_1 \mu_2} , { \partial \phi \over \partial z}\rangle . $$ From \cite{bc3}, if $\mu_2<\mu_1$, we have $ - \langle C(z),{ \partial \phi \over \partial z}\rangle \geq 0$, and since $ \mu_1,\mu_2 >0$, hence $$- \langle C(z) \big({1 \over \mu_1} + {1 \over \mu_2}\big) , { \partial \phi \over \partial z}\rangle \geq 0. $$ Also $ \langle {C(z)^2 \over 2\mu_1 \mu_2} ,{ \partial \phi \over \partial z} \rangle \geq 0$; therefore, $ { \partial d^\varepsilon \over \partial z} \geq 0$ in the distribution sense. \hfill$\Box$ \begin{remark} \label{rmk6.5} \rm Propositions \ref{prop6.1} and \ref{prop6.3} allow us to compute explicitly the coefficients of the tensors $\mathcal{A}$ and $\mathcal{B}$. \end{remark} For tensor $\mathcal{A}$ (cf \cite{bc3}), we have \begin{gather*} a_{1111}= a_{2222}= \begin{cases} 2\mu_1 & \mbox{if } h_1