\documentclass[twoside]{article} \usepackage{graphics} \usepackage{graphicx} \usepackage{latexsym} \usepackage{amssymb} \usepackage{amsmath} \usepackage{BSPMsample} \numberwithin{equation}{section} \newcommand{\ree}{\mathbb{R}} \begin{document} \title[Asymptotic modeling of thin plastic oscillating layer] {Asymptotic modeling of thin plastic oscillating layer}%\thanks{LaRI Department ***}} %between [] goes the title that will appear on the top of every %odd page, between {} goes the real title. %\thanks{...} put your grant \author[A. Ait Moussa ~and ~M. Verid ABDELKADER]{A. Ait Moussa ~and ~M. Verid ABDELKADER} %between [] goes the authors that will appear on the top of every %even page, between {} goes the real author. \address{Ait Moussa Abdlaziz\\Department of Mathematics. \\ University of Mohammed 1st, \\60000 Oujda, Morroco\\ \\ ABDELKADER Mohamed Verid\\Department of Mathematics. \\ University of Mohammed 1st, \\60000 Oujda, Morroco\\ \email: ab.verid@gmail.com} \maketitle \begin{abstract} In this paper we study the asymptotic behavior of solutions to a elasticity problem, of a containing structure a plastic thin oscillating layer of thickness and rigidity depending of small parameters $\varepsilon$. We use the epi-convergence method to find the limit problems with interface conditions. \end{abstract} %%put here Key words \keywords Limit behavior, plasticity problem, thin oscillating layer, epi-convergence method, subadditive theorem. \tableofcontents %% put here the subject class \2000mathclass{35B40, 35L70} \section{Introduction}\label{sec1} The inclusion of a very thin layer of very rigid material into a given elastic body has been widely considered, and in the classic literature. For more details, we can refer to \cite{Eljaroudi}, \cite{VB}, \cite{Suquet} and \cite{Ait}. In general, the computation of solution using numerical methods is very difficult. In one hind, this is because the thickness of the adhesive requires a fine mesh, which in turn implies an increase of the degrees of flexible than the adherents, and this produces numerical instabilities in the stiffness matrix. To overcame this difficulties, thanks to Goland and Reissner \cite{Goland} find a limit problem in which the adhesive is treated on this theoretical approach, see for example A. Ait Moussa and J. Messaho \cite{AitJ}, Acerbi, Buttazzo and Perceivable \cite{AcerbiBP}, Licht and Michail \cite{LichtM} and A. Ait Moussa and L. Zla\"{\i}ji \cite{AitL}. In this present work, we consider a structure containing a plastic thin oscillating layer of thickness, rigidity, and periodicity parameter depending on $\varepsilon$, where $\varepsilon$ being a parameter intended to tend towards 0. In a such structure, we have treated the scalar case for a thermal conductivity problem in \cite{AitV}. The aim of this work is to study the limit behavior of an elasticity problem with a convex energy functional posed in a such structure. This paper is organized in the following way. In section 2, we express the problem to study, and we give some notation and we define functional spaces for this study in the section 3. In the section 4, we study the problem (4.1). The section 5 is reserved to the determination of the limits problems and our main result. %\section{Statement of the problem }\label{sec2} \section{Statement of the problem}\label{sec2} We consider a structure, occupying a bonded domain $\Omega\in\mathbb{R}^3$ with Lipschitzian boundary $\partial\Omega$. It is constituted of two elastic bodies joined together by an incompressible rigid thin layer with oscillating boundary, the latter obeys to nonlinear elastic low of power type. More precisely, the stress field is related to the displacement's field by \begin{equation*} \sigma^\varepsilon= \frac{1}{\varepsilon}~|e(u^\varepsilon)|^{-1}~e(u^\varepsilon),~~\varepsilon>0. \end{equation*} The structure occupies the regular domain $\Omega=B_\varepsilon\cup\Omega_\varepsilon$, where $B_\varepsilon$ is given by $B_\varepsilon = \{x=(x',x_3) ~/~ |x_3|<\frac{\varepsilon}{2}\}$, and $\Omega_\varepsilon=\Omega\setminus B_\varepsilon$ represent the regions occupied by the thin plate and the two elastic bodies, see you Figure 1, $\varepsilon$ being a positive parameter intended to approach 0, and $\Sigma=\{x=(x',x_3) ~/~ |x_3|=0\}.$ The structure is subjected to a density of forces of volume $f$, $f:\Omega\rightarrow\mathbb{R}^3$, and it is fixed on the boundary $\partial\Omega.$ Equations which relate the stress field $\sigma^\varepsilon$, $\sigma^\varepsilon:\Omega\rightarrow\mathbb{R}_S^9$, and the field of displacement $u^\varepsilon$, $u^\varepsilon:\Omega\rightarrow\mathbb{R}^3$ are $$\left\{\begin{array}{cccc} div(\sigma^\varepsilon)+ f = 0 & \text{in} & \Omega \\~~\\ \sigma_{ij}^\varepsilon= a_{ijkh}e_{kh}(u^\varepsilon) & \text{in} & \Omega_\varepsilon \\~~\\ \sigma^\varepsilon=\frac{1}{\varepsilon}{|e(u^\varepsilon)|}^{-1}e(u^\varepsilon) & \text{in} & B_\varepsilon \\~~\\ u^\varepsilon=0 & \text{on} & \partial\Omega \end{array} ~~ ~~ ~~~ ~~ ~~ ~~ ~~~ ~~ ~~(\mathbb{P}_\epsilon)\right.$$~~ Where $a_{ijkh}$ are the elasticity coefficients, and $\mathbb{R}^9_S$ the vector space of the square symmetrical matrices of order three, $e_{ij}(u)$ are the components of the linearized tensor of deformation $e(u)$. \begin{figure} \begin{center} % Requires \usepackage{graphicx}\ \includegraphics[width=5in,height=3in]{domain1.jpg} \caption{Domain}\label{fig1} \end{center} \end{figure} $\varphi_\varepsilon$ being a bounded real function and ${]0,\varepsilon[}^2$-periodic. In the sequel, we assume that the elasticity coefficients $a_{ijkh}$ satisfy to the following hypotheses : \begin{eqnarray}\label{sec2} % \nonumber to remove numbering (before each equation) a_{ijkh}\in L^\infty(\Omega) \\ a_{ijkh}~=~a_{jikh}~=~a_{khij} \\ a_{ijkh}\tau_{ij}\tau_{kh}\geq C\tau_{ij}\tau_{ij},~\forall\tau\in\mathbb{R}^9_S \end{eqnarray} \section{Notation and Function Setting}\label{sec3} \subsection{Notations} We begin by introducing some notation which is used throughout the paper~ $x=(x',x_3)$, where $x'=(x_1,x_2)$, $\tau\otimes\zeta=(\tau_i\zeta_i)_{1\leq i,j\leq 3}$ and \begin{equation*} \tau\otimes_s\zeta=\frac{\tau\otimes\zeta+\zeta\otimes\tau}{2},~\forall\tau,\zeta\in\mathbb{R}^3;~e^*(.)=(\nabla'+\nabla^{'t})(.);\text{ or}~\nabla'=(\frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2}). \end{equation*} We set $Y=]0,1[\times]0,1[$, $\varphi:\mathbb{R}^2\rightarrow [a_1,a_2]$, such as $a_2\geq a_1>0$, and $\varphi$ is $Y$-periodic, $\varphi_\varepsilon(x')=\varphi(\frac{x'}{\varepsilon})$, and \begin{equation*} m(\varphi)=\frac{1}{\int_Ydx'}\int_Y\varphi(x')dx'. \end{equation*} In the following $C$ will denote any constant with respect to $\varepsilon$, $[v]$ is the jump of displacement field $v$ through $\Sigma,$ and $\nu$, $H_2$ respectively the Lebesgue Hausdorff measures. Also, we use the convention $0.(+\infty).$ \subsection{Functions } First, we introduce the following space : \begin{equation*} V^\varepsilon=\{u\in L^1(\Omega) / e(u)\in L^2(\Omega_\varepsilon,\mathbb{R}_s^9),~~u\in LD_0(B_\varepsilon,\mathbb{R}_s^3),~ {[u]}^\varepsilon=0~\text{in }\Sigma^{\pm}_\varepsilon,\text{ and} ~~u=0 \text{ on }\partial\Omega~ \} \end{equation*} where ${[u]}^\varepsilon$ is the jump of $u$ on $\Sigma^{\pm}_\varepsilon$ defined by \begin{equation*} {[u]}^\varepsilon=\pm u_{|_{\Omega_\varepsilon^{\pm}}}\mp u_{|_{B_\varepsilon^{\pm}}}, \end{equation*} and \begin{equation*} LD_0\{u\in L^1(\Omega,\mathbb{R}^3) / e(u)\in L^1(\Omega,\mathbb{R}_s^9), ~\text{and}~ u=0~ \text{on}~ \partial\Omega\} \end{equation*} we easily show that $V^\varepsilon$ is a Banach space with respect to the norm \begin{equation*} u\rightarrow\|e(u)\|_{L^2(\Omega_\varepsilon,\mathbb{R}_s^9)}+\|e(u)\|_{L^1(B_\varepsilon,\mathbb{R}_s^9)}. \end{equation*} Our goal in this work is to study the problem $(P_\varepsilon)$, and its limit behavior when $\varepsilon$ tends to zero. \section{Study of Problem}\label{sec4} The problem $\mathbb{P}_\varepsilon$ is equivalent of the minimization problem \begin{equation}\label{4.1} inf_{v\in V^\varepsilon}\{\frac{1}{2}\int_{\Omega_\varepsilon}a_{ijhk}e_{ij}(v)e_{hk}(v)dx + \frac{1}{\varepsilon}\int_{B_\varepsilon}|e(v)| - \int_\Omega fvdx\} \end{equation} To study problem $\mathbb{P}_\varepsilon$, we will study the minimization problem (4.1). The existence and uniqueness of solutions to (4.1) is given in the following proposition. \begin{prop} %\label{prop4.1} Under the hypotheses $(2.1)$, $(2.2)$, $(2.3)$ and for $f\in L^\infty$, problem $(4.1)$ admits an unique solution. \end{prop} \begin{pf}%\label{prop4.1} From $(2.1)$ and $(2.3)$, we show easily that the energy functional in $(4.1)$ is weakly lower semicontinuous, strictly convex and coercive over $V^\varepsilon$, Since $V^\varepsilon$ is not reflexive, so we may not apply directly result given in Dacorogna \cite{DA}, but we can follow our proof by using the compact imbedding of Sobolev for the $LD_0(\Omega)$ space in the reflexivity space $L^q(\Omega)$, or $q\in]1,\frac{3}{2}]$ for more information see Temam (\cite{Temam} p.117).\\ On the other hand, let $u_n$ be a minimizing sequence for $(4.1)$, to simplify the writing let \begin{equation*} F^\varepsilon(v)=\frac{1}{2}\int_{\Omega_\varepsilon}a_{ijhk}e_{ij}(v)e_{hk}(v)dx + \frac{1}{\varepsilon}\int_{B_\varepsilon}|e(v)| - \int_\Omega fvdx \end{equation*} so, we have $F^\varepsilon(u_n)\rightarrow\inf_{v\in V^\varepsilon}F(v)$. Using the coercivity of $F^\varepsilon$, we may then deduce that there exists a constant $C>0$, independent of $n$, such that \begin{equation*}\|u_n\|_{V^\varepsilon}\leq C,\end{equation*} then $u_n$ bounded in $L^q$, therefore a subsequence of $u_n$, still denoted by $u_n$, there exists $u_0\in V^\varepsilon$ such that $u_n\rightharpoonup u_0$ in $V^\varepsilon$. The weak lower semi-continuity and the strict convexity of $F^\varepsilon$ imply then the result. \end{pf} \begin{lem}%\label{prop4.1} Assuming that for any sequence ${(u^\varepsilon)}_\varepsilon\subset V^\varepsilon$, there exists a constant $C>0$ such that $F^\varepsilon(u^\varepsilon)\leq C$, under $(2.1)$, $(2.3)$ and for $f\in L^\infty(\Omega,\ree^3)$, ${(u^\varepsilon)}_{\varepsilon>0}$ satisfies \begin{equation}%\label{sec4} \|e(u^\varepsilon)\|^2_{L^2(\Omega_\varepsilon,\ree^9_s)}\leq C \end{equation} \begin{equation} \frac{1}{\varepsilon} \int_{B_\varepsilon}|e(u^\varepsilon)|\leq C. \end{equation} \end{lem} moreover $u^\varepsilon$ is bounded in $W^{1,1}_0(\Omega,\ree^3).$ \begin{pf} Science $F^\varepsilon(u^\varepsilon)\leq C$, we have \begin{equation*} \frac{1}{2}\int_{\Omega_\varepsilon}a_{ijhk}e_{ij}(u^\varepsilon)e_{hk}(u^\varepsilon)dx + \frac{1}{\varepsilon}\int_{B_\varepsilon}|e(u^\varepsilon)|- \int_\Omega fu^\varepsilon dx \leq C \end{equation*} according to (2.3), we have \begin{equation}%\lable{sec4} \|e(u^\varepsilon)\|^2_{_{L^2{(\Omega_\varepsilon,\mathbb{R}_s^9)}}} + \frac{1}{\varepsilon}\int_{B_\varepsilon}|e(u^\varepsilon)| \leq C+C\int_\Omega fu^\varepsilon dx. \end{equation} Then, taking advantage of the fact that $u^\varepsilon$ vanishes on $\partial\Omega$ : \begin{equation*} \int_\Omega fu^\varepsilon dx= \int_{\Omega_\varepsilon} fu^\varepsilon dx + \int_{B_\varepsilon} fu^\varepsilon dx. \end{equation*} Where \begin{equation} \int_{\Omega_\varepsilon} fu^\varepsilon dx=\int_{\Omega} \chi_{\Omega_\varepsilon}fu^\varepsilon dx\leq C\|e(u^\varepsilon)\|_{L^2_{(\Omega_\varepsilon,\mathbb{R}_s^9)}} \end{equation} otherwise since $LD_0\hookrightarrow L^q(\Omega,\mathbb{R}^3)$ for all $q\in[1,\frac{3}{2}]$, in particular for $q_0=\frac{3}{2}$, we denote by $q'_0$ the conjugate of $q_0$, by H\"{o}lder inequality, we obtain \begin{equation*} \int_{B_\varepsilon}fu^\varepsilon\leq \|f\|_{L^{q'_0}(B_\varepsilon,\mathbb{R}^3)}\|u^\varepsilon\|_{L^{q_0}(B_\varepsilon,\mathbb{R}^3)} \end{equation*} since $u^\varepsilon=0~\text{on}~\partial B_\varepsilon$, one has, according to Pioncar้'s type inequality see you (\cite{Temam}, p.121), \begin{equation}\label{*} \|f\|_{L^{q'_0}(B_\varepsilon,\mathbb{R}^3)}\|u^\varepsilon\|_{L^{q_0}(B_\varepsilon,\mathbb{R}^3)} \leq C{(\varepsilon\varphi_\varepsilon)}^{\frac{1}{q'_0}}\int_\Omega|e(u^\varepsilon)|\leq C{(\varepsilon\varphi_\varepsilon)}^{\frac{1}{q'_0}}(\int_{\Omega_\varepsilon}|e(u^\varepsilon)|+\int_{B_\varepsilon}|e(u^\varepsilon)|) \end{equation} such as $\varphi_\varepsilon$ is $Y$-periodic and for a small enough $\varepsilon$, than we have : \begin{equation*} \varphi_\varepsilon<\frac{1}{\varepsilon{(1+C)}^{q'_0}},~~~~\text{ let }~~ c_\varepsilon=\frac{C}{\varepsilon(1+C)}. \end{equation*} According to $(4.4)$, and using $(4.5)$, $(4.6)$, then we obtain \begin{equation*} \|e(u^\varepsilon)\|^2_{L^2(\Omega_\varepsilon,\ree^9_s)}+\frac{1}{\varepsilon}\int_{B_\varepsilon}|e(u^\varepsilon)|\leq C+C\|e(u^\varepsilon)\|_{L^2(\Omega_\varepsilon,\ree^9_s)} +c_\varepsilon\int_{B_\varepsilon}|e(u^\varepsilon)| \end{equation*} Using Young inequality, \begin{equation*} \leq C+\frac{1}{2}\|e(u^\varepsilon)\|^2_{L^2(\Omega_\varepsilon,\ree^9_s)} +c_\varepsilon\int_{B_\varepsilon}|e(u^\varepsilon)| \end{equation*} so that \begin{equation*} \|e(u^\varepsilon)\|^2_{L^2(\Omega_\varepsilon,\ree^9_s)}+(\frac{1}{\varepsilon}-c_\varepsilon)\int_{B_\varepsilon}|e(u^\varepsilon)| \leq C \end{equation*} we obtain : \begin{equation*} \|e(u^\varepsilon)\|^2_{L^2(\Omega_\varepsilon,\ree^9_s)}+\frac{1}{\varepsilon}(1-\frac{C}{1+C})\int_{B_\varepsilon}|e(u^\varepsilon)| \leq C. \end{equation*} Therefore, we will have $(4.2)$ and $(4.3)$. According to (4.2), (4.3) and for a small enough $\varepsilon$ the sequence $u^\varepsilon$ is bounded in $LD_0(\Omega)$. \end{pf} We give some lemmas that will be used in the sequel. \begin{lem} Let $g\in C^\infty(\Sigma,\ree^9)$ and $u\in\mathcal{D}(\Sigma,\ree^3)$, so we have \begin{equation*} \int_\Sigma \tau e(u)=-\int_\Sigma div_T(\tau)u \end{equation*} with $div_T(\tau)=div(\frac{\tau+\tau^T}{2}).$ \end{lem} \begin{lem} Let $u$ be a regular function defined in a neighborhood of $\Sigma$, then \begin{equation*} \delta_j(\int_{0}^{\varepsilon\varphi_\varepsilon}u)=\varepsilon u(x',\varepsilon\varphi_\varepsilon)\delta_j\varphi_\varepsilon + \int_0^{\varepsilon\varphi_\varepsilon}\delta_ju. \end{equation*} \end{lem} This lemme is a consequence of (\cite{AcerbiBP} Proposition 2).\\ To apply the epi-convergence method, we need to characterize the topological spaces containing any cluster point of the solution of the problem $(4.1)$ with respect to the used topology, therefore the weak topology to use is insured by the lemma 4.1. So the topological spaces characterization is given in the following proposition. Let us \begin{equation*} w^\varepsilon=\frac{1}{2\varepsilon\varphi_\varepsilon}\int_{-\varepsilon\varphi_\varepsilon}^{\varepsilon\varphi_\varepsilon} u^\varepsilon \end{equation*} \begin{prop} The solution $u^\varepsilon $ of the problem (4.1) possess a cluster point $u^*$ in $LD_0(\Omega)$, with respect to the weak topology and $u^*_{|_\Sigma}$ is a weak cluster point of $w^\varepsilon$ in $LD_0(\Sigma,\mathbb{R}^3).$ \end{prop} \begin{pf} According to a $(4.2)$, $(4.3)$ and for a small $\varepsilon$, the solution $u^\varepsilon$ is bounded in $LD_0(\Omega)$, then It's relatively compact in $L^1(\Omega)$, this is consequence of (\cite{Temam}, Theorem1.4 p.117), and $e(u^\varepsilon)$ so for a subsequences of $u^\varepsilon$, still denoted by $u^\varepsilon$, there exists there exists $u^*\in L^1(\Omega)$, such that \begin{equation*} u^\varepsilon\rightarrow u^*~\text{ in}~L^1(\Omega), \end{equation*} %and $$e(u^\varepsilon)\rightharpoonup e(u^*) \text{ in}~W^{1,1}_0(\Omega)$$ we have \begin{equation*} u^\varepsilon\rightharpoonup u^*~\text{ in}~LD_0(\Omega) \end{equation*} other hand \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \int_\Sigma|w^\varepsilon-u^\varepsilon_{|_\Sigma}|dx'\leq \int_\Sigma\frac{1}{2\varepsilon\varphi_\varepsilon}\int_{-\varepsilon\varphi_\varepsilon}^{\varepsilon\varphi_\varepsilon} |u^\varepsilon(x)-u^\varepsilon(x',0)|dx_3dx' \\ \leq \frac{C}{\varepsilon}\int_\Sigma\int_{-\varepsilon\varphi_\varepsilon}^{\varepsilon\varphi_\varepsilon} |\int^{x_3}_0\frac{\partial u^\varepsilon}{\partial x_3}(x',t)dt|dx_3dx' \\ \leq \frac{C}{\varepsilon}\int_\Sigma{(\varepsilon\varphi_\varepsilon)}^2 \int_{-\varepsilon\varphi_\varepsilon}^{\varepsilon\varphi_\varepsilon} |\frac{\partial u^\varepsilon}{\partial x_3}(x',x_3)|dx_3 dx' \\ \leq C\varepsilon\int_{B_\varepsilon}\nabla u^\varepsilon dx\leq C\varepsilon\int_{\Omega}e( u^\varepsilon)dx \end{eqnarray*} Thanks to Lemma 4.1 and the Young's inequality, so we have $$\int_\Sigma|w^\varepsilon-u^\varepsilon_{|_\Sigma}|\leq C\varepsilon(\int_{\Omega_\varepsilon}e(u^\varepsilon)+\int_{B_\varepsilon}e(u^\varepsilon))$$ $$\leq C\varepsilon(C+C).$$ Then \begin{equation*} \lim_{\varepsilon\rightarrow 0}\int_\Sigma|w^\varepsilon-u^\varepsilon_{|_\Sigma}|=0 \end{equation*} since $u^\varepsilon_{|_\Sigma}\rightharpoonup u^*_{|_\Sigma}$ in $LD_0(\Sigma)$, so $w^\varepsilon\rightharpoonup u^*_{|_\Sigma}$ in $LD_0(\Sigma).$ \end{pf} \begin{rem} The Proposition 4.2 remains true for any weak cluster point $u$ of a sequence $u^\varepsilon$ in $LD_0(\Omega,\ree^3)$ satisfies (4.2) and (4.3). \end{rem} \vspace*{0.3cm} To study the limit behavior of the solution of the problem (4.1), we will use the epi-convergence method, (see Annex, definition ). \section{Limit Behavior} In this section, we are interested to the asymptotic behavior of the solution of the problem (4.1) when e close to zero. In the sequel, we consider the following functionals \begin{equation}\label{} F_\varepsilon(v)= \begin{cases} \frac{1}{2}\int_{\Omega_\varepsilon}a_{ijhk}e_{ij}(v)e_{hk}(v)dx + \frac{1}{\varepsilon}\int_{B_\varepsilon}|e(v)| &\mbox{if }v\in V^{\varepsilon}\\ +\infty &\mbox{if } v \not\in V^{\varepsilon} \end{cases} \end{equation} \begin{equation*} G(v)=-\int_\Omega fvdx, ~~ \forall v\in V^\varepsilon \end{equation*} We design by $\tau_f$ the weak topology on the space. In the sequel, we shall characterize, the epi-limit of the energy functional given by $(5.1)$ in the following theorem : \begin{thm} Under $(2.1)$, $(2.2$), $(2.3$) and for $f\in L^\infty(\Omega,\ree^3)$, there exists a functional $F:W^{1,1}(\Omega)\rightarrow\ree\cup \{+\infty\}$ such that \begin{equation*} \tau_f-\lim_\varepsilon F^\varepsilon=F ~~\text{ in } W^{1,1}_0(\Omega) \end{equation*} where $F$ is given by \begin{equation*} F(u)= \begin{cases} \frac{1}{2}\int_{\Omega} a_{ijhk}e_{ij}(u)e_{hk}(u)dx + m(\varphi)\int_{\Sigma}|e^*(u_{|_\Sigma})| &\mbox{if }u\in W^{1,1}_0(\Omega)\\ +\infty &\mbox{if } u\not\in W^{1,1}_0(\Omega) \end{cases}. \end{equation*} \end{thm} \begin{pf} %%%%%%%%%%%%%%%%%%%%%%%%่่่่่่่่่่่่่่่่่77777777777777777777777777777777777777777 $\bullet-(a)$ We are now in position to determine the upper epi-limit.\\ Let $u\in LD_0(\Omega)$, as $C^\infty(\Omega)$ is dense in $LD_0(\Omega)$ see you (\cite{Temam}, p.116), so there exists a sequence $(u^n)$ in $C^\infty(\Omega)$ such that $(u^n)\rightharpoonup u$ weakly in $LD_0(\Omega)$.\\ Let us consider the sequence \begin{equation*} u^{n,\varepsilon}=\theta_\varepsilon(x)u^n_{|_\Sigma}+(1-\theta_\varepsilon(x))u^{n} \end{equation*} where $\theta$ is a regular function satisfies : \begin{equation*} \theta(x_3)=1\text{ if }|x_3|\leq1,~~\theta(x_3)=0\text{ if }|x_3|\geq2~~\text{and}~~|\theta'(x_3)|\leq2~\forall x\in\mathbb{R} \end{equation*} we set \begin{equation*} \theta_\varepsilon=\theta(\frac{x_3}{\varepsilon\varphi_\varepsilon}) \end{equation*} we have \begin{equation*} F^\varepsilon(u^{\varepsilon,n})=\frac{1}{2}\int_{\Omega_\varepsilon}a_{ijhk}e_{ij}(u^{\varepsilon,n})e_{hk}(u^{\varepsilon,n})dx + \frac{1}{\varepsilon}\int_{B_\varepsilon}|e(u^{\varepsilon,n})| \end{equation*} which implies that $$F^\varepsilon(u^{\varepsilon,n})=\frac{1}{2}\int_{|x_3|>2\varepsilon\varphi_\varepsilon}a_{ijhk}e_{ij}(u^{\varepsilon,n})e_{hk} (u^{\varepsilon,n})dx$$$$\hspace{4cm}+ \frac{1}{2}\int_{\varepsilon\varphi_\varepsilon<|x_3|<2\varepsilon\varphi_\varepsilon}a_{ijhk}e_{ij}(u^{\varepsilon,n})e_{hk} (u^{\varepsilon,n})dx $$$$+\frac{1}{\varepsilon}\int_{B_\varepsilon}|e(u^{\varepsilon,n})|$$ $$=\int_{|x_3|>2\varepsilon\varphi_\varepsilon}a_{ijhk}e_{ij}(u^{\varepsilon,n})e_{hk} (u^{\varepsilon,n})dx$$ $$\hspace{4cm}+ \frac{1}{2}\int_{\varepsilon\varphi_\varepsilon<|x_3|<2\varepsilon\varphi_\varepsilon}a_{ijhk}e_{ij}(u^{\varepsilon,n}) e_{hk}(u^{\varepsilon,n})dx $$ $$\hspace{2cm}+\frac{1}{\varepsilon}\int_{\Sigma}\varepsilon\varphi_\varepsilon|e^*(u^{\varepsilon,n}_{|_\Sigma})|.$$ As $\varphi_\varepsilon$ is bounded, then it's easy to prove that \begin{equation*} \lim_{\varepsilon\rightarrow0}\{~\frac{1}{2}\int_{\varepsilon\varphi_\varepsilon<|x_3|<2\varepsilon\varphi_\varepsilon}a_{ijhk}e_{ij} (u^{\varepsilon,n})e_{hk}(u^{\varepsilon,n})dx~\}=0. \end{equation*} Otherwise $\varphi_\varepsilon\rightarrow m(\varphi)$ in $L^1(\Omega)$ see you Annex so by passing to the upper limit, we obtain : \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \limsup_{\varepsilon\rightarrow0}F^\varepsilon(u^{\varepsilon,n})= \limsup_{\varepsilon\rightarrow0}\{~ \frac{1}{2}\int_{|x_3|>2\varepsilon\varphi_\varepsilon}a_{ijhk}e_{ij}(u^{\varepsilon,n})e_{hk} (u^{\varepsilon,n})dx +\int_{\Sigma}\varphi_\varepsilon|e^*(u^{\varepsilon,n}_{|_\Sigma})|~\} \\ =\frac{1}{2}\int_{\Omega}a_{ijhk}e_{ij}(u^{n})e_{hk} (u^n)dx + m(\varphi)\int_{\Sigma}|e^*(u^n_{|_\Sigma})|. \end{eqnarray*} Since $u^n\rightarrow u$ in $C^\infty(\Omega)$, there fore according to the classic result, diagonalization's Lemma, see (\cite{HA} p.32), there exists a real function $n(\varepsilon): \ree^{+}\rightarrow\mathbb{N}$ increasing to $+\infty$, such that $u^{\varepsilon,n(\varepsilon)}\rightharpoonup u$ in $C^\infty(\Omega)$ when $\varepsilon\rightarrow 0.$\newline Consequently, we have \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \limsup_{\varepsilon\rightarrow0}F^\varepsilon(u^{\varepsilon,n(\varepsilon)})\leq \limsup_{n\rightarrow0}\limsup_{\varepsilon\rightarrow0}F^\varepsilon(u^{\varepsilon,n}) \\ \leq \frac{1}{2}\int_{\Omega} a_{ijhk}e_{ij}(u)e_{hk}(u)dx+ m(\varphi)\int_{\Sigma}|e^*(u_{|_\Sigma})| \end{eqnarray*} \\ %รงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรง PROOF OF (B) %9999999999999รงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรงรง $\bullet-(b)$ We are now in position to determine the lower epi-limit.\\ Let as $(u^\varepsilon)\subset V^\varepsilon$ such as $u^\varepsilon\rightharpoonup u$ in $LD_0(\Omega),$ If $\liminf_{\varepsilon\rightarrow0}F^\varepsilon(u^\varepsilon)=+\infty$ there is nothing to prove, because \begin{eqnarray*} \frac{1}{2}\int_{\Omega} a_{ijhk}e_{ij}(u)e_{hk}(u)dx+ m(\varphi)\int_{\Sigma}|e^*(u_{|_\Sigma})|\leq+\infty. \end{eqnarray*} Otherwise, we suppose $\liminf_{\varepsilon\rightarrow0}F^\varepsilon(u^\varepsilon)<+\infty,$ there exists a subsequence of $F^\varepsilon(u^\varepsilon)$, still denoted by $F^\varepsilon(u^\varepsilon)$ and a constant $C >0$, such that \\ $F^\varepsilon(u^\varepsilon)\leq C$, which implies that \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \|e(u^\varepsilon)\|^2_{L^2(\Omega_\varepsilon,\ree^9_s)}\leq C \\ \frac{1}{\varepsilon}\int_{B_\varepsilon}|e(u^\varepsilon)|\leq C. \end{eqnarray*} Then $\chi_{\Omega_\varepsilon}e(u^\varepsilon)$ is bounded in $L^2(\Omega)$, so for a subsequence of $\chi_{\Omega_\varepsilon}e(u^\varepsilon)$, still denoted by $\chi_{\Omega_\varepsilon}e(u^\varepsilon)$ we then show easily, like in the proof of the above proposition, that \begin{eqnarray*} \chi_{\Omega_\varepsilon}e(u^\varepsilon)\rightharpoonup e(u)~ \text{in}~L^2(\Omega) \end{eqnarray*} From the subdifferentiability's inequality of $u\rightarrow\frac{1}{2}\int_{\Omega_\varepsilon} a_{ijhk}e_{ij}(u)e_{hk}(u)dx$ and passing to the lower limit, we obtain \begin{equation}\label{} \liminf_{\varepsilon\rightarrow0}\frac{1}{2}\int_{\Omega_\varepsilon} a_{ijhk}e_{ij}(u^\varepsilon)e_{hk}(u^\varepsilon)dx\geq \frac{1}{2}\int_{\Omega} a_{ijhk}e_{ij}(u)e_{hk}(u)dx. \end{equation} Let us set, for $\eta<\frac{\varepsilon}{2}$ \begin{eqnarray*} B_\eta=\{x\in \Omega : |x_3|<\eta\}. \end{eqnarray*} According to the diagonalization's lemma (\cite{HA}, Lemma1.15 p.32), there exists a function $\eta(\varepsilon):\mathbb{R}^+\rightarrow\mathbb{R}^+$ decreasing to 0 when $\varepsilon\rightarrow0$ such that \begin{equation}\label{} \liminf_{\varepsilon\rightarrow0}\int_{B^{\eta(\varepsilon)}}|e(u^\varepsilon)|\geq \liminf_{\eta\rightarrow0}\liminf_{\varepsilon\rightarrow0}\int_{B_\eta}|e(u^\varepsilon)| \end{equation} since \begin{eqnarray*} \int_{B^{\eta}}|e(u^\varepsilon)|\geq\int_{B_\eta}\phi(e(u^\varepsilon)-e(u))+\int_{B^{\eta}}\phi(e(u))~~\forall \phi\in C^\infty_0(B_\eta,\mathbb{R}^9_s) \end{eqnarray*} it follow that $$\liminf_{\varepsilon\rightarrow0}\int_{B^{\eta}}|e(u^\varepsilon)|\geq \int_{B^{\eta}}\phi(e(u))~~\forall \phi\in C^\infty_0(B_\eta,\mathbb{R}^9_s)$$ therefore, $$\liminf_{\varepsilon\rightarrow0}\int_{B^{\eta}}|e(u^\varepsilon)|\geq \int_{B^{\eta}}e(u),$$ According to a Lemma 4.3, and let $w^\varepsilon$ be the sequence define before the Proposition 4.3, we have \begin{equation} \liminf_{\varepsilon\rightarrow0}\int_{B^{\eta}}|e(u^\varepsilon)|\geq\int_{\Sigma}e^*(w^\varepsilon)-\int_{B_\eta} \varepsilon\varphi_\varepsilon\delta\varphi_\varepsilon\otimes_sU^\varepsilon \end{equation} Or \begin{equation*} U^\varepsilon=[u^\varepsilon(x',\varepsilon\varphi_\varepsilon)+ u^\varepsilon(x',-\varepsilon\varphi_\varepsilon)] \end{equation*} thanks to a Lemma 4.2, let $g\in \mathcal{D}(\Sigma,\ree^9)$ we have \begin{equation*} \int_{\Sigma}g e^*(w^\varepsilon)=-\int_{\Sigma}div_Tg(\frac{1}{\varepsilon\varphi_\varepsilon}\int^{ \varepsilon\varphi_\varepsilon}_{\varepsilon\varphi_\varepsilon}u^\varepsilon) \end{equation*} thanks to a Proposition 4.2 and $\varphi_\varepsilon\rightarrow m(\varphi)$ in $L^1(\Sigma)$, so passing to limit, we obtain \begin{equation} \int_{\Sigma}g e^*(w^\varepsilon)=-m(\varphi)\int_{\Sigma}div_Tgu_{|_\Sigma} = m(\varphi)\int_{\Sigma}ge^*(u_{|_\Sigma})~~\forall g\in \mathcal{D}(\Sigma,\ree^9) \end{equation} By passing to the limit, $(\eta\rightarrow0)$, in (5.4) we have \begin{equation*} \liminf_{\eta\rightarrow0} \liminf_{\varepsilon\rightarrow0}\int_{B^{\eta}}|e(u^\varepsilon)|\geq m(\varphi)\int_{\Sigma}e^*(u_{|_\Sigma}) \end{equation*} According to the definition of $B_\eta$, (5.3), we deduce that \begin{equation}%\label{} \liminf_{\varepsilon\rightarrow0}\int_{B^{\varepsilon}}|e(u^\varepsilon)|\geq m(\varphi)\int_{\Sigma}e^*(u_{|_\Sigma}) \end{equation} Hence (5.2) and (5.6), \begin{equation*} \liminf_{\varepsilon\rightarrow0}F^\varepsilon(u^\varepsilon)\geq \frac{1}{2}\int_{\Omega} a_{ijhk}e_{ij}(u)e_{hk}(u)dx +m(\varphi)\int_{\Sigma}e^*(u_{|_\Sigma}) \end{equation*} For $u\in LD_0(\Omega)$ and $u^\varepsilon\in V^\varepsilon$, such that $u^\varepsilon\rightharpoonup u$ in $LD_0(\Omega)$, Assume that $$\liminf_{\varepsilon\rightarrow 0}F^\varepsilon(u^\varepsilon)<+\infty.$$ So there exists a constant $C > 0$ and a subsequence of $F^\varepsilon (u^\varepsilon)$, still de noted by $F^\varepsilon (u^\varepsilon)$, such that $$F^\varepsilon(u^\varepsilon)0. \end{equation*} So that \begin{equation*} \varphi_\varepsilon\rightarrow m(\varphi) \text{ in } L^s(\Sigma) \text{ for } 1\leq s\leq\infty,\\ \varphi_\varepsilon\rightharpoonup^* m(\varphi) \text{ in } L^\infty(\Sigma). \end{equation*} \end{lem} \begin{pf} Since $\varphi_\varepsilon$ is a $\varepsilon Y$-periodic, so one has \begin{equation*} \varphi_\varepsilon\rightharpoonup m(\varphi) \text{ in } L^s(\Sigma) \text{ for } 1\leq s\leq\infty,\\ \varphi_\varepsilon\rightharpoonup^* m(\varphi) \text{ in } L^\infty(\Sigma). \end{equation*} Since $\varphi$ is bounded in $\Sigma$, so for evry $s\geq1$, there existes a constant $C>0$, such that \begin{equation*} \int_\Sigma{|\varphi_\varepsilon-m(\varphi)|}^s\leq C\int_\Sigma{|\varphi_\varepsilon-m(\varphi)|} \end{equation*} \begin{equation} \leq C[\int_{\varphi\geq m(\varphi)}{(\varphi_\varepsilon-m(\varphi))}-\int_{\varphi\leq m(\varphi)}{(m(\varphi)-\varphi_\varepsilon)}]. \end{equation} Passing to the limit in (7.1), one has $\varphi_\varepsilon\rightarrow m(\varphi)$ in $L^s(\Sigma)$ for $1\leq s\leq\infty.$ \end{pf} %\begin{cor}\label{corollary1} %to create the corollary %This is the environment to corollary %\end{cor} %\begin{Rem} %to create the remark %This is the environment to remarks. %\end{Rem} %%% to create the Acknowledgments \begin{thebibliography}{99} \bibitem{AitJ} A. 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