\documentclass[reqno]{amsart}
\usepackage{graphicx}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 34, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/34\hfil Asymptotic behavior of solutions]
{Asymptotic behavior of solutions on a thin plastic plate}

\author[A. Ait Moussa, J. Messaho\hfil EJDE-2009/34\hfilneg]
{Abdelaziz Ait Moussa, Jamal Messaho}  % in alphabetical order

\address{Abdelaziz Ait Moussa \newline
D\'epartement de math\'ematiques et informatique\\
Facult\'e des sciences, Universit\'e Mohammed 1er\\
 Oujda, Morocco}
\email{a\_aitmoussa@yahoo.fr}

\address{Jamal Messaho \newline
D\'epartement de math\'ematiques et informatique\\
Facult\'e des sciences, Universit\'e Mohammed 1er\\
 Oujda, Morocco}
\email{j\_messaho@yahoo.fr}


\thanks{Submitted July 30, 2008. Published February 23, 2009.}
\subjclass[2000]{35B40, 82B24, 76M50}
\keywords{Asymptotic behavior; plasticity problem;
epiconvergence method; \hfill\break\indent limit problems}

\begin{abstract}
 In the present work, we study the asymptotic behavior of  solutions
 to a plasticity problem in a containing  structure, a thin plastic
 plate of thickness that tends to zero. To find the limit problems
 with interface conditions we use the epiconvergence method.
\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]{\|#1\|}
\newcommand{\abs}[1]{|#1|}


\section{Introduction}

The study of the inclusion between two elastic bodies involves
introducing a very thin third body between them. A very similar
situation occurs when taking into account the effects of a thin
layer which has been bonded onto the surface of a body to prevent
wear caused by the contact with another solid. It is, therefore of
interest to study the asymptotic behavior of thin layer between
the two bodies, assuming various contact laws between them. In the
case of a thin plate, the thermal conductivity problems were
treated by Brillard et al and Sanchez-Palencia et al in
\cite{brd1, sanch}. The elasticity problems, linear and nonlinear
case, were widely studied by Ait Moussa et al, Ait moussa,
Brillard et al, Geymonat et al and Lenci et al in
 \cite{aitmoussa, moussa, brd2, Gey, lenci}.
In the case of an oscillating layer, we
have treated the scalar case for a thermal conductivity problem in
Messaho et al in \cite{messaho}. In the present work, we consider
a structure containing a thin plastic plate of thickness depending
on a parameter $\varepsilon$ intended to tend towards 0.
The aim of this work is to study the asymptotic
behavior of the solution of a plasticity problem posed on a such
structure.

This paper is organized in the following way. In section
\ref{sec:2}, we express the problem to study, and we give some notation
and we define functional spaces for this
study in the section \ref{sec:3}. In the section
\ref{sec:4}, we study the problem \eqref{pbmin}.
The section \ref{sec:5} is
reserved to the determination of the limits problems and our main result.

\section{Statement of the problem} \label{sec:2}

We consider a structure constituted of two linear elastics bodies,
joined together by a thin plastic plate of thickness
$\varepsilon$,  the latter obeys to a nonlinear plastic law of
power type. More precisely the stress field is related to the
displacement's field by
\begin{align*}
\sigma^{\varepsilon}&=&
\lambda|e(u^{\varepsilon})|^{-1}e(u^{\varepsilon}),\quad \lambda>
0.
\end{align*}
The structure occupies the regular domain
$\Omega=B_{\varepsilon}\cup\Omega_{\varepsilon}$, where $
B_{\varepsilon}$ is given by
$ B_{\varepsilon}=\{x=(x',x_3)/\abs{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 figure \ref{fig1}). $\varepsilon$ being a positive parameter
intended to approach 0.


\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1}
\caption{The domain $\Omega$.}\label{fig1}
\end{center}
\end{figure}

The structure is subjected to a density of forces of volume $f$,
$f\colon \Omega\to\mathbb{R}^3$, and it is fixed on the boundary
$\partial\Omega$. Equations which relate the stress field
$\sigma^{\varepsilon}$, $\sigma^{\varepsilon}\colon
\Omega\to\mathbb{R}^9_{S}$, and the field of displacement
$u^\varepsilon$, $u^\varepsilon\colon \Omega\to\mathbb{R}^3$ are
\begin{equation} \label{prob}
\begin{gathered}
\mathop{\rm div}\sigma^\varepsilon+f=0\quad \text{in }\Omega,\\
\sigma_{ij}^{\varepsilon}= a_{ijkh} e_{kh}(u^\varepsilon)
\quad \text{in }\Omega_\varepsilon,\\
\sigma^{\varepsilon}=\lambda|e(u^{\varepsilon})|^{-1}e(u^{\varepsilon})
\quad\text{in }B_\varepsilon,\\
u^\varepsilon=0\quad \text{on }\partial \Omega.
\end{gathered}
\end{equation}
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 $e(u)$.
In the sequel, we assume that the elasticity coefficients
$a_{ijkh}$ satisfy to the following hypotheses:
\begin{gather}
    a_{ijkh}\in L^\infty(\Omega),\label{h1} \\
    a_{ijkh}=a_{jikh}=a_{khij},\label{h2}\\
    a_{ijkh}\tau_{ij}\tau_{kh}\ge C \tau_{ij}\tau_{ij},\quad
  \forall \tau\in \mathbb{R}^9_{S}\label{h3}.
\end{gather}

\section{Notation and functional setting}\label{sec:3}

Here is the notation that will be used in the sequel:\\
$ x=(x',x_3)$ where $x'=(x_1,x_2)$,
$\tau\otimes\zeta= (\tau_{i}\zeta_{j})_{1\le i,j\le 3}$
and $\tau\otimes_{S}\zeta=\frac{\tau\otimes\zeta+\zeta\otimes\tau}{2}$
for all $\tau,\zeta\in \mathbb{R}^3$.

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  space
\begin{align*}
V^\varepsilon=\big\{& u\in L^{1}(\Omega,\mathbb{R}^3):
 e(u)\in L^2(\Omega_\varepsilon,\mathbb{R}^9_S),\; u\in
BD(B_{\varepsilon}),\\
 &[u]^{\varepsilon}=0 \text{ in }\Sigma_\varepsilon^{\pm}
\text{ and }u=0 \text{ in }\partial\Omega \big\},
\end{align*}
where $[u]^{\varepsilon} $ is  the jump of $u$ on
$\Sigma_\varepsilon^{\pm}$ defined by
\begin{gather*}
[u]^{\varepsilon}
=\pm u_{|_{\Omega_{\varepsilon}^{\pm}}}\mp u_{|_{B_\varepsilon^{\pm}}},\\
BD(B_{\varepsilon})=\big\{ u\in L^{1}(\Omega,\mathbb{R}^3):
 e(u)\in M_{1}(B_\varepsilon,\mathbb{R}^9_S)\big\},\\
BD_0(\Omega)=\big\{ u\in BD(\Omega,\mathbb{R}^3):
 u= 0 \text{ in } \partial\Omega\big\},
\end{gather*}
and $M_{1}(.)$  is a bounded measure space, for more information we
can refer the reader to \cite{temam}.
We show easily that $V^\varepsilon$ is a Banach space with the
 norm
$$u\to \norm{e(u)}_{L^2(\Omega_{\varepsilon},\mathbb{R}^9_{S})}
+\norm{e(u)}_{M_{1}(B_{\varepsilon},\mathbb{R}^9_{S})}.
$$
Where
\[
\norm{e(u)}_{M_{1}(B_{\varepsilon},\mathbb{R}^9_{S})}
=\int_{B_\varepsilon}\abs{e(u)}
=\sup_{\tau\in\mathcal{C}^{\infty}_0{(B_\varepsilon)},\,
\abs{\tau(x)}\le 1.}
\big\langle e(u), \tau \big\rangle.
\]
We remark that $V^\varepsilon\subset BD_0(\Omega)$.

Our goal in this work is to study the problem
(\ref{prob}), and its limit behavior
when $\varepsilon$ tends to zero.

\section{Study of  problem \eqref{prob}}\label{sec:4}

Problem (\ref{prob}) is equivalent to the minimization problem
\begin{equation} \label{pbmin}
\inf_{v\in V^{\varepsilon}}\big\{\frac{1}{2}
\int_{\Omega_\varepsilon}  a_{ijhk}e_{hk}(v)e_{ij}(v)dx
+\lambda\int_{B_\varepsilon}\abs{e( v)}
-\int_{\Omega}fvdx\big\}
\end{equation}
To study problem \eqref{prob}, we will study the minimization
problem \eqref{pbmin}. The existence and uniqueness of solutions to
\eqref{pbmin} is given in the following proposition.

\begin{proposition}\label{existe}
Under the hypotheses \eqref{h1}, \eqref{h2}, \eqref{h3} and for $f\in L^{\infty}(\Omega,\mathbb{R}^3)$,
problem \eqref{pbmin} admits an unique solution
$u^\varepsilon$ in $V^\varepsilon$.
\end{proposition}

\begin{proof}
 From \eqref{h1} and \eqref{h3}, we show easily that the energy
functional in \eqref{pbmin} 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[theorem 1.1 p.48]{daco}, but we can follow our proof by using
the compact imbedding of Sobolev for the BD space, for more information
we can refer the reader to \cite{daco}. Indeed, let $u_n$ be a minimizing
sequence for \eqref{pbmin}, to simplify the writing let
$$
\mathbb{F}^\varepsilon(u)=\frac{1}{2}
\int_{\Omega_\varepsilon}
 a_{ijhk}e_{hk}(u)e_{ij}(u)dx
+\lambda\int_{B_\varepsilon}\abs{e(u)}
-\int_{\Omega}fudx,
$$
so, we have $\displaystyle\mathbb{F}^\varepsilon(u_n)\to \inf_{v\in V^\varepsilon}\mathbb{F}^\varepsilon(v)$.
Using the coercivity of $\mathbb{F}^\varepsilon$, we may then deduce that
there exists a constant $C>0$, independent of $n$, such that
$$
\norm{u_n}_{V^\varepsilon}\le C,
$$
according to the reflexivity of $H^1(\Omega_\varepsilon)$ and using the given
result in \cite[p.158]{temam} for $BD(B_\varepsilon)$, then for 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
$\mathbb{F}^\varepsilon$ imply then the result.
\end{proof}

\begin{lemma}\label{lemme1}
Assuming that for any sequence
$(u^\varepsilon)_{\varepsilon>0}\subset V^\varepsilon$, there exists a
constant $C>0$ such that $\mathbb{F}^\varepsilon(u^\varepsilon)\le C$,
under \eqref{h1}, \eqref{h3} and for $f\in L^{\infty}(\Omega,\mathbb{R}^3)$,
$(u^\varepsilon)_{\varepsilon>0}$ satisfies
\begin{gather}
\norm{e(u^\varepsilon)}_{L^2(\Omega_\varepsilon,
\mathbb{R}^9_{S})}^2\leq C, \label{assert:1}\\
\norm{e(u^\varepsilon)}_{M_1(B_\varepsilon,
\mathbb{R}^9_{S})} \leq  C,\label{assert:2}
\end{gather}
moreover $u^\varepsilon$ is bounded in $BD_0(\Omega,\mathbb{R}^3)$.
\end{lemma}

\begin{proof}
Since
$\mathbb{F}^\varepsilon(u^\varepsilon)\le C$, we have
$$
\frac{1}{2}
\int_{\Omega_\varepsilon}
 a_{ijhk}e_{hk}(u^\varepsilon)e_{ij}(u^\varepsilon)dx
+\lambda\int_{B_\varepsilon}\abs{e(u^\varepsilon)}
-\int_{\Omega}fu^\varepsilon dx\le C\,.
$$
Then
\[
\frac{1}{2} \int_{\Omega_\varepsilon}
 a_{ijhk}e_{hk}(u^\varepsilon)e_{ij}(u^\varepsilon)dx
+\lambda\int_{B_\varepsilon}\abs{e(u^\varepsilon)}
\le C + \int_{\Omega}fu^\varepsilon dx \,.
\]
According to \eqref{h3},   H\"{o}lder and Young the inequalities,
we have
\begin{align*}
\norm{e(
u^\varepsilon)}^2_{L^2(\Omega_\varepsilon,\mathbb{R}^9_{S})}
+\int_{B_\varepsilon}\abs{e(u^\varepsilon)}
&\le  C + C\int_{\Omega}fu^\varepsilon dx ,\\
&\le  C + C\norm{e(u^\varepsilon)}_{L^2(\Omega_\varepsilon,\mathbb{R}^9_{S})}
+ \int_{B_\varepsilon}fu^\varepsilon dx,
\end{align*}
since $BD(\Omega)\hookrightarrow L^{q}(\Omega,\mathbb{R}^3)$ for all
$q\in[1,\frac{3}{2}]$, (with a continuous imbedding, see for example
\cite{temam}). In particular
$BD(\Omega)\hookrightarrow L^{q_0}(\Omega,\mathbb{R}^3)$ with
$1<q_0\le\frac{3}{2}$, according to the H\"{o}lder inequality,
we then have
\begin{align*}
\int_{B_\varepsilon}fu^\varepsilon&\leq \norm{f}_{L^{q_0'}(B_{\varepsilon},\mathbb{R}^3)}
\norm{u^\varepsilon}_{L^{q_0}(B_{_{\varepsilon}},\mathbb{R}^3)},\\
&\leq C\varepsilon^{1/q_0'}\int_{\Omega}\abs{e(u^\varepsilon)},\\
&\leq C\varepsilon^{1/q_0'}\Big(\norm{e(
u^\varepsilon)}_{L^2(\Omega_\varepsilon,\mathbb{R}^9_{S})}
+\int_{B_{\varepsilon}}\abs{e(u^\varepsilon)}\Big),
\end{align*}
so for $\varepsilon<\big(\frac{1}{1+C}\big)^{q_0'}$, let $\widetilde{C}=\frac{C}{1+C}$, we then have
\begin{align*}
\norm{e(u^\varepsilon)}^2_{L^2(\Omega_\varepsilon,\mathbb{R}^9_{S})}
+\int_{B_\varepsilon}\abs{e(u^\varepsilon)}
&\leq  C + C\norm{e(u^\varepsilon)}_{L^2(\Omega_\varepsilon,
  \mathbb{R}^9_{S})} + \widetilde{C}\int_{B_\varepsilon}\abs{e(u^\varepsilon)},\\
&\leq C + \frac{1}{2}\norm{e(u^\varepsilon)}^2
_{L^2(\Omega_\varepsilon,\mathbb{R}^9_{S})}
 +\widetilde{C}\int_{B_\varepsilon}\abs{e(u^\varepsilon)},
\end{align*}
so that
\[
\frac{1}{2}\norm{e(
u^\varepsilon)}^2_{L^2(\Omega_\varepsilon,\mathbb{R}^9_{S})}
+(1-\widetilde{C})\int_{B_\varepsilon}\abs{e(u^\varepsilon)}
\leq  C.
\]
Therefore, we will have (\ref{assert:1}) and (\ref{assert:2}).
According to (\ref{assert:1}) and (\ref{assert:2}) and for a small
enough $\varepsilon$ the sequence ($u^\varepsilon$) is bounded in
$BD_0(\Omega,\mathbb{R}^3)$.
\end{proof}

\begin{remark}\label{rem2} \rm
The solution $u^\varepsilon$ of the problem \eqref{pbmin} satisfy
to the lemma \ref{lemme1}.
\end{remark}

To apply the epiconvergence method, we need to characterize the
topological spaces containing any cluster point of the solution of
the problem \eqref{pbmin} with respect to the used topology, therefore
the weak topology to use is insured by the lemma \ref{lemme1}.
So the topological spaces characterization is given in the following
proposition.

\begin{proposition}\label{prop2}
The solution $u^\varepsilon$ of the problem \eqref{pbmin} possess
a cluster point $u^*$ in
$BD_0(\Omega)\cap H^1(\Omega\setminus\Sigma,\mathbb{R}^3)$
with respect to the weak topology of $BD_0(\Omega)$.
\end{proposition}

\begin{proof}
According to the remark \ref{rem2} and lemma \ref{lemme1},
for a small enough $\varepsilon$, $u^\varepsilon$ is bounded in
$BD_0(\Omega)$, so for a subsequences of
$u^\varepsilon$, still denoted by $u^\varepsilon$, there exists
$u^*\in BD_0(\Omega)$, (see \cite[p. 158]{temam}), such that
\begin{equation}
u^\varepsilon \rightharpoonup u^* \text{in
}BD_0(\Omega,\mathbb{R}^3),\label{eq:a}
\end{equation}
so that
\begin{equation}
\lim_{\varepsilon\to 0}\int_{\Omega}v e(u^\varepsilon)
=\int_{\Omega}v e(u^*), \quad
\forall v\in \mathcal{C}^{\infty}_0(\Omega,\mathbb{R}^9_S).\label{eq:b}
\end{equation}
For $\varepsilon$ a small enough, let $\eta>0$ and
$\Omega^\eta=\big\{ x\in\Omega: \abs{x_3}>\eta\big\}$,
 such that $\varepsilon<\eta$.
From \eqref{assert:1}, we then have
\[
\norm{e(u^\varepsilon)}_{L^2(\Omega^\eta,
\mathbb{R}^9_{S})}^2\leq C,
\]
Therefore,
$e(u^\varepsilon)$ is bounded in
$L^2(\Omega^\eta,\mathbb{R}^9_S)$, so for a subsequence of
$e(u^\varepsilon)$, still denoted by $e(u^\varepsilon)$, there exists
$w\in L^2(\Omega^\eta,\mathbb{R}^9_S)$, such that
$$
e(u^\varepsilon)\rightharpoonup w
\quad \text{in }L^2(\Omega^\eta,\mathbb{R}^9_S),
$$
according \eqref{eq:a} and \eqref{eq:b} remains true in
$\mathcal{C}^{\infty}_0(\Omega^\eta,\mathbb{R}^9_S)$,
we then deduce $e(u^*)=w$, hence
$e(u^*)\in L^2(\Omega^\eta,\mathbb{R}^9_S)$  for all $\eta>0$,
so by passing to the limit ($\eta\to 0$), we then have
$e(u^*)\in L^2(\Omega\setminus\Sigma,\mathbb{R}^9_S)$.
 According to the classical result
 \cite[proposition 1.2, p. 16]{temam}, we have $u^*\in
H^1(\Omega\setminus\Sigma,\mathbb{R}^3)$.
\end{proof}

In the following, let
\begin{gather*}
\mathbb{H}^{1}_0=\big\{u\in H^1(\Omega\setminus\Sigma,\mathbb{R}^3):
u=0 \text{ on }\partial\Omega\big\}. \\
\mathbb{C}^{\infty}_0=\big\{u\in \mathcal{C}^\infty(\Omega\setminus\Sigma,\mathbb{R}^3)
: u=0 \text{ on }\partial\Omega\big\}.
\end{gather*}

\begin{remark}\label{remark1} \rm
Proposition \ref{prop2} remains valid for any weak cluster point $u$ of
a sequence $u_\varepsilon$ in $V^\varepsilon$,that  satisfies
\eqref{assert:1} and \eqref{assert:2}.
\end{remark}

To study the limit behavior of the solution of the problem \eqref{pbmin},
we will use the epiconvergence method, (see Annex, definition \ref{def:epi}).

\section{Limit behavior}\label{sec:5}

Let
\begin{equation} \label{F0}
\begin{gathered}
F^{\varepsilon}(u)= \begin{cases}
\displaystyle\frac{1}{2} \int_{\Omega_\varepsilon}
 a_{ijkh} e_{kh}(u)e_{ij}(u)dx
+\lambda\int_{B_\varepsilon}\abs{e(
u)} &\text{if }u\in V^\varepsilon,\\
+\infty &\text{if }u\in BD_0(\Omega)\setminus V^\varepsilon.
\end{cases} \\
G(u)=-\int_{\Omega}fu dx, \quad \forall u\in BD_0(\Omega).
\end{gathered}
\end{equation}
We design by $\tau_f$ the weak topology on the space
${BD}_0(\Omega)$. In the sequel, we shall characterize, the epi-limit
of the energy functional given by (\ref{F0}) in the following theorem.

\begin{theorem}\label{cal:epi}
Under \eqref{h1}, \eqref{h2}, \eqref{h3} and for
$f\in L^{\infty}(\Omega,\mathbb{R}^3)$, there exists a functional
$F\colon BD_0(\Omega)\to \mathbb{R}\cup\{+\infty\}$
such that
\[
\tau_f-lim _{e}F^{\varepsilon}{=}F\quad \text{in }BD_0(\Omega),
\]
where $F$ is given by
\[
F(u)= \begin{cases}
\displaystyle\frac{1}{2} \int_{\Omega}
 a_{ijkh} e_{kh}(u)e_{ij}(u)+\lambda\int_{\Sigma}\abs{[u]\otimes_Se_3}
&\text{if } u\in \mathbb{H}^{1}_0,\\
+\infty &\text{if }u\in BD_0(\Omega)\setminus\mathbb{H}^{1}_0.
\end{cases}
\]
\end{theorem}

\begin{proof}
(a) We are now in position to determine the upper epi-limit.\\
Let $u\in \mathbb{H}^{1}_0\subset BD_0(\Omega)$, so there exists
a sequence $(u^n)$ in $\mathbb{C}^{\infty}_0$ such that $u^n \to u$ in
$\mathbb{H}^{1}_0\text{when } n\to +\infty$, so $u^n \rightharpoonup u$
 weakly in $BD_0(\Omega)$.
Let us consider the  sequence
\[
u^{\varepsilon,n}=\begin{cases}
u^{n}(x',x_3)& \text{if }\abs{x_3}>\frac{\varepsilon}{2},\\[3pt]
\frac{1}{2}\big(u^{n}(x',\frac{\varepsilon}{2})+u^{n}(x',-\frac{\varepsilon}{2})\big)\\
+\frac{x_3}{\varepsilon} \big(u^{n}(x',\frac{\varepsilon}{2})-u^{n}(x',-\frac{\varepsilon}{2})\big)
& \text{if }\abs{x_3}<\frac{\varepsilon}{2}.
\end{cases}
\]
We have $u^{\varepsilon,n}\in V^\varepsilon$ and we prove easily
that $u^{\varepsilon,n}\rightharpoonup u^n$ in
$\mathbb{H}^{1}_0$ when $\varepsilon\to 0$.
As
\begin{align*}
F^\varepsilon(u^{\varepsilon,n})&=&
\frac{1}{2} \int_{\Omega_\varepsilon}
a_{ijkh}e_{kh}(u^{\varepsilon,n})e_{ij}(u^{\varepsilon,n})
+\lambda\int_{B_\varepsilon}\abs{
e(u^{\varepsilon,n})}.
\end{align*}
It implies that
\[
F^\varepsilon(u^{\varepsilon,n})
= \frac{1}{2} \int_{\Omega_\varepsilon}
a_{ijkh}e_{kh}(u^{n})e_{ij}(u^{n})
+\lambda \int_{B_\varepsilon}\abs{e(
u^{\varepsilon,n})}
=: S_1+S_2.
\]
So that
\[
\lim_{\varepsilon\to 0}S_1= \frac{1}{2}
\int_{\Omega}a_{ijkh}e_{kh}(u^{n})e_{ij}(u^{n}).
\]
we have
\begin{equation}\label{s3}
S_2=\lambda\int_{B_\varepsilon}\abs{e(u^{\varepsilon,n})},
\end{equation}
As in \cite{moussa} we show  that
\[ %\label{s3}
\lim_{\varepsilon\to 0} \int_{B_\varepsilon} | e(u^{\varepsilon,n})
-\frac{1}{\varepsilon}[u^n]\otimes_{S}e_{3} |=0.
\]
Consequently,
\[
\limsup_{\varepsilon\to 0}F^\varepsilon(u^{\varepsilon,n})
=\frac{1}{2} \int_{\Omega}
a_{ijkh}e_{kh}(u^{n})e_{ij}(u^{n})+
\lambda\int_{\Sigma} \abs{[u^n]\otimes_{S}e_{3}}.
\]
Since
$u^{n}\to u$ in $\mathbb{H}^{1}_0$ when $n\to+\infty$,
therefore according to a classic result, diagonalization's lemma,
(see, \cite[Lemma 1.15 p. 32]{attouch}), there exists
a function $n(\varepsilon):\mathbb{R}^+\to\mathbb{N} $
increasing to $+\infty$ when $\varepsilon\to 0$ such that
$u^{\varepsilon,n(\varepsilon)}\rightharpoonup u$ in
$\mathbb{H}^{1}_0$ when $\varepsilon\to 0$.
and while $n\to+\infty$, consequently we have
\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}{2} \int_{\Omega}
a_{ijkh}e_{kh}(u)e_{ij}(u)+
\lambda\int_{\Sigma}
\abs{[u]\otimes_{S}e_{3}}.
\end{align*}
For $u\in BD_0(\Omega,\mathbb{R}^3)\setminus\mathbb{H}^{1}_0$,
so for any sequence $u^\varepsilon\rightharpoonup u$ in
$BD_0(\Omega)$,we obtain
$$
\limsup_{\varepsilon\to
0}F^\varepsilon(u^{\varepsilon})\leq +\infty.
$$

(b) We are now in position to determine the lower epi-limit.
Let $u\in \mathbb{H}^{1}_0$ and $(u^\varepsilon)\subset V^\varepsilon$
such that
$u^\varepsilon{\rightharpoonup}u$ in $BD_0(\Omega)$.
If $\liminf_{\varepsilon\to
0}F^\varepsilon(u^\varepsilon)=+\infty$, there is nothing to prove,
because
$$
\frac{1}{2}  \int_{\Omega}
a_{ijkh}e_{kh}(u)e_{ij}(u)+\lambda
\int_{\Sigma}\abs{[u]\otimes_{S}e_{3}}\leq
+\infty.
$$
otherwise, $\liminf_{\varepsilon\to 0}F^\varepsilon(u^\varepsilon)<+\infty$,
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{gather*}
\norm{e(u^\varepsilon}_{L^2(\Omega_\varepsilon,\mathbb{R}^9_S)} \leq  C,\\
\int_{B_{\varepsilon}}\abs{e(u^\varepsilon)} \leq  C,
\end{gather*}
then $\chi_{{\Omega_{\varepsilon}}}e( u^\varepsilon)$ is bounded
in $L^2(\Omega,\mathbb{R}^9_S)$, 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{equation}
\chi_{{\Omega_{\varepsilon}}}e(u^\varepsilon)
\rightharpoonup e( u)\quad \text{in }L^2(\Omega,\mathbb{R}^9_S))
\label{eq:liminf2}
\end{equation}
From the subdifferentiability's inequality of
$u\to \frac{1}{2}  \int_{\Omega_\varepsilon}
a_{ijkh}e_{kh}(u)e_{ij}(u)$,
and passing to the lower limit, we obtain
\[
\liminf_{\varepsilon\to 0}\frac{1}{2}  \int_{\Omega_\varepsilon}
a_{ijkh}e_{kh}(u^\varepsilon)e_{ij}(u^\varepsilon)
\geq \frac{1}{2}  \int_{\Omega}
a_{ijkh}e_{kh}(u)e_{ij}(u).
\]
For $\eta<\varepsilon/2$, let us set
$$
B^\eta=\big\{x\in\Omega: \abs{x_3}<\eta\big\}.
$$
According to the diagonalization's lemma  \cite[Lemma 1.15 p. 32]{attouch},
 there exists a function $\eta(\varepsilon):\mathbb{R}^+\to\mathbb{R}^+ $
decreasing to $0$ when $\varepsilon\to 0$ such that
\begin{equation}
\liminf_{\varepsilon\to 0}\int_{B^{\eta(\varepsilon)}}\abs{e(u^\varepsilon)}
\ge\liminf_{\eta\to 0}\liminf_{\varepsilon\to 0}\int_{B^{\eta}}
\abs{e(u^\varepsilon)}.\label{ddiag}
\end{equation}
Since
$$
\int_{B^{\eta}}\abs{e(u^\varepsilon)}\ge \int_{B^{\eta}}\phi
\big(e(u^\varepsilon)-e(u)\big)+\int_{B^{\eta}}\phi e(u)
,\quad \forall \phi\in \mathcal{C}^\infty_0(B^\eta,\mathbb{R}^9_S),
$$
it follows that
$$
\liminf_{\varepsilon\to 0}\int_{B^{\eta}}\abs{e(u^\varepsilon)}\ge\int_{B^{\eta}}\phi e(u)
,\quad \forall \phi\in \mathcal{C}^\infty_0(B^\eta,\mathbb{R}^9_S).
$$
Therefore,
$$
\liminf_{\varepsilon\to 0}\int_{B^{\eta}}\abs{e(u^\varepsilon)}
\ge\int_{B^{\eta}}\abs{e(u)}.
$$
According to a classic result \cite[Lemma 2.2 p. 145]{temam}),
we then have
$$
\liminf_{\varepsilon\to 0}\int_{B^{\eta}}\abs{e(u^\varepsilon)}
\ge\int_{B^{\eta}}\phi e(u)+\int_{\Sigma}\phi [u]\otimes_{S}e_{3}dx',
\quad \forall \phi\in \mathcal{C}^\infty_0(\Omega,\mathbb{R}^9_S).
$$
By passing to the limit, ($\eta\to 0$), we have
$$
\liminf_{\eta\to 0}\liminf_{\varepsilon\to 0}\int_{B^{\eta}}\abs{e(u^\varepsilon)}
\ge\int_{\Sigma}\abs{[u]\otimes_{S}e_{3}}dx'.
$$
According to the definition of $B^{\eta}$ and \eqref{ddiag}, we deduce that
$$
\liminf_{\varepsilon\to 0}\int_{B_{\varepsilon}}\abs{e(u^\varepsilon)}
\ge\int_{\Sigma}\abs{[u]\otimes_{S}e_{3}}dx'.
$$
Hence
$$
\liminf_{\varepsilon\to 0}F^\varepsilon(u^\varepsilon)\ge\frac{1}{2}  \int_{\Omega}
a_{ijkh}e_{kh}(u)e_{ij}(u)+\int_{\Sigma}
\abs{[u]\otimes_{S}e_{3}}dx'.
$$
For $u\in BD_0(\Omega)\setminus\mathbb{H}^{1}_0$ and
$u^\varepsilon\in V^\varepsilon$, such that
$u^\varepsilon\rightharpoonup u$ in $BD_0(\Omega)$.
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{equation} \label{eqliminf333}
F^\varepsilon(u^{\varepsilon})<C.
\end{equation}
So $u^{\varepsilon}$ verifies the following evaluations \eqref{assert:1}
and \eqref{assert:2}, as
$u^\varepsilon\rightharpoonup u$ in $BD_0(\Omega)$, thanks to
the remark \ref{remark1}, we have $u\in \mathbb{H}^{1}_0$, what
contradicts the fact that
$u\in BD_0(\Omega)\setminus\mathbb{H}^{1}_0$, consequently we have
\[
\liminf_{\varepsilon\to 0}F^\varepsilon(u^{\varepsilon})=+\infty.
\]
Hence the proof is complete.
\end{proof}

In the sequel, we determine the limit problem  linked
to \eqref{pbmin}, when $\varepsilon$ approaches to
zero. Thanks to the epi-convergence results,
(see Annex, theorem \ref{th:epi}, proposition \ref{stab:epi}) and the theorem
\ref{cal:epi}, according to the $\tau_f$-continuity of the functional $G$ in
$BD_0(\Omega)$, we have $F^\varepsilon +G$
$\tau_f$-epiconverges to $F+G$ in $BD_0(\Omega)$.

\begin{proposition} \label{prop5.2}
For any $f\in L^2(\Omega,\mathbb{R}^3)$, there exists
$u^*\in BD_0(\Omega)$ satisfying:
$u^\varepsilon \rightharpoonup u^*$ in $BD_0(\Omega)$ and
\[
 F(u^*)+G(u^*)=\inf_{v\in \mathbb{H}^{1}_0}
 \{F(v)+G(v)\}.
\]
\end{proposition}

\begin{proof}
Thanks to lemma \ref{lemme1}, the  family $(u^\varepsilon)_{_{\_{\varepsilon}}}$ is
bounded in $BD_0(\Omega)$, therefore it possess a
$\tau_f$-cluster point $u^*$ in $BD_0(\Omega)$. And thanks to a
classical epi-convergence result,  theorem \ref{th:epi},
it follows that $u^*$ is a solution of the problem: Find
\begin{equation}\label{pblm}
 \inf_{v\in BD_0(\Omega)}\bigl\{F(v)+G(v)\bigr\}.
\end{equation}
Since  $F=+\infty$ on $BD_0(\Omega)\setminus
\mathbb{H}^{1}_0$, so (\ref{pblm}) becomes
\begin{align*} \label{Pi}
\inf_{v\in \mathbb{H}^{1}_0}\bigl\{F(v)+G(v)\bigr\}.
\end{align*}
According to the uniqueness of solutions of problem (\ref{pblm}),
so $u^\varepsilon$ admits an unique $\tau_f$-cluster point $u^*$,
and therefore $u^\varepsilon\rightharpoonup u^*$ in $BD_0(\Omega)$.
 \end{proof}

\subsection*{Conclusion}
 Using the epiconvergence method, we showed that the question of finding
the limit problem, composed of a classical linear elasticity problem posed
over $\Omega\setminus\Sigma$, contains an interface condition which
depends on the displacement field jump.
We found the same result of Ait Moussa, with $p=1$, in \cite{these}.

\section{Annex}

\begin{definition}[{\cite[Definition 1.9]{attouch}}] \label{def:epi} \rm
 Let $(\mathbb{X},\tau)$ be a metric space and
$(F^{\varepsilon})_\varepsilon$
 and $F$ be functionals defined on $\mathbb{X}$ and with value in
 $\mathbb{R}\cup\{+\infty\}$. $F^{\varepsilon}$
 epi-converges to $F$ in $(\mathbb{X},\tau)$, noted
$\tau-lim_e F^\varepsilon=F$,
 if the following assertions are satisfied
 \begin{itemize}
 \item For all $x\in \mathbb{X}$, there exists $x_\varepsilon^0$,
$x_\varepsilon^0\stackrel{\tau}  {\to} x$ such that
$\displaystyle\limsup_{\varepsilon\to 0}
F^\varepsilon(x_\varepsilon^0)\leq F(x)$.

 \item For all $x\in \mathbb{X}$ and all
 $x_\varepsilon$ with $x_\varepsilon\stackrel{\tau}{\to} x$,
 $\displaystyle\liminf_{\varepsilon\to 0} F^\varepsilon(x_\varepsilon)\geq F(x)$.
 \end{itemize}
\end{definition}

We have the following stability result for  epi-convergence.

 \begin{proposition}[{\cite[p. 40]{attouch}}] \label{stab:epi}
 Suppose that $F^{\varepsilon}$ epi-converges to $F$ in
$(\mathbb{X},\tau)$  and that $G\colon \mathbb{X}\to \mathbb{R}
\cup\{+\infty\}$, is $\tau-continuous$. Then
 $F^{\varepsilon}+G$ epi-converges to $F+G$ in $(\mathbb{X},\tau)$
 \end{proposition}

 This epi-convergence is a special case of the
 $\Gamma-$convergence introduced by De Giorgi (1979) \cite{degiorgi}.
It is well suited to the asymptotic analysis of sequences of
minimization problems since one has the following fundamental result.

\begin{theorem}[{\cite[theorem 1.10]{attouch}}] \label{th:epi}
 Suppose that
 \begin{enumerate}
 \item  $F^{\varepsilon}$ admits a minimizer on
 $\mathbb{X}$,
 \item The sequence $(\overline u^{\varepsilon})$ is
$\tau$-relatively  compact,
 \item The sequence $F^\varepsilon$ epi-converges to
 $F$ in this topology $\tau$.
 \end{enumerate}
 Then every cluster point $\overline u$ of the sequence
$(\overline{u}^{\varepsilon})$ minimizes $F$
 on $\mathbb{X}$ and
 $$
 \lim_{\varepsilon'\to  0}F^{\varepsilon'}(\overline{u}^{\varepsilon'})
=F(\overline u),
 $$
 where $(\overline{u}^{\varepsilon'})_{{\varepsilon'}}$ denotes
 any subsequence of
 $(\overline{u}^{\varepsilon})_{{\varepsilon}}$ which
 converges to $\overline u$.
 \end{theorem}

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\end{document}