\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{graphicx,amssymb} \AtBeginDocument{{\noindent\small 2005-Oujda International Conference on Nonlinear Analysis. \newline {\em Electronic Journal of Differential Equations}, Conference 14, 2006, pp. 21--33.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \setcounter{page}{21} \begin{document} \title[\hfilneg EJDE/Conf/14 \hfil Limit behavior of an oscillating thin layer] {Limit behavior of an oscillating thin layer} \author[A. Ait Moussa, J. Messaho \hfil EJDE/Conf/14 \hfilneg] {Abdelaziz Ait Moussa, Jamal Messaho} % in alphabetical order \address{Abdelaziz Ait Moussa and Jamal Messaho\newline D\'epartement de math\'ematiques et informatique, Facult\'e des sciences, Universit\'e Mohammed 1er, 60000 Oujda, Morocco.} \email{moussa@sciences.univ-oujda.ac.ma} \email{j\_messaho@yahoo.fr} \date{} \thanks{Published September 20, 2006.} \subjclass[2000]{35B40, 82B24, 76M50} \keywords{Limit behavior; epi-convergence method; limit problems} \begin{abstract} We study the limit behavior of a thermal problem, of a containing structure, an oscillating thin layer of thickness and conductivity depending of $\varepsilon$. We use the the epi-convergence method to find the limit problems with interface conditions. The obtained results are tested numerically. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}{Lemma}[section] \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newcommand{\norm}[1]{\left\Vert#1\right\Vert} \newcommand{\abs}[1]{\left\vert#1\right\vert} %\renewcommand{\qedsymbol}{$\boxtimes$} \section{Introduction} In mathematical physics, one meets several kinds of boundary problems, the heat conduction, electrostatic, electromagnetic, mechanical of the continuous mediums, where the unknown $u$ satisfies the transmission conditions on the surface of separation between two domains $\Omega_1$ and $\Omega_2$: \begin{gather} u_{|_{\Omega_1}}=u_{|_{\Omega_2}} \label{1}\\ \sigma_1|\nabla u |^{p-2}{\frac{\partial u}{\partial n}}\big|_{\Omega_1} = \sigma_2|\nabla u |^{p-2}{\frac{\partial u}{\partial n}}|_{\Omega_2} \label{2} \end{gather} where $p>1$ and $n$ represents the outward normal vector to the surface of separation, $\sigma_1$ and $\sigma_2$ are the associated constants to each domain $\Omega_1$ and $\Omega_2$ respectively. The boundary conditions of type \eqref{1} and \eqref{2} are met in thermal conductivity problems, where $\sigma_1$ and $\sigma_2$ designate the conductivities of two bodies. In the electrostatic or magnetostatic problems $\sigma_1$ and $\sigma_2$ are the dielectric or permeability constants respectively. A transmission problem with the conditions of type (\ref{1}) and (\ref{2}) and $p=2$, was studied by Sanchez-Palencia in \cite{sanch}. Our aim in this work is to study the limit behavior of solutions of a thermal conductivity problem, this last is in a structure containing an oscillating layer of thickness and conductivity depending of $\varepsilon$, $\varepsilon$ being a parameter intended to tend towards $0$. A similar problems are found in Brillard and al in \cite{brd1}. The vectorial case one finds it in Ait Moussa and al, and Brillard and al in \cite{aitmoussa, brd2}. This paper is organized in the following way. In section \ref{sec:2}, one expresses the problem to study, and one defines functional spaces for this study in the section \ref{sec:3}. In the section \ref{sec:4}, one studies the problem \eqref{Pe}. The section \ref{cal:epii} is reserved to the determination of the limits problems. Finally in the section \ref{sim}, one will give a numerical test illustrating the obtained theoretical results. \section{Position of the problem}\label{sec:2} One considers a problem of nonlinear thermal conduction in a body which occupies a bonded domain $\Omega\subset\mathbb{R}^3$, with a Lipschitz border $\partial\Omega$, composed of a layer $B_\varepsilon$, with oscillating border $\Sigma^{^\pm}_\varepsilon$, of average interface $\sigma$, of very high conductivity, and a remaining region $\Omega_\varepsilon$ with a constant conductivity ( see figure \ref{fig:1}). The body occupying the domain $\Omega$, is subject to an outside temperature $f$, $f\colon \Omega\to\mathbb{R}$, and cooled at the boundary $\partial\Omega$. The equations of the problem are: $$\label{Pe} \begin{gathered} \mathop{\rm div}(|\nabla u^\varepsilon|^{p-2}\nabla u^\varepsilon)+f =0 \quad \mbox{in }\Omega_\varepsilon ,\\ \frac{1}{\varepsilon^{\alpha}}\mathop{\rm div}(|\nabla u^\varepsilon|^{p-2}\nabla u^\varepsilon)+f=0 \quad\mbox{in }B_\varepsilon, \\ [ u^{\varepsilon}] =0 \quad \mbox{on }\Sigma^{^\pm}_\varepsilon,\\ |\nabla u^\varepsilon|^{p-2}\frac{\partial u^{\varepsilon}} {\partial n}\big|_{\Omega_\varepsilon}=\frac{1}{\varepsilon^{\alpha}} |\nabla u^\varepsilon|^{p-2}\frac{\partial u^{\varepsilon}}{\partial n}\big|_{{B_\varepsilon}} \quad \mbox{on }\Sigma^{^\pm}_\varepsilon,\\ u^\varepsilon = 0 \quad\mbox{on }\partial\Omega, \end{gathered}$$ where $n$ the outward normal to $\partial \Omega$, $p>1$ and $\alpha \geq 0$. \begin{figure}[ht] \begin{center} \includegraphics[width=0.6\textwidth,height=0.6\textheight,angle=-90]{fig1} \caption{Domain $\Omega$.} \label{fig:1} \end{center} \end{figure} Where $\varepsilon$ being a positive parameter intended to tend towards zero and $\varphi_\varepsilon$ is a bounded real function, $]0,\varepsilon [^ {2}$-periodic. \section{Notation and functional setting}\label{sec:3} Here is the notation that will be used in the sequel:\\ $x=(x',x_3)$ where $x'=(x_1,x_2)$, $\lambda=1,2$, $\nabla'=(\frac{\partial}{\partial x_1}$, $\frac{\partial}{\partial x_2}), Y=]0,1[\times]0,1[$,\\ $\varphi\colon \mathbb{R}^{2}\to [a_1,a_2]$ where $\varphi$ is $Y$-periodic and $a_2\geq a_1>0$, $\varphi_{_\varepsilon}(x')=\varphi(\frac{x'}{\varepsilon})$,\\ $\frac{\partial \varphi}{\partial x_{\lambda}}\in \mathcal{C}(\Sigma) \cap L^{^{\infty}}(\Sigma)$, $m(\varphi)= (\frac{1}{\int_{Y}dx'})\int_{Y}\varphi(x')dx',$ $\displaystyle\eta(t)=\lim_{\varepsilon\to 0}\varepsilon^{1-t}$ with $t\geq 0$. In the following $C$ will denote any constant with respect to $\varepsilon$. Also we use the convention $0.+\infty=0$. \subsection{Functional setting} First, we introduce the Banach space $V^\varepsilon=W^{1,p}_0(\Omega)$. Let \begin{gather*} V^{p}(\Sigma)=\Big\{ u\in W^{1,p}_0(\Omega) : u\big|_{{\Sigma}}\in W^{1,p} (\Sigma)\Big\},\\ V^{C}(\Sigma)=\Big\{ u\in W^{1,p}_0(\Omega) : u\big|_{{\Sigma}}=C\Big\}. \end{gather*} The set $V^{C}(\Sigma)$ is a Banach space with the norm of $W^{1,p}_0(\Omega)$. we show easily that $\displaystyle V^{p}(\Sigma)$ is a Banach space with the norm $$u\mapsto \norm{\nabla u}_{L^{^p}(\Omega)^3} +\norm{ \nabla' u_{|_{\Sigma}}}_{L^{^p}(\Sigma)^2}.$$ Let \begin{gather*} \mathbb{G}^{\alpha}=\begin{cases}\Big\{ u\in W^{1,p}_0(\Omega): \eta(\alpha)u\big|_{{\Sigma}}\in W^{1,p}(\Sigma)\Big\} &\text{if } \alpha\leq 1 ,\\ V^C&\text{if }\alpha>1.\end{cases} \\ \mathbb{D}^{\alpha}=\begin{cases} \mathcal{D}({\Omega})&\text{if }\alpha\leq 1 ,\\ \Big\{u\in \mathcal{D}({\Omega}): u\big|_{{\Sigma}}=C \Big\}&\text{if } \alpha>1.\end{cases} \end{gather*} It is known that $\displaystyle \overline{\mathbb{D}^{\alpha}}=\mathbb{G}^{\alpha}$. Our goal in this work, is to study the problem \eqref{Pe} and its limit behavior. \section{Study of the problem \eqref{Pe}}\label{sec:4} The problem \eqref{Pe} is equivalent to the minimization problem $$\label{Pie} \inf_{v\in V^{\varepsilon}}\Big\{\frac{1}{p} \int_{\Omega_\varepsilon} \abs{\nabla v}^p +\frac{1}{p\varepsilon^{\alpha}}\int_{B_\varepsilon} \abs{\nabla v}^{p} -\int_{\Omega}f.v\Big\}.$$ \begin{proposition}\label{existe} For $f\in L^{p'}(\Omega)$, the problem \eqref{Pie} admits an unique solution $u^\varepsilon$ in $V^\varepsilon$. \end{proposition} The proof of this proposition is based on classical convexity arguments see for example \cite{brez}. \begin{lemma} \label{lemme1} For every $f\in L^{p'}(\Omega)$, the family $(u^\varepsilon)_{\varepsilon>0}$ satisfies: \begin{gather} \norm{\nabla u^\varepsilon}_{{L^{p}(\Omega_\varepsilon)}}^p \leq C\label{assert:1},\\ \frac{1}{\varepsilon^{\alpha}}\norm{\nabla u^\varepsilon}_{{L^{^p}(B_\varepsilon)}}^p \leq C.\label{assert:2} \end{gather} Moreover $u^\varepsilon$ is bounded in $W^{1,p}_0(\Omega)$. \end{lemma} \begin{proof} Since $u^\varepsilon$ is the solution of the problem \eqref{Pie}, we have $\int_{\Omega_{\varepsilon}} \abs{\nabla u^\varepsilon}^{p-2}\nabla u^\varepsilon\nabla v+\frac{1}{\varepsilon^{\alpha}} \int_{B_{\varepsilon}}\abs{\nabla u^\varepsilon}^{p-2}\nabla u^\varepsilon\nabla v=\int_{\Omega}f v, \quad \forall v\in V^\varepsilon.$ In particular, by taking $v=u^\varepsilon$, one obtains $\norm{ \nabla u^\varepsilon}^p_{{L^{p}(\Omega_\varepsilon)}} +\frac{1}{\varepsilon^{\alpha}}\norm{\nabla u^\varepsilon}^p_{{L^p(B_\varepsilon)}}=\int_{{\Omega}}fu^\varepsilon.$ According to the inequalities of H\"older and Young, one has \begin{align*} \norm{ \nabla u^\varepsilon}^p_{{L^{p}(\Omega_\varepsilon)}}+\frac{1}{\varepsilon^{\alpha}}\norm{\nabla u^\varepsilon}^p _{{L^p(B_\varepsilon)}} &\leq C\norm{ \nabla u^\varepsilon}_{{L^{p}(\Omega)}}\\ &\leq C(\norm{ \nabla u^\varepsilon}_{{L^{p}(\Omega_\varepsilon)}}+\norm{\nabla u^\varepsilon}_{{L^p(B_\varepsilon)}})\\ &\leq C+\frac{1}{p}\norm{ \nabla u^\varepsilon}^p_{{L^{p}(\Omega_\varepsilon)}}+\frac{1}{p}\norm{\nabla u^\varepsilon}^p_{{L^p(B_\varepsilon)}}\\ &\leq C+\frac{1}{p}\norm{ \nabla u^\varepsilon}^p_{{L^{p}(\Omega_\varepsilon)}}+\frac{1}{\varepsilon^{\alpha}}\frac{1}{p}\norm{\nabla u^\varepsilon}^p_{{L^p(B_\varepsilon)}}. \end{align*} So that $\norm{\nabla u^\varepsilon}^p_{{L^{p}(\Omega_\varepsilon)}}+\frac{1}{\varepsilon^{\alpha}}\norm{\nabla u^\varepsilon}^p _{{L^p(B_\varepsilon)}} \leq C.$ Therefore, one will have the assertions \eqref{assert:1} and \eqref{assert:2}. It is clair that for a small enough $\varepsilon$, the solution ($u^\varepsilon$) is bounded in $W^{1,p}_0(\Omega)$. \end{proof} Let us define the operator $m^\varepsilon$'' which transforms the definite functions $u$ on $B_\varepsilon$ into functions definite on $\Sigma$ by $$\label{moyenne} {m^\varepsilon}u(x_1,x_2)=\frac{1}{2\varepsilon\varphi_{\varepsilon}} \int_{-\varepsilon\varphi_{\varepsilon}}^{\varepsilon\varphi_{\varepsilon}} u(x_1,x_2,x_3)dx_3.$$ \begin{lemma}\label{lemme2} The operator $m^\varepsilon$ definite by \eqref{moyenne} is linear and bounded of $L^p(B_\varepsilon)$ (respectively $W^{1,p}(B_\varepsilon)$) in $L^p(\Sigma)$ (respectively $W^{1,p}(\Sigma)$), with norm $\leq C {\varepsilon}^{-\frac{1}{p}}$, moreover, for all $u\in W^{1,p}(B_\varepsilon)$, one has $$\label{fort0} \norm{m^\varepsilon u-u_{|_{\Sigma}}}_{|_{L^{^p}(\Sigma)}}^{^{p}} \leq C \varepsilon^{p-1}\int_{B_{{\varepsilon}}}\abs{\nabla u}^p.$$ \end{lemma} \begin{proof} One has $$\label{etiq16} \int_{\Sigma}\abs{{m^{\varepsilon}}u}^p dx_1 dx_2=\int_{\Sigma}(\frac{1}{2\varepsilon\varphi_\varepsilon})^p \abs{\int_{-\varepsilon\varphi_\varepsilon}^{\varepsilon\varphi_\varepsilon} u dx_3}^{p}dx_1dx_2,$$ since $00}}\subset V^\varepsilon$ which satisfies \eqref{assert:1} and \eqref{assert:2}. Then $$\label{etiq21'} \norm{\nabla^{'}({m^{\varepsilon}}u^{\varepsilon})}_{{({L^p(\Sigma))^2}}}^p \leq C\varepsilon^{\alpha-1}.$$ Moreover $m^\varepsilon u^\varepsilon$ possess a bounded subsequence in $L^p(\Sigma)$. \end{lemma} \begin{proof} Thanks to lemma \ref{lemme2}, one has $\big\|\frac{\partial ({m^{\varepsilon}}u^{\varepsilon})} {\partial x_{\lambda}}\big\|_{{L^p(\Sigma)^2}}^p \leq C\varepsilon^{-1} \int_{B_\varepsilon}\abs{\nabla u^{\varepsilon}}^pdx\,.$ According to \eqref{assert:2}, one has $\big\|\frac{\partial ({m^{\varepsilon}}u^{\varepsilon})}{\partial x_{\lambda}} \big\|_{{L^p(\Sigma)^2}}^p \leq C\varepsilon^{\alpha-1}.$ Let us show that (${m^{\varepsilon}}u^\varepsilon$) is a bounded sequence in $L^p(\Sigma)$. From (\ref{fort0}), (see, lemma \ref{lemme2}), one has $\big\|{m^{\varepsilon}}u^\varepsilon-{u^\varepsilon}_{|_{\Sigma}} \big\|_{{L^p(\Sigma)}}^p \leq C\varepsilon^{p-1} \int_{B_\varepsilon} \abs{\nabla u^{\varepsilon} }^pdx.$ According to \eqref{assert:2}, one obtains $\norm{{m^{\varepsilon}}u^\varepsilon-{u^\varepsilon}_{|_{\Sigma}} }_{{L^p(\Sigma)}}^p \leq C\varepsilon^{\alpha+p-1}.$ As $u^\varepsilon$ satisfies \eqref{assert:1} and \eqref{assert:2}, so $u^\varepsilon$ is bounded in $W^{1,p}_0(\Omega)$, it follows that there exists $u^*\in W^{1,p}_0(\Omega)$ and a subsequence of $u^\varepsilon$, still denoted by $u^\varepsilon$, such that $u^\varepsilon \rightharpoonup u^*$ in $W^{1,p}_0(\Omega)$, so ${u^\varepsilon}_{|_{\Sigma}}$ is a bounded sequence in $L^p(\Sigma)$. Since $$\label{etiq29} \norm{{m^{\varepsilon}}u^\varepsilon}_{{L^p(\Sigma)}} \leq \norm{{m^{\varepsilon}}u^\varepsilon-{u^\varepsilon}_{|_{\Sigma}} }_{{L^p(\Sigma)}}+\norm{{u^\varepsilon}_{|_{\Sigma}} }_{{L^p(\Sigma)}},$$ from (\ref{etiq29}), there exists a constant $C>0$ such that $\norm{{m^{\varepsilon}}u^\varepsilon }_{{L^p(\Sigma)}}^p\leq C$. \end{proof} \begin{proposition} \label{prop1} The solution of the problem \eqref{Pie}, $(u^\varepsilon)_\varepsilon$, possess a subsequence weakly convergent toward an element $u^*$ in $W^{1,p}_0(\Omega)$ satisfying \begin{enumerate} \item If $\alpha=1$: ${u^*}\big|_{{\Sigma}}\in W^{1,p}(\Sigma)$. \item If $\alpha>1$: ${u^*}\big|_{{\Sigma}}=C$. \end{enumerate} \end{proposition} \begin{proof} According to lemma \ref{lemme1}, the sequence $u^{\varepsilon}$ is bounded in $W^{1,p}_0(\Omega)$, it follows that there exists an element $u^*\in W^{1,p}_0(\Omega)$ and a subsequence of $u^{\varepsilon}$, still denoted by $u^{\varepsilon}$ such that $u^{\varepsilon}\rightharpoonup u^*$ in $W^{1,p}_0(\Omega)$. One has $\norm{{m^{\varepsilon}}u^\varepsilon-{u^\varepsilon}_{|_{\Sigma}} }_{{L^p(\Sigma)}} \leq C\varepsilon^{\frac{\alpha+p-1}{p}}\quad\text{and } {u^\varepsilon}_{|_{\Sigma}}\rightharpoonup{u^*}_{|_{\Sigma}}\text{ in }L^p(\Sigma).$ For $\alpha=1$, according to the evaluation (\ref{etiq21'}), the sequence $\nabla' m^\varepsilon u^{\varepsilon}$ possess a subsequence, still denoted by $\nabla' m^\varepsilon u^{\varepsilon}$ weakly convergent to an element $u^2$ in $L^{p}(\Sigma)^2$, as $m^\varepsilon u^{\varepsilon}\rightharpoonup {u^*}_{|_{\Sigma}}\text{ in }L^{p}(\Sigma)$, so one concludes that $m^\varepsilon u^{\varepsilon}\rightharpoonup {u^*}_{|_{\Sigma}}\text{ in }W^{1,p}(\Sigma)$ and $\nabla' {u^*}_{|_{\Sigma}}=u^2$. Hence ${u^*}_{|_{\Sigma}}\in W^{1,p}(\Sigma)$. For $\alpha>1$, one shows, as in the case $\alpha=1$ and taking $u^2=0$, that ${u^*}_{|_{\Sigma}}=C$. Hence the proof of proposition \ref{prop1} is complete. \end{proof} The limit behavior of the problem \eqref{Pie}, will be derived with the epi-convergence method, (see definition \ref{def:epi}). \section{Limit behavior}\label{cal:epii} \setcounter{equation}{0} Let \begin{gather}\label{F0} F^{\varepsilon}(u)= \frac{1}{p} \int_{\Omega_\varepsilon} \abs{\nabla u}^p +\frac{1}{p\varepsilon^{\alpha}}\int_{B_\varepsilon}\abs{\nabla u}^{p},\quad \forall u\in W^{1,p}_0(\Omega),\\ G(u)=-\int_{{\Omega}}f u,\quad \forall u\in W^{1,p}_0(\Omega). \end{gather} One denotes by $\tau_f$ the weak topology on $W^{1,p}_0(\Omega)$. \begin{theorem}\label{cal:epi} According to the values of $\alpha$, there exists a functional $F^{\alpha}$ defined on $W^{1,p}_0(\Omega)$ with value in $\mathbb{R}\cup\{+\infty\}$ such that $\tau_{_f}-\lim_{e}F^{\varepsilon}= F^{\alpha}\quad \mbox{in }W^{1,p}_0(\Omega),$ where the functional $F^{\alpha}$ is given by \begin{enumerate} \item If $0\leq\alpha<1$: $F^{\alpha}(u)= \frac{1}{p} \int_{\Omega} \abs{\nabla u}^p, \quad\forall u\in W^{1,p}_0(\Omega).$ \item If $\alpha\geq 1$: $F^{\alpha}(u)= \begin{cases}\displaystyle \frac{1}{p} \int_{\Omega} \abs{\nabla u}^p +\frac{2m(\varphi)\;\eta(\alpha)}{p}\int_{\Sigma}\abs{\nabla' u_{|_{\Sigma}}}^{p} &\mbox{if } u\in \mathbb{G}^{\alpha},\\ +\infty &\mbox{if }u\in W^{1,p}_0(\Omega)\setminus \mathbb{G}^{\alpha}. \end{cases}$ \end{enumerate} \end{theorem} \begin{proof} (a) One is going to determine the upper epi-limit: Let $u\in \mathbb{G}^{\alpha}\subset W^{1,p}_0(\Omega)$, there exists a sequence $(u^n)$ in $\mathbb{D}^{\alpha}$ such that $$u^n \to u \text{ in } \mathbb{G}^{\alpha}\mbox{, when } n\to +\infty.$$ So that $u^n \to u$ in $W^{1,p}_0(\Omega)$. Let $\theta$ be a smooth function satisfying \begin{gather*} \theta(x_3)=1 \text{ if }\abs{x_3}\leq 1,\; \theta(x_3)=0\text{ if }\abs{x_3}\geq 2\; \text{ and }\abs{\theta'(x_3)}\leq 2 \;\forall x\in\mathbb{R}, \end{gather*} and set \begin{gather*} \theta_{\varepsilon}(x)=\theta(\frac{x_3}{\varepsilon\varphi_{\varepsilon}}); \end{gather*} we define \begin{gather*} u^{\varepsilon,n}=\theta_{\varepsilon}(x){u^n}_{|_{\Sigma}}+(1-\theta_{\varepsilon}(x))u^n, \end{gather*} One shows easily that $u^{\varepsilon,n}\in V^\varepsilon$ and $u^{\varepsilon,n}\to u^n$ in $\mathbb{G}^{\alpha}$, when $\varepsilon\to 0$. Since $F^\varepsilon(u^{\varepsilon,n})= \frac{1}{p} \int_{\Omega_\varepsilon} \abs{\nabla u^{\varepsilon,n}}^p +\frac{1}{p\varepsilon^{\alpha}}\int_{B_\varepsilon}\abs{\nabla u^{\varepsilon,n}}^{p},$ so that \begin{align} F^\varepsilon(u^{\varepsilon,n}) &= \frac{1}{p} \int_{\abs{x_3}>2\varepsilon\varphi_{\varepsilon}} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\abs{\nabla u^{\varepsilon,n}}^p +\frac{1}{p} \int_{\varepsilon\varphi_{\varepsilon}<\abs{x_3}<2\varepsilon\varphi_{\varepsilon}} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\abs{\nabla u^{\varepsilon,n}}^p +\frac{1}{p\varepsilon^{\alpha}}\int_{B_\varepsilon}\abs{\nabla u^{\varepsilon,n}}^{p}\nonumber\\ &= \frac{1}{p} \int_{\abs{x_3}>2\varepsilon\varphi_{\varepsilon}} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\abs{\nabla u^{n}}^p +\frac{1}{p} \int_{\varepsilon\varphi_{\varepsilon}<\abs{x_3}<2\varepsilon\varphi_{\varepsilon}} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\abs{\nabla u^{\varepsilon,n}}^p +\frac{2\varepsilon^{1-\alpha}}{p}\int_{\Sigma}\varphi_{{\varepsilon}} \abs{\nabla' {u^{n}}_{|_{\Sigma}}}^{p}.\label{sup1} \end{align} Since $\varphi_{\varepsilon}$ is bounded, one verifies easily that \begin{align} \lim_{\varepsilon\to 0}\Big\{ \frac{1}{p} \int_{\varepsilon\varphi_{\varepsilon}<\abs{x_3}<2\varepsilon\varphi_{\varepsilon}} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\abs{\nabla u^{\varepsilon,n}}^p\Big\}=0.\label{sup2} \end{align} \begin{enumerate} \item If $\alpha\leq 1$: Since $\varphi_{_\varepsilon}\rightharpoonup^* m(\varphi)$ in $L^{\infty}(\Sigma)$ and $\varepsilon^{1-\alpha}\to \eta(\alpha)$, it follows that $\lim_{\varepsilon\to 0}\frac{2\varepsilon^{1-\alpha}}{p}\int_{\Sigma}\varphi_{{\varepsilon}}\abs{\nabla' {u^{n}}_{|_{\Sigma}}}^{p} =\frac{2m(\varphi) \eta(\alpha)}{p}\int_{\Sigma}\abs{\nabla' {u^{n}}_{|_{\Sigma}}}^{p}.$ By passage to the upper limit, one has \begin{align*} \limsup_{\varepsilon\to 0}F^\varepsilon(u^{\varepsilon,n}) &= \limsup_{\varepsilon\to 0}\Big(\frac{1}{p} \int_{\abs{x_3}>2\varepsilon\varphi_{\varepsilon}}\!\!\!\!\!\!\!\!\!\!\! \abs{\nabla u^{n}}^p +\frac{2\varepsilon^{1-\alpha}}{p}\int_{\Sigma}\varphi_{{\varepsilon}}\abs{\nabla' {u^{n}}_{|_{\Sigma}}}^{p}\Big)\\ &= \frac{1}{p} \int_{\Omega} \abs{\nabla u^{n}}^p+\frac{2m(\varphi)\;\eta(\alpha)}{p}\int_{\Sigma}\abs{\nabla' {u^{n}}_{|_{\Sigma}}}^{p}. \end{align*} \item If $\alpha>1$: By passage to the upper limit, one has \begin{align*} \limsup_{\varepsilon\to 0}F^\varepsilon(u^{\varepsilon,n}) &= \limsup_{\varepsilon\to 0}\Big(\frac{1}{p} \int_{\abs{x_3}>2\varepsilon\varphi_{\varepsilon}}\!\!\!\!\!\!\!\!\!\!\! \abs{\nabla u^{n}}^p \Big)\\ &= \frac{1}{p} \int_{\Omega} \abs{\nabla u^{n}}^p. \end{align*} \end{enumerate} Since $u^{n}\to u$ in $\mathbb{G}^{\alpha}$, when $n\to+\infty$. According to the classical result, diagonalization's lemma \cite[Lemma 1.15]{attouch}, there exists a function $n(\varepsilon):\mathbb{R}^+\to\mathbb{N}$ increasing to $+\infty$ when $\varepsilon\to 0$, such that $u^{\varepsilon,n(\varepsilon)}\to u$ in $\mathbb{G}^{\alpha}$ when $\varepsilon\to 0$. While $n$ approaches $+\infty$, one will have \begin{enumerate} \item If $\alpha\neq1$: \begin{align*} \limsup_{\varepsilon\to 0}F^\varepsilon(u^{\varepsilon,n(\varepsilon)}) &\leq \limsup_{n\to +\infty}\limsup_{\varepsilon\to 0} F^\varepsilon(u^{\varepsilon,n})\\ &\leq\frac{1}{p} \int_{\Omega} \abs{\nabla u}^p. \end{align*} \item If $\alpha=1$: \begin{align*} \limsup_{\varepsilon\to 0}F^\varepsilon(u^{\varepsilon,n(\varepsilon)}) &\leq \limsup_{n\to +\infty}\limsup_{\varepsilon\to 0}F^\varepsilon(u^{\varepsilon,n})\\ &\leq \frac{1}{p} \int_{\Omega} \abs{\nabla u}^p+\frac{2m(\varphi) \eta(\alpha)}{p}\int_{\Sigma}\abs{\nabla'{u}_{|_{\Sigma}}}^{p}. \end{align*} \end{enumerate} If $u\in W^{1,p}_0(\Omega)\setminus\mathbb{G}^{\alpha}$, it is clear that, for every $u^\varepsilon\in W^{1,p}_0(\Omega)$, $u^\varepsilon\rightharpoonup u$ in $W^{1,p}_0(\Omega)$, one has $\limsup_{\varepsilon\to 0}F^\varepsilon(u^{\varepsilon}) \leq +\infty.$ (b) One is going to determine the lower epi-limit. Let $u\in \mathbb{G}^{\alpha}$ and $(u^\varepsilon)$ be a sequence in $W^{1,p}_0(\Omega)$ such that $u^\varepsilon{\rightharpoonup}u$ in $W^{1,p}_0(\Omega)$, so that $$\chi_{{\Omega_{\varepsilon}}}\nabla u^\varepsilon \rightharpoonup \nabla u\quad \text{ in }L^{^p}(\Omega)^3.\label{eq:liminf2}$$ \begin{enumerate} \item If $\alpha\neq 1$: Since $F^\varepsilon(u^{\varepsilon}) \geq \frac{1}{p} \int_{\Omega_\varepsilon} \abs{\nabla u^{\varepsilon}}^p.$ According to \eqref{eq:liminf2} and by passage to the lower limit, one obtains $\liminf_{\varepsilon\to 0}F^\varepsilon(u^{\varepsilon}) \geq \frac{1}{p} \int_{\Omega} \abs{\nabla u}^p.$ \item If $\alpha=1$: If $\displaystyle\liminf_{\varepsilon\to 0}F^\varepsilon(u^\varepsilon)=+\infty$, there is nothing to prove, because $$\frac{1}{p} \int_{\Omega} \abs{\nabla u}^p+\frac{2m(\varphi) \eta(\alpha)}{p}\int_{\Sigma}\abs{\nabla' {u}_{|_{\Sigma}}}^{p}\leq +\infty.$$ Otherwise, $\displaystyle\liminf_{\varepsilon\to 0}F^\varepsilon(u^\varepsilon)<+\infty$, 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 $$\begin{gathered} \frac{1}{p\varepsilon^{\alpha}}\int_{B_{\varepsilon}} \abs{\nabla u^\varepsilon}^p \leq C. \end{gathered}\label{eqliminf33}$$ So $u^\varepsilon$ satisfies the hypothesis of the lemma \ref{lemme3}, and according to this last, $\nabla' m^\varepsilon u^\varepsilon$ is bounded in $L^{^p}(\Sigma)^2$, so there exists an element $u_1\in L^{^p}(\Sigma)^2$ and a subsequence of $\nabla' m^\varepsilon u^\varepsilon$, still denoted by $\nabla' m^\varepsilon u^\varepsilon$, such that $\nabla' m^\varepsilon u^\varepsilon\rightharpoonup u_1$ in $L^{^p}(\Sigma)^2$, since ${u^\varepsilon}_{|_{\Sigma}}\rightharpoonup u_{|_{\Sigma}}$ in $L^{^p}(\Sigma)$, and thanks to (\ref{fort0}) and (\ref{eqliminf33}), one has $m^\varepsilon{u^\varepsilon}\rightharpoonup u_{|_{\Sigma}}$ in $L^{^p}(\Sigma)$, then $m^\varepsilon{u^\varepsilon}\rightharpoonup u_{|_{\Sigma}}$ in $W^{1,p}(\Sigma)$, therefore $u_1=\nabla' u_{|_{\Sigma}}$, so that $\nabla' m^\varepsilon u^\varepsilon\rightharpoonup \nabla' u_{|_{\Sigma}}$ in $L^{^p}(\Sigma)^2$. One has \begin{align*} F^\varepsilon(u^{\varepsilon}) &\geq \frac{1}{p} \int_{\Omega_\varepsilon} \abs{\nabla u^{\varepsilon}}^p +\frac{1}{p\varepsilon^{\alpha}} \int_{B_\varepsilon} \abs{\nabla u^{\varepsilon}}^p\\ &\geq \frac{1}{p} \int_{\Omega_\varepsilon} \abs{\nabla u^{\varepsilon}}^p +\frac{2\varepsilon^{1-\alpha}}{p} \int_{\Sigma}\varphi_{{\varepsilon}} \abs{\nabla' m^\varepsilon u^{\varepsilon}}^p. \end{align*} Using the sub-differential inequality of $$v\to \frac{2\varepsilon^{1-\alpha}}{p} \int_{\Sigma}\varphi_{{\varepsilon}} \abs{v}^p, \forall v\in L^{^p}(\Sigma)^2,$$ one has \begin{align*} F^\varepsilon(u^{\varepsilon}) &\geq \frac{1}{p} \int_{\Omega_\varepsilon} \abs{\nabla u^{\varepsilon}}^p +\frac{2\varepsilon^{1-\alpha}}{p} \int_{\Sigma}\varphi_{{\varepsilon}} \abs{ \nabla' u_{|_{\Sigma}}}^p\\ &\quad +\frac{2\varepsilon^{1-\alpha}}{p} \int_{\Sigma}\varphi_{{\varepsilon}} \abs{ \nabla' u_{|_{\Sigma}}}^{p-2}\nabla' u_{|_{\Sigma}} (\nabla' m^\varepsilon u^{\varepsilon}-\nabla' u_{|_{\Sigma}}). \end{align*} Thanks to lemma \ref{period}, one has $\varphi_{{\varepsilon}}\to m(\varphi)$ in $L^{^{p'}}(\Sigma)$, so according to (\ref{eq:liminf2}) and by passage to the lower limit, one obtains $\liminf_{\varepsilon\to 0}F^\varepsilon(u^{\varepsilon}) \geq \frac{1}{p} \int_{\Omega} \abs{\nabla u}^p +\frac{2m(\varphi)\;\eta(\alpha)}{p} \int_{\Sigma} \abs{ \nabla' u_{|_{\Sigma}}}^p.$ \end{enumerate} If $u\in W^{1,p}_0(\Omega)\setminus\mathbb{G}^{\alpha}$ and $u^\varepsilon\in W^{1,p}_0(\Omega)$, such that $u^\varepsilon\rightharpoonup u$ in $W^{1,p}_0(\Omega)$. \\Assume that $\liminf_{\varepsilon\to 0}F^\varepsilon(u^{\varepsilon})<+\infty.$ So there exists a constant $C>0$ and a subsequence of $F^\varepsilon(u^{\varepsilon})$, still denoted by $F^\varepsilon(u^{\varepsilon})$, such that \begin{align}\label{eqliminf333} F^\varepsilon(u^{\varepsilon})0.  So that \begin{gather*} \varphi_{{\varepsilon}}\to m(\varphi) \quad\mbox{in $L^s(\Sigma)$ for } 1\leq s<\infty, \\ \varphi_{{\varepsilon}}\rightharpoonup^* m(\varphi)\quad \mbox{in }L^\infty(\Sigma). \end{gather*} \end{lemma} \begin{proof} Since $\varphi_\varepsilon$ is a $\varepsilon Y$-periodic, so one has $$\label{lem:ann} \begin{gathered} \varphi_{\varepsilon}\rightharpoonup m(\varphi) \quad \mbox{in } L^s(\Sigma) \mbox{ for } 1\leq s<\infty, \\ \varphi_{\varepsilon}\rightharpoonup^* m(\varphi)\quad \mbox{ in }L^\infty(\Sigma). \end{gathered}$$ Since $\varphi$ is bounded a.e. in $\Sigma$, so for every $s\geq 1$, there exists a constant $C>0$, such that \label{lem:annn} \begin{aligned} \int_{\Sigma}\abs{\varphi_{{\varepsilon}}-m(\varphi)}^s &\leq C \int_{\Sigma} \abs{ \varphi_{{\varepsilon}}-m(\varphi)} \\ &\leq C \Big(\int_{\varphi \geq m(\varphi)} (\varphi_{{\varepsilon}}-m(\varphi))-\int_{\varphi \leq m(\varphi)} (\varphi_{{\varepsilon}}-m(\varphi))\Big). \end{aligned} Passing to the limit in (\ref{lem:annn}), one has $\varphi_{{\varepsilon}}\to m(\varphi)$ in $L^s(\Sigma)$ for $1\leq s<\infty$. \end{proof} \begin{thebibliography}{99} \bibitem{aitmoussa} Ait Moussa A. et Licht C., \emph{Comportement asymptotique d'une plaque mince non linéaire}, J. 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