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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 137, pp. 1--11.\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{8mm}}

\begin{document}
\title[\hfilneg EJDE-2006/137\hfil A property of the $H$-convergence]
{A property of the $H$-convergence for elasticity in perforated domains}

\author[H. Haddadou\hfil EJDE-2006/137\hfilneg]
{Hamid Haddadou}

\address{Hamid Haddadou \newline
Institut Nationale de formation en Informatique,
BP 68, Oued Smar, El Harrach, Alger, Alg\'{e}rie}
\email{h\_haddadou@ini.dz}

\date{}
\thanks{Submitted July 6, 2006. Published October 31, 2006.}
\subjclass[2000]{35B40, 74B05}
\keywords{Homogenization; H-convergence; linearized
elasticity system; \hfill\break\indent perforated domains}

\begin{abstract}
In this article, we obtain the $H_{e}^{0}$-convergence as a limit case
of the $H_{e}$-convergence. More precisely, if
$\Omega_{\varepsilon}$ is a perforated domain with (admissible)
holes $T_{\varepsilon}$ and $\chi_{\varepsilon}$ denote its
characteristic function and if
$(A^{\varepsilon},T_{\varepsilon})\overset{H_{e}^{0}}{\rightharpoonup}A^{0}$,
we show how the behavior as $(\varepsilon,\delta)\to(0,0)$
of the double sequence of tensors
$A^{\varepsilon}_{\delta}=(\chi_{\varepsilon}+\delta(1-\chi_{\varepsilon}))
A^{\varepsilon}$ is connected to $A^{0}$. These results  extend
those given by  Cioranescu,  Damlamian,  Donato and
Mascarenhas in \cite{ciodam} for the $H$-convergence of the scalar
second elliptic operators to the linearized elasticity systems.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

The  notion of  $H$-convergence was introduced  by  Murat
and Tartar \cite{mur,murta,tar}   for the
second-order elliptic  operators (non necessary symmetric) and
extended to the case of holes by    Briane, Damlamian and
Donato in \cite{bri} and  called  $H^{0}$-convergence.
 Cioranescu, Damlamian, Donato and Mascarenhas \cite{ciodam}
obtain the $H^{0}$-convergence as a limit case of the
$H$-convergence with a vanishing coercivity constant in the holes.

In this work, we show that a similar property holds  for the
linearized elasticity systems, namely between the
$H_{e}$-convergence studied by  Francfort and  Murat in
\cite{fra} and  its generalization to the case of holes, denoted by
$H_{e}^{0}$-convergence, which has been developed by Donato and
 El Hajji in \cite{don}. The $H_{e}$-convergence deals with the
convergence of the solutions of a system of linearized
elasticity whose tensor coefficients $\{A^{\varepsilon}\}$ are
equibounded and uniformly definite positive. The
$H_{e}^{0}$-convergence treat the same problem  in a perforated
domain $\Omega_{\varepsilon}$ with a traction condition on the
holes for which uniform Korn estimates hold.

Let us briefly describe here the main results of this paper. Let
$\Omega$ a bounded open subset of $\mathbb{R}^{n}$,
$\{T_{\varepsilon}\}$  a sequence of (admissible) holes, denote
$\Omega_{\varepsilon}=\Omega\setminus T_{\varepsilon}$ the
perforated domain and $\chi^{\varepsilon}$ the characteristic
function of $\Omega_{\varepsilon}$. Let also $\{A^{\varepsilon }\}$
a sequence of linearized elasticity tensors on $\Omega$  such that
$(A^{\varepsilon},T_{\varepsilon})\overset{H_{e}^{0}}{\rightharpoonup}A^{0}$.
We prove (Theorem \ref{li10}) that if we set for every $\delta>0$
$$
A^{\varepsilon}_{\delta}=(\chi_{\varepsilon}+\delta(1-\chi_{\varepsilon}))
A^{\varepsilon}\quad\text{ a.e. in }\Omega
$$
and if
$A^{\varepsilon}_{\delta}$ $H_{e}$-converges to a tensor
$A_{\delta}$ (for some subsequence), then  $A_{\delta} \to
A^{0}$ strongly in $L^{p}(\Omega)$ for any $p\geq1$, and weakly
$\star$ in
$L^{\infty}(\Omega)$. Moreover, under suitable assumption
(see  \eqref{e22} below), we have also (Theorem \ref{li11})
\begin{equation*}
(A_{\delta}^{\varepsilon}
\overset{\delta\to0}{\longrightarrow}
(A^{\varepsilon},T_{\varepsilon} ))\text{ in the sense }\left\{
\begin{array}{l}
u^{\varepsilon}_{\delta}\to u^{\varepsilon}\text{ strongly in }
H^{1}_{0}(\Omega_{\varepsilon})^{n},\\[3pt]
A^{\varepsilon}_{\delta}e(u^{\varepsilon}_{\delta})\to
A^{\varepsilon} \widetilde{e(u^{\varepsilon})}\text{ strongly in }
\;L^{2}(\Omega)^{n\times n}
\end{array}
\right.
\end{equation*}
and  (Theorem \ref{li12})\begin{equation*}
(A^{\varepsilon}_{\delta}\overset{(\varepsilon,\delta)\to(0,0)}{\longrightarrow}
A^{0})\text{ in the sense }\left\{
\begin{array}{l}
u^{\varepsilon}_{\delta}\to u^{0}\text{ weakly in }
H^{1}_{0}(\Omega)^{n},\\
A^{\varepsilon}_{\delta}e(u^{\varepsilon}_{\delta})\to
A^{0}e(u^{0})\text{ weakly in }L^{2}(\Omega)^{n\times n},
\end{array}
\right.
\end{equation*}
where $u^{\varepsilon}$, $u^{\varepsilon}_{\delta}$ and $u^{0}$ are
the solutions of \eqref{e2}, \eqref{e9} and \eqref{e4}
respectively. This results can be resumed by the following
commutative schema:
\begin{equation*}
  \begin{array}{ccc}
 A^{\varepsilon}_{\delta} &  \overset{H_{e}}{\rightharpoonup}&A_{\delta} \\
\downarrow &  \searrow& \downarrow\; \\
  (A^{\varepsilon},T_{\varepsilon} )& \overset{H_{e}^{0}}{\rightharpoonup} & A^{0}. \\
  \end{array}
\end{equation*}

The definition and the main properties of the
$H_{e}^{0}$-convergence are recalled in Section 2. In Section 3 we
give some preliminary results and in Section 4 we
 state and prove the main results.

\section{The $H_{e}^{0}$-convergence}

We use the following notation:\\
$\bullet$  If $A =(A _{ijkl}) _{1\leq i,j,k,l\leq n}$ is a forth
order tensors and  $\Lambda\in\mathbb{R}^{n\times n}$, we set
\begin{gather*}
A\Lambda=\sum_{1\leq i,j,k,l\leq n} A_{ijkl}
\Lambda_{pq},\\
\Lambda.\Upsilon=\sum_{1\leq i,j\leq n}\Lambda_{ij}\Upsilon_{ij},\\
|\Lambda|= (\sum_{1\leq i,j\leq n} |\Lambda_{ij}|^{2})^\frac{1}{2},
\end{gather*}
$\bullet$ $\Omega $ is a domain of $\mathbb{R}^{n}$,\\
$\bullet$ if $F$ is a set of matrices fields, $F_{S}=\{M\in F\;s.\;t.\;M\;is \;symmetric\}$,\\
$\bullet$ $\{\varepsilon\} $ and $\{\delta\} $ denote a two strictly
decreasing sequence converging to zero,
\newline
$\bullet$ if ${v} =({v} _{1}, $\dots  $, {v} _{n})$ is a vector valued
function and $\zeta =( \zeta _{ij})_{1\leq i, j\leq n}$
is a second order tensor of variable $x=(x_{1}, $\dots  , $x_{n})$,  we
set
\begin{gather*}
( \nabla {v} ) _{ij}=\frac {\partial v_{i}}{  \partial x_{j}}\equiv D_{x_{j}}v_{i},  \\
e({v} )=\frac{1}{2}(\nabla {v} +^{t}\nabla {v} ),  \\
( \mathop{\rm div}\xi )_{i}=\frac{\partial \xi _{ij}}{
\partial x_{j}};
 \end{gather*}
$\bullet$ for two real numbers $\alpha $ and $\beta$ such that
$0<\alpha <\beta $, $M_{e}( \alpha , \beta , \Omega) $ is
the set of the tensors $A=( A_{ijkl}) _{1\leq i, j, k,
l\leq n}$ defined on $\Omega$, such that a.e.   on $\Omega$ we have
\begin{itemize}
\item[(i)] $A_{ijkl}\in L^{\infty }( \Omega)$,   for
any $i, j, k, l=1,\dots  , n$,
\item[(ii)] $A_{ijkl}=A_{jikl}=A_{klij}$,  for any $i, j, k,l=1,\dots  , n $,
\item[(iii)] $\alpha | \Lambda | ^{2}\leq A\Lambda .\Lambda$,
 for any symmetric matrix $\Lambda$,
\item[(iv)] $| A\Lambda|\leq \beta | \Lambda|$,  for any
matrix $\Lambda$,
\end{itemize}
$\bullet$ if $f\in H^{-1}(\Omega)^{n}$ and $u\in
H^{1}_{0}(\Omega)^{n}$, we set
 $\langle f,u\rangle=\langle f,u\rangle
_{H^{-1}(\Omega)^{n},H^{1}_{0}(\Omega)^{n}}$.


Let us recall first the main results concerning the
$H^{0}_{e}$-convergence introduced by  Donato and  El Hajji
\cite{don}. We introduce the perforated domain
\begin{equation*}
\Omega_{\varepsilon}=\Omega \backslash T_{\varepsilon},
\end{equation*}
where $T_{\varepsilon}$  is a sequence of compact subsets of
$\Omega$ and set
\begin{equation*}
V_{\varepsilon}=\{ v \in H^{1}( \Omega_{\varepsilon}) ^{n}
\text{ s. t.  $v=0$ on }\partial \Omega\}.
\end{equation*}

In the following, we denote by $\nu $ the outward normal unit vector
on the boundary  of $\Omega _{\varepsilon}$ and $\widetilde{.}$ the
extension by 0 from $\Omega_{\varepsilon}$ to $\Omega $ and set
$\chi^{\varepsilon}=\chi _{_{\Omega _{\varepsilon}}}$.

\begin{definition}[\cite{don}]\label{li1} \rm
The set $T_{\varepsilon}$ is said to be admissible (in $\Omega )$
for the linearized elasticity (or e-admissible),  if and only
if:
\begin{itemize}
\item[(i)] Every $L^{\infty }$ weak $\ast $-limit point of
$\{ \chi _{_{\Omega _{\varepsilon}}}\} _{\varepsilon}$ is
positive a.e.   in $\Omega $;

\item[(ii)] there exists a positive real $C$,  independent of ${\varepsilon}
$,  and a sequence $\{ P_{\varepsilon}\} _{\varepsilon}$
of linear extension operators such that for each ${\varepsilon} $
\begin{equation}\label{e1}
\begin{gathered}
P_{\varepsilon}\in \mathcal{L}(V_{\varepsilon}, H_{0}^{1}(\Omega )^{n}), \\
( P_{\varepsilon}v ) |_{\Omega_{\varepsilon}}={v} ,\quad \forall {v}
\in V_{\varepsilon},  \\
\|e( P_{\varepsilon}v ) \| _{0,\Omega}\leq
C\|e({v} )\| _{0, \Omega _{\varepsilon}}, \quad
\forall {v} \in V_{\varepsilon}.
\end{gathered}
\end{equation}
\end{itemize}
\end{definition}

We denote by $P_{\varepsilon}^{\star }$ the adjoint operator of
$P_{\varepsilon}$,  which is defined from $H^{-1}(\Omega)^{n}$ to
$V'_{\varepsilon}$  with $P_{\varepsilon}^{\star }$ given for every
$f\in H^{-1}(\Omega) ^{n}$ by
\begin{equation*}
\forall v\in V_{\varepsilon},\;\langle
P_{\varepsilon}^{\star}f,v\rangle_{V'_{\varepsilon},V_{\varepsilon}}=
\langle
f,P_{\varepsilon}v\rangle_{H^{-1}(\Omega)^{n},H_{0}^{1}(\Omega)^{n}}.
\end{equation*}

\begin{definition}[\cite{don}]\label{li2} \rm
Let $ A^{{\varepsilon} }\in M_{e}( \alpha , \beta , {\Omega}) $,
 $T_{\varepsilon}$ be e-admissible in $\Omega $.
The pair $(A^{{\varepsilon} }, T_{\varepsilon})$ is said
$H_{e}^{0}$-converge to the tensor $A^{0}\in M_{e}( \alpha
', \beta ', \Omega) $ and denoted by $(A^{{\varepsilon} }, T_{\varepsilon})
\overset{H_{e}^{0}}{\rightharpoonup }A^{0}$ if and only if for each
function $f^{\varepsilon}\in H^{-1}(\Omega)^{n}$ such that
$f^{\varepsilon}\to f$ strongly in $H^{-1}(\Omega)^{n}$, the
solution
 $u^{\varepsilon}$ of
\begin{equation}\label{e2}
\begin{gathered}
-\mathop{\rm div}\;(A^{{\varepsilon} }e( u^{{\varepsilon} })
)=P_{\varepsilon}^{\star }f^{\varepsilon}\quad\text{in }\Omega _{\varepsilon},
\\
(A^{{\varepsilon} }e( u^{{\varepsilon} }) )\nu =0\quad \text{on }
\partial T_{\varepsilon},  \\
u^{{\varepsilon} }=0\quad\text{on }\partial \Omega ,
\end{gathered}
\end{equation}
satisfies the weak convergence
\begin{equation}\label{e3}
\begin{gathered}
 P_{\varepsilon}(u^{{\varepsilon} }){
\rightharpoonup }u^{0}\quad \text{weakly in }H_{0}^{1}(\Omega )^{n}, \\
 A^{{\varepsilon} }\widetilde{e( u^{{\varepsilon}
}) } { \rightharpoonup }A^{0}e(u^{0})\quad \text{weakly in }
L^{2}(\Omega )^{n\times n},
\end{gathered}
\end{equation}
where $u^{0}$ is the unique solution of the problem
\begin{equation}\label{e4}
\begin{gathered}
-\mathop{\rm div}(A^{0}e( u^{0}) )=f\quad\text{in }\Omega ,\\
u^{0}=0\quad\text{on }\partial \Omega.
\end{gathered}
\end{equation}
\end{definition}

\begin{remark}\label{li3} \rm
(1) In \cite{don} the definition is  given  for fixed
$f_{\varepsilon}\doteq f$. The two definitions are equivalent
in view of \cite[Proposition 2]{don}.\\
(2) In the case where $T_{\varepsilon}=\emptyset$, this definition
reduces  to the definition of the $H_{e}$-convergence \cite{fra}.
\end{remark}

This notion of convergence makes sense in view of the following
compactness theorem:

\begin{theorem}[\cite{don}] \label{li4}
Let $A^{{\varepsilon} }\in M_{e}( \alpha , \beta ,
\Omega) $  and $T_{\varepsilon}$ be e-admissible in $\Omega $.
Then there exists a subsequence, still denoted by $\{\varepsilon\}$,
 and a tensor
$A^{0}\in M_{e}(\frac{\alpha}{C^{2}}, \beta , {\Omega})$,
 such that the sequence $\{ (A^{{\varepsilon} },
T_{\varepsilon})\} _{\varepsilon}$ $H_{e}^{0}$-converge to
$A^{0}$.
\end{theorem}

 \begin{remark}\label{li5} \rm
The fact that $A^{0}$ belongs to
$M_{e}( \frac{\alpha}{C^{2}}, \beta , {\Omega}) $,  does
not  appears  explicitly   in the statement given in \cite{don},
 but  can be easily deduced with
the same arguments as that used in the non perforated case.
\end{remark}

Let us recall also a property  proved recently in \cite{do-ha}.

\begin{theorem}[\cite{do-ha}] \label{li6bis}
Let $\{A^{\varepsilon }\}\in \mathcal{M}_{e}( \alpha , \beta
,\Omega)$ and $\{B^{\varepsilon}\}\in \mathcal{M}_{e}( \alpha' ,
\beta ',\Omega)$ such that
$A^{\varepsilon}\overset{H_{e}}{\rightharpoonup}A^{0}$ and
$B^{\varepsilon}\overset{H_{e}}{\rightharpoonup}B^{0}$. Assume that
there exists two functions
$h^{\varepsilon},\;h^{0}\in L^{1}(\Omega)$ such that
\begin{equation*}
|A^{\varepsilon}-B^{\varepsilon}|\leq h^{\varepsilon}\longrightarrow
h^{0}\quad\text{strongly in }L^{1}(\Omega).
\end{equation*}
Then
\begin{equation*}|A^{0}(x)-B^{0}(x)|\leq
\sqrt{\frac {\beta\beta'}{\alpha\alpha'}}\;h^{0}(x)\quad\text{a.e. in }
\Omega.
\end{equation*}
\end{theorem}

The following proposition completes a result given in
\cite{don}:

\begin{proposition} \label{li6} One has
\begin{enumerate}
\item If $\{v ^{\varepsilon}\}$  is a bounded sequence in
$H^{1}_{0}(\Omega)$, then
\begin{equation*}
( v ^{\varepsilon}\;\rightharpoonup \; v \text{ weakly in }
H_{0}^{1}(\Omega )^{n}) \Leftrightarrow
\Big(P_{\varepsilon}({v^{\varepsilon} } |_{\Omega_{\varepsilon}} )
\;\rightharpoonup \; {v} \text{ weakly in } H_{0}^{1}(\Omega )^{n}\Big).
\end{equation*}

\item If $(\varepsilon,\delta)$ is a sequence of
$\mathbb{R}_{+}^{\star}\times\mathbb{R}_{+}^{\star}$ such that
$(\varepsilon,\delta)\to(0,0)$ and $\{v
^{\varepsilon}_{\delta}\}$ is a sequence in $H^{1}_{0}(\Omega)$
bounded independently of $\varepsilon$ and $\delta$, then
\begin{equation*}
( v ^{\varepsilon}_{\delta}\;\rightharpoonup \; v
\text{ weakly in }H_{0}^{1}(\Omega )^{n}) \Leftrightarrow
\Big( P_{\varepsilon}({v^{\varepsilon}_{\delta}} |_{\Omega
_{\varepsilon}} ) \;\rightharpoonup \; {v} \text{ weakly in }
 H_{0}^{1}(\Omega )^{n}\Big).
\end{equation*}
\end{enumerate}
\end{proposition}

\begin{proof}
Suppose that \begin{equation}\label{e4bis2}
P_{\varepsilon}({v^{\varepsilon} } |_{\Omega
_{\varepsilon}} ){ \;\rightharpoonup \; }{v} \quad
\text{weakly in } H_{0}^{1}(\Omega )^{n}.
\end{equation}
Observe first that
\begin{equation}\label{e4bis1}
v^{\varepsilon}\chi^{\varepsilon}=P_{\varepsilon}({v^{\varepsilon}}
_{| _{\Omega _{\varepsilon}}})\chi^{\varepsilon}.
\end{equation}
On the other hand, since $\{v ^{\varepsilon}\}$ is a bounded
sequence in $H^{1}_{0}(\Omega)$,  there exists a
$\{\varepsilon'\}\subset\{\varepsilon\}$ and $w\in H_{0}^{1}(\Omega
)^{n}$ such that
\begin{equation}\label{e4bis3}
v^{\varepsilon'}\rightharpoonup w\quad\text{weakly in }
H_{0}^{1}(\Omega)^{n}.
\end{equation}
But $|\chi^{\varepsilon'}|\leq 1$, hence there exists
$\chi^{0}\in L^{\infty}(\Omega)$ and
$\{\varepsilon''\}\subset \{\varepsilon'\}$
such that
\begin{equation}\label{e4bis4}
 \chi^{\varepsilon''}\rightharpoonup \chi^{0}\quad
\text{weakly $\star$ in }L^{\infty}(\Omega).
\end{equation}
Passing to the limit (in $D'(\Omega)$) in (\ref{e4bis1}) by using
(\ref{e4bis2}), (\ref{e4bis3}) and (\ref{e4bis4}), we find
$$
\chi^{0}w=\chi^{0}v.
$$
Taking now into account the fact that (in view of Definition
\ref{li1}) $\chi^{0}>0$, we obtain $w=v$. This, together with
(\ref{e4bis3}), implies that the whole sequence
$P_{\varepsilon}({v_{\delta}^{\varepsilon}} _{| _{\Omega
_{\varepsilon}}})$ converge weakly to $v$.
We refer to \cite{don} for the converse implication.
The proof of (2) follows by the  same arguments.
\end{proof}

\section{Preliminary results}

In this paper, $\{A^{\varepsilon}\}$ is a sequence of
fourth-order tensors of  $\mathcal{M}_{e}(\alpha,\beta,\Omega)$ and
$\{T_{\varepsilon}\}$ is a sequence of holes e-admissible in
$\Omega$
 such that
\begin{equation}\label{e5}
A^{\varepsilon}\overset{H^{0}_{e}}{\rightharpoonup}A^{0}.
\end{equation}
Set, for every $\delta>0$,
\begin{equation}\label{e6}A^{\varepsilon}_{\delta}=(\chi_{\varepsilon}+\delta(1-\chi_{\varepsilon}))
A^{\varepsilon}\quad\text{a.e. in }\Omega.
\end{equation}
Since, for fixed $\delta>0$,
$A^{\varepsilon}_{\delta}\in\mathcal{M}_{e}(\min(1,\delta)\alpha,
\max(1,\delta)\beta,\Omega)$,
in view of the compactness properties of $H_{e}$-convergence,
there exists a subsequence $ \{\varepsilon_{m}\}$  of
$\{\varepsilon\}$ and
$A_{\delta}\in\mathcal{M}_{e}(\min(1,\delta)\alpha,\max(1,\delta)\beta,\Omega)$
such that
$A_{\delta}^{\varepsilon_{m}}\overset{H_{e}}{\rightharpoonup}A_{\delta}$
as $\varepsilon_{m}\to0.$ Hence, for every $\delta>0$, the
set
\begin{equation}\label{e7}
W_{\delta}=\{A_{\delta}
;\;\exists\{\varepsilon_{m}\}_{m\in\mathbb{N}}\subset\{\varepsilon\}
\text{ s. t. }
A_{\delta}^{\varepsilon_{m}}\overset{H_{e}}{\rightharpoonup}A_{\delta}\}\\
\end{equation}
is not empty.
Let $\{f^{\varepsilon}\}$ be a sequence in $H^{-1}(\Omega)^{n}$ such
that
\begin{equation}\label{e8}f^{\varepsilon}\to
f\\quad \text{strongly in } H^{-1}(\Omega)^{n}
\end{equation}
and  let $A_{\delta}$ be in $ W_{\delta}$. Let $u^{\varepsilon}_{\delta}$
and $u_{\delta}$ the solutions of
\begin{equation}\label{e9}
\begin{gathered}
-\mathop{\rm div}(A^{\varepsilon}_{\delta}e(u^{\varepsilon}_{\delta})
)=f^{\varepsilon}\quad\text{in } \Omega, \\
u^{\varepsilon}_{\delta}=0\quad\text{on }\partial\Omega
\end{gathered}
\end{equation}
and
\begin{equation}\label{e10}
\begin{gathered}
-\mathop{\rm div}(A_{\delta}e(u_{\delta}) )=f\quad\text{in } \Omega, \\
u_{\delta}=0\quad\text{on }\partial\Omega
\end{gathered}
\end{equation}
respectively. We consider now the following sets:
\begin{equation}\label{e11}
\begin{gathered}
U_{\delta}=\{u_{\delta}: u_{\delta}\text{ is  the  solution of \eqref{e10}
for some }A_{\delta}\in W_{\delta}\},\\
V_{\delta} =\{\text{The set of weak limit
points of } A^{\varepsilon}_{\delta}e(u^{\varepsilon}_{\delta})
\text{ in $L^{2}(\Omega)^{n}$ as $\varepsilon\to 0$}\}.
\end{gathered}
\end{equation}
One has the following result:

\begin{lemma}\label{li7}
One has
$$
V_{\delta}=\{A_{\delta}e(u_{\delta}) : A_{\delta}\in
W_{\delta}\text{ and $u_{\delta}$ is the solution
of \eqref{e10}}\}.
$$
\end{lemma}

\begin{proof}
It is clear that, if $ A_{\delta}\in W_{\delta}$ and $u_{\delta}$ is
the  solution of \eqref{e10}, then  $A_{\delta}e(u_{\delta})$
belongs to  $V_{\delta}$. On the other hand, let $v\in V_{\delta}$.
Then, there exists  a subsequence $\{\varepsilon_{m}\}$ of
$\varepsilon$ such that
\begin{equation}\label{e12}
A^{\varepsilon_{m}}_{\delta}e(u^{\varepsilon_{m}}_{\delta})\rightharpoonup
v\quad\text{weakly in }L^{2}(\Omega)^{n},
\end{equation}
as $\varepsilon_{m}\to0$. But the compactness property of
the $H_{e}$-convergence shows that there exists  a subsequence
$\{\varepsilon'_{m}\}$ of $\{\varepsilon_{m}\}$ and a forth-order
tensor $A_{\delta}$ such that
$$
A_{\delta}^{\varepsilon'_{m}}\overset{H_{e}}{\rightharpoonup}A_{\delta}.
$$
This  implies in particular
$$
A^{\varepsilon'_{m}}_{\delta}e(u^{\varepsilon'_{m}}_{\delta})\rightharpoonup
A_{\delta}e(u_{\delta})\quad\text{weakly in }L^{2}(\Omega)^{n}.
$$
This, together with (\ref{e12}), gives  $v=A_{\delta}e(u_{\delta})$.
\end{proof}

\begin{remark}\label{e12bis} \rm
Let us show that in view of  Theorem \ref{li6bis},
there exists $\{\varepsilon_{m}\}\subset\{\varepsilon\}$
 and for all $\delta>0$  a tensor
$\widehat{A_{\delta}}$  such that

\begin{equation} \label{e12bis0}
A_{\delta}^{\varepsilon_{m}}\overset{H_{e}}{\rightharpoonup}
\widehat{A_{\delta}}.
\end{equation}
Let us also point out that in Theorem \ref{li10}  we will consider a
more general situation, where for every $\delta>0$, there exists
$\{\varepsilon_{\delta}\}$ and a tensor $A_{\delta}$ such that
$A_{\delta}^{\varepsilon_{\delta}}\overset{H_{e}}{\rightharpoonup}A_{\delta}$.
\end{remark}

Let us prove (\ref{e12bis0}). Using the diagonal subsequence
procedure  and the compactness property of the $H_{e}$-convergence,
one extracts a subsequence $\{\varepsilon_{m}\}$ of
$\{\varepsilon\}$ such that, for every
$\delta\in\mathbb{Q}^{\star}_{+}$, one has
\begin{equation}\label{e12bis1}
A^{\varepsilon_{m}}_{\delta} \;H_{e}\text{-convergences to a
 limit }A_{\delta}.
\end{equation}
Since a.e in $\Omega$ one has
\begin{gather*}
|A^{\varepsilon_m}_{\delta_{1}}-A^{\varepsilon_m}_{\delta_{2}}|\leq
\beta|\delta_{1}-\delta_{2}|,\quad \forall
\delta_{1},\;\delta_{2}\in\mathbb{Q}^{\star}_{+},\\
A^{\varepsilon_m}_{\delta_{1}}\in\mathcal{M}_{e}(\min(1,\delta_{1})\alpha,
\max(1,\delta_{1})\beta,\Omega),\\
A^{\varepsilon_m}_{\delta_{2}}\in\mathcal{M}_{e}(\min(1,\delta_{2})\alpha,
\max(1,\delta_{2})\beta,\Omega).
\end{gather*}
Then, from Theorem \ref{li6bis}, it follows
\begin{equation*}
|A_{\delta_{1}}-A_{\delta_{2}}|\leq
\frac{\beta^{2}}{\alpha}\;
\sqrt{\frac{\max(1,\delta_{1})\max(1,\delta_{2})}{\min(1,\delta_{1})
\min(1,\delta_{2})}}\;|\delta_{1}-\delta_{2}|.
\end{equation*}
This implies that the mapping
$\delta\in\mathbb{Q}^{\star}_{+}\mapsto
A_{\delta}\in\mathbb{L}^{\infty}(\Omega)$ is uniformly continuous.
Hence, it can be extended to a mapping (still denoted by
$\delta\mapsto A_{\delta}$) defined and uniformly continuous on
all $\mathbb{R}^{\star}_{+}$ (since $\mathbb{Q}^{\star}_{+}$ is
dense in $\mathbb{R}^{\star}_{+}$).

Let now $\delta$ be a strictly positive real and $\{\delta_{s}\}$ be
a sequence of $\mathbb{Q}^{\star}_{+}$ which converges to $\delta$
as $s\to\infty$. Then, there exists a sub-subsequence
$\{\varepsilon'_{m}\}$ of $\{\varepsilon_{m}\}$  such that
\begin{equation}\label{e12bis2}
A^{\varepsilon'_{m}}_{\delta}\;H_{e}\text{-converges to some }A.
\end{equation}
In view of  Theorem \ref{li6bis} this give, together with
(\ref{e12bis1}) and the fact that
$|A^{\varepsilon'_{m}}_{\delta}-A^{\varepsilon'_{m}}_{\delta_{s}}|\leq
\beta|\delta-\delta_{s}|$, the following inequality:
\[
|A-A_{\delta_{s}}|\leq
\frac{\beta^{2}}{\alpha}\;\sqrt{\frac{\max(1,\delta)\max(1,\delta_{s})}
{\min(1,\delta)\min(1,\delta_{s})}}\;|\delta-\delta_{s}|\quad
\text{a.e. in } \Omega.
\]
Using the continuity of the mapping $\delta\mapsto A_{\delta}$ on
$\mathbb{R}^{\star}_{+}$ and passing to the limit in this inequality
as $s\to\infty$, one finds
\begin{equation*}
A=A_{\delta},\quad\text{a.e. in }\Omega.
\end{equation*}
The uniqueness  of the limit implies then that the whole subsequence
$A^{\varepsilon_{m}}_{\delta}$ $H_{e}$-converges to $A_{\delta}$,
for every $\delta>0$.

The following results state some a priori estimates that we will
need in the following:

\begin{proposition}\label{li8}
Let $u^{\varepsilon}$ and $u^{\varepsilon}_{\delta}$ the solutions
of \eqref{e2} and \eqref{e9} respectively. Then, there exists
$c>0$ independent of $\varepsilon$ and $\delta$ such that
\begin{equation}\label{e13}
\begin{gathered}
\|P^{\varepsilon}({u^{\varepsilon}_{\delta}}_{|_{\Omega_{\varepsilon}}})
-P^{\varepsilon}u^{\varepsilon}
\|_{H^{1}_{0}(\Omega)^{n}} \leq
c(\delta^{1/2}+|\langle
f_{\varepsilon},P^{\varepsilon}({u^{\varepsilon}_{\delta}}_
{|_{\Omega_{\varepsilon}}})-u_{\delta}^{\varepsilon}\rangle|^{1/2}
),\\
\|e(u^{\varepsilon}_{\delta})\|_{L^{2}(T_{\varepsilon})^{n\times
n}} \leq c(1+\delta^{^{-\frac{1}{2}}}|\langle
f_{\varepsilon},P^{\varepsilon}({u^{\varepsilon}_{\delta}}_{|_{\Omega_
{\varepsilon}}})-u_{\delta}^{\varepsilon}\rangle|^{1/2}
),\\
\|A^{\varepsilon}_{\delta}e(u^{\varepsilon}_{\delta})-A^{{\varepsilon}
}\widetilde{e( u^{{\varepsilon} }) }\ \|_{L^{2}(\Omega)^{n\times n}}
\leq c(\delta^{1/2}+|\langle
f_{\varepsilon},P^{\varepsilon}({u^{\varepsilon}_{\delta}}_
{|_{\Omega_{\varepsilon}}})-u_{\delta}^{\varepsilon}\rangle|^{1/2}).
\end{gathered}
\end{equation}
\end{proposition}

\begin{proof}
Observe that the variational formulations of problems \eqref{e2}
and \eqref{e9} are
\begin{equation}\label{e13bis}
\forall w\in H^{1}_{0}(\Omega)^{n},\;\;\;\int_{\Omega_{\varepsilon}}
A^{\varepsilon}e(u^{\varepsilon}).e(w)dx=\langle
f_{\varepsilon},P^{\varepsilon}(w_{|_{\Omega_{\varepsilon}}})\rangle
\end{equation}
and
\begin{equation*}
\forall w\in H^{1}_{0}(\Omega)^{n},\;\int_{\Omega_{\varepsilon}}
A^{\varepsilon}e(u^{\varepsilon}_{\delta}).e(w)dx+
\delta\int_{T_{\varepsilon}}A^{\varepsilon}e(u^{\varepsilon}_{\delta}).e(w)dx=\langle
f_{\varepsilon},w\rangle
\end{equation*}
respectively. Then, for
every $w\in H^{1}_{0}(\Omega)^{n}$, one has
\begin{equation*}
\int_{\Omega_{\varepsilon}}
A^{\varepsilon}(e(u^{\varepsilon}_{\delta})-e(u^{\varepsilon})).e(w)dx+
\delta\int_{T_{\varepsilon}}A^{\varepsilon}e(u^{\varepsilon}_{\delta}).e(w)dx=-\langle
f_{\varepsilon},P^{\varepsilon}(w_{|_{\Omega_{\varepsilon}}})-w\rangle
.\end{equation*}
In particular, for
$w=u^{\varepsilon}_{\delta}-P^{\varepsilon}u^{\varepsilon}$, this
gives
\begin{align*}
&\int_{\Omega_{\varepsilon}}
A^{\varepsilon}(e(u^{\varepsilon}_{\delta})-e(u^{\varepsilon})).
(e(u^{\varepsilon}_{\delta})-e(P^{\varepsilon}u^{\varepsilon}))dx+
\delta\int_{T_{\varepsilon}}A^{\varepsilon}e(u^{\varepsilon}_{\delta}).
(e(u^{\varepsilon}_{\delta})-e(P^{\varepsilon}u^{\varepsilon}))dx\\
&=-\langle f_{\varepsilon},P^{\varepsilon}((u^{\varepsilon}_{\delta}
-P^{\varepsilon}u^{\varepsilon})_{|_{\Omega_{\varepsilon}}})
-u^{\varepsilon}_{\delta}-P^{\varepsilon}u^{\varepsilon}\rangle.
\end{align*}
Using  that ${P^{\varepsilon}u^{\varepsilon}}
_{|_{\Omega_{\varepsilon}}}=u^{\varepsilon}$, one deduces
\begin{align*}
&\int_{\Omega_{\varepsilon}}
A^{\varepsilon}(e(u^{\varepsilon}_{\delta})-e(u^{\varepsilon})).
(e(u^{\varepsilon}_{\delta})-e(u^{\varepsilon}))dx+
\delta\int_{T_{\varepsilon}}A^{\varepsilon}e(u^{\varepsilon}_{\delta}).
e(u^{\varepsilon}_{\delta})dx\\
&=\delta\int_{T_{\varepsilon}}A^{\varepsilon}e(u^{\varepsilon}_{\delta}).
e(P^{\varepsilon}u^{\varepsilon})dx-\langle
f_{\varepsilon},P^{\varepsilon}(u^{\varepsilon}_{\delta})
_{|_{\Omega_{\varepsilon}}}-u^{\varepsilon}_{\delta}\rangle.
\end{align*}
In view of the fact that $A^{\varepsilon}\in
M_{e}(\alpha, \beta, {\Omega})$, this gives
\begin{equation}\label{e13bis1}
\begin{aligned}
&\alpha\int_{\Omega_{\varepsilon}}
|e(u^{\varepsilon}_{\delta})-e(u^{\varepsilon}))|^{2}dx
+\alpha\delta\int_{T_{\varepsilon}}
|e(u^{\varepsilon}_{\delta})|^{2}dx\\
&\leq \delta|\int_{T_{\varepsilon}}A^{\varepsilon}e(u^{\varepsilon}_{\delta}).
e(P^{\varepsilon}u^{\varepsilon})dx|+|\langle
f_{\varepsilon},P^{\varepsilon}({u^{\varepsilon}_{\delta}}_{|_{\Omega_{\varepsilon}}})-u^{\varepsilon}_{\delta}\rangle|.
\end{aligned}
\end{equation}
 Using the Young's inequality, one obtains
\begin{align*}
\big|\int_{T_{\varepsilon}}A^{\varepsilon}e(u^{\varepsilon}_{\delta}).e(P^{\varepsilon}
u^{\varepsilon})dx\big|
&\leq \beta\int_{T_{\varepsilon}}|e(u^{\varepsilon}_{\delta})|
|e(P^{\varepsilon} u^{\varepsilon})|dx\\
&\leq \frac{\alpha}{2}\int_{T_{\varepsilon}}
|e(u^{\varepsilon}_{\delta})|^{2}dx+
\frac{\beta^{2}}{2\alpha}\int_{T_{\varepsilon}}
|e(P^{\varepsilon}u^{\varepsilon})|^{2}dx\\
&\leq \frac{\alpha}{2}\int_{T_{\varepsilon}}|e(u^{\varepsilon}_{\delta})|^{2}dx
+ \frac{\beta^{2}}{2\alpha}\int_{\Omega}|e(P^{\varepsilon}u^{\varepsilon})|^{2}
dx.
\end{align*}
But, taking  $w=P^{\varepsilon}u^{\varepsilon}$ in
(\ref{e13bis}), one finds
$$
\int_{\Omega}|e(P^{\varepsilon}u^{\varepsilon})|dx\leq c_{1},
$$
with $c_{1}>0$ independent of $\varepsilon$ and $\delta$. Then
\begin{equation*}
\big|\int_{T_{\varepsilon}}A^{\varepsilon}e(u^{\varepsilon}_{\delta})
.e(P^{\varepsilon} u^{\varepsilon})dx\big| \leq
\frac{\alpha}{2}\int_{T_{\varepsilon}}|e(u^{\varepsilon}_{\delta})|^{2}dx+
c_{2},
\end{equation*}
where $c_{2}>0$ independent of $\varepsilon$ and $\delta$. This,
together with (\ref{e13bis1}), gives
\begin{equation}\label{e14}
\alpha\int_{\Omega_{\varepsilon}}
|e(u^{\varepsilon}_{\delta})-e(u^{\varepsilon}))|^{2}dx+\frac{\alpha\delta}{2}\int_{T_{\varepsilon}}
|e(u^{\varepsilon}_{\delta})|^{2}dx\leq c_{3}(\delta+|\langle
f_{\varepsilon},P^{\varepsilon}({u^{\varepsilon}_{\delta}}_{|_{\Omega_{\varepsilon}}})-u^{\varepsilon}_{\delta}\rangle|).
\end{equation}
with $c_{3}>0$ independent of $\varepsilon$ and $\delta$. From this,
\eqref{e13}ii) follows  immediately.
 Moreover, since
${u^{\varepsilon}_{\delta}}_{|_{\Omega_{\varepsilon}}}-u^{\varepsilon}\in
H^{1}_{0}(\Omega_{\varepsilon})^{n}$, Definition \ref{li1} shows
that
$$
\|e(P^{\varepsilon}({u^{\varepsilon}_{\delta}}{|_{\Omega_{\varepsilon}}}
-u^{\varepsilon}))\|_{0,\Omega}\leq
C \|e(u^{\varepsilon}_{\delta}-u^{\varepsilon})\|_{0,\Omega_{\varepsilon}}.
$$
Hence, by virtue of the Korn inequality, (\ref{e14}) gives also \eqref{e13}i).\medskip\\
On the other hand, since  $A^{\varepsilon}_{\delta}=A^{\varepsilon}$
a.e. in $\Omega_{\varepsilon}$ and $A^{\varepsilon}_{\delta}=\delta
A^{\varepsilon}$ a.e. in $T_{\varepsilon}$, one has
\begin{align*}
\int_{\Omega} |A_{\delta}^{\varepsilon}e(u^{\varepsilon}_{\delta})
-A^{\varepsilon}\widetilde{e(u^{{\varepsilon} }) }|^{2} dx
&\leq \int_{\Omega_{\varepsilon}}
|A^{\varepsilon}e(u^{\varepsilon}_{\delta})-A^{\varepsilon}e(
u^{{\varepsilon} })|^{2}
dx+\delta^{2}\int_{T_{\varepsilon}}|A^{\varepsilon}e(u^{\varepsilon}_{\delta})|^{2}
dx\\
&\leq \beta^{2}\int_{\Omega_{\varepsilon}}
|e(u^{\varepsilon}_{\delta})-e( u^{{\varepsilon} })|^{2}
dx+\beta^{2}\delta^{2}\int_{T_{\varepsilon}}|e(u^{\varepsilon}_{\delta})|^{2}dx,
\end{align*}
which, together with \eqref{e13}(i) and \eqref{e13}(ii), gives
\eqref{e13}(iii).
\end{proof}

\begin{proposition}\label{li9}
Let $u^{0}$ the solution  of \eqref{e4}. Then
\begin{equation}\label{e15}
\begin{gathered}
\sup_{u\in U_{\delta}} \|u-u^{0} \|_{H^{1}_{0}(\Omega)^{n}} \leq
c\;\delta^{1/2},\\
\sup_{v\in V_{\delta}}\|v-A^{0}e(u^{0})\|_{L^{2}(\Omega)^{n\times n}} \leq
c\;\delta^{1/2} .
\end{gathered}
\end{equation}
\end{proposition}

\begin{proof}
Let be  $u_{\delta}$ in $ U_{\delta}$. This means that there exists
$A_{\delta}\in W_{\delta}$ such that  $u_{\delta}$ is the solution
of \eqref{e10}. But the fact that $A_{\delta}$ is in $
W_{\delta}$ implies that there exists a subsequence
$\{\varepsilon_{m}\}$ of $\varepsilon$ such that
$A_{\delta}^{\varepsilon_{m}}$ $H_{e}$-converges to $A_{\delta}$.
Hence, the solution $u^{\varepsilon}_{\delta}$ of
\begin{equation}\label{e16}
\begin{gathered}
-\mathop{\rm div}(A^{\varepsilon_{m}}_{\delta}e(u^{\varepsilon_{m}}_{\delta})
)=f^{\varepsilon_{m}}\quad\text{in } \Omega, \\
u^{\varepsilon_{m}}_{\delta}=0\quad\text{on }\partial\Omega
\end{gathered}
\end{equation}
satisfies as $\varepsilon_{m}\to0$
\begin{equation}\label{e17}
\begin{gathered}
u^{\varepsilon_{m}}_{\delta}\rightharpoonup
u_{\delta}\quad\text{weakly in }H^{1}_{0}(\Omega)^{n},\\
A^{\varepsilon_{m}}_{\delta}e(u^{\varepsilon_{m}}_{\delta})\rightharpoonup
A_{\delta} e(u_{\delta})\quad\text{weakly  in }L^{2}(\Omega)^{n\times n}.
\end{gathered}
\end{equation}

\noindent\textbf{Estimate \eqref{e15}(i):}
By Lemma \ref{li6}, (\ref{e17})i) implies that, for every fixed
$\delta>0$,
\begin{equation}\label{e18}
P_{\varepsilon_{m}} ({u^{\varepsilon_{m}}_{\delta}}{|_{\Omega_{\varepsilon_{m}}}})
\rightharpoonup u_{\delta}\quad\text{weakly in }H^{1}_{0}(\Omega)^{n}.
\end{equation}
Hence, by  \eqref{e8}, one has
\begin{equation}\label{e19}
\lim_{\varepsilon_{m}\to0} \langle
f_{\varepsilon_{m}},P^{\varepsilon_{m}}({u^{\varepsilon_{m}}_{\delta}}_
{|_{\Omega_{\varepsilon_{m}}}})-u_{\delta}\rangle=0.
\end{equation}
 From this and \eqref{e13}(i), it comes
\begin{equation}\label{e19bis}
\lim_{\varepsilon_m\to0} \|P^{\varepsilon_{m}}
({u^{\varepsilon_{m}}_{\delta}}_{|_{\Omega_{\varepsilon_{m}}}})
-P^{\varepsilon_{m}} u^{\varepsilon_{m}} \|_{H^{1}_{0}(\Omega)^{n}} \leq
c\delta^{1/2}.
\end{equation}
But, (\ref{e18}) and (\ref{e3})i) imply
\begin{equation*}
P_{\varepsilon_{m}}({u^{\varepsilon_{m}}_{\delta}}_{|_{\Omega_{\varepsilon}}})
-P_{\varepsilon_{m}}(u^{\varepsilon_{m}})\rightharpoonup
u_{\delta}-u^{0}\quad\text{weakly in }H^{1}_{0}(\Omega)^{n}.
\end{equation*}
This gives, by using the weak lower semi-continuity of the
$H^{1}_{0}$-norm,
\begin{equation*}
\|u_{\delta}-u^{0} \|_{H^{1}_{0}(\Omega)^{n}} \leq
\lim_{\varepsilon_m\to0} \|P^{\varepsilon_{m}}
({u^{\varepsilon_{m}}_{\delta}}_{|_{\Omega_{\varepsilon_{m}}}})
-P^{\varepsilon_{m}}u^{\varepsilon_{m}}
\|_{H^{1}_{0}(\Omega)^{n}},
\end{equation*}
where $u^{0}$ is the solution of \eqref{e4}. Hence,
(\ref{e19bis}) gives
\begin{equation*}
\|u_{\delta}-u^{0} \|_{H^{1}_{0}(\Omega)^{n}}\leq
c\delta^{1/2}.
\end{equation*}
This is still valid for every $u_{\delta}\in U_{\delta}$, which
implies \eqref{e15}(i).

\noindent\textbf{Estimate \ref{e15}(ii):}
  From \eqref{e13}(iii) and (\ref{e19}), it comes
$$
\lim_{\varepsilon_m\to0}
\|A^{\varepsilon_{m}}_{\delta}e(u^{\varepsilon_{m}}_{\delta})
-A^{{\varepsilon_{m}}
}\widetilde{e( u^{{\varepsilon_{m}} }) }\
\|_{H^{1}_{0}(\Omega)^{n\times n}} \leq c\delta^{1/2}.
$$
But  \eqref{e3}(ii) and \eqref{e17}(ii) imply, as $\varepsilon_{m}\to0$, that
$$
(A^{\varepsilon_{m}}_{\delta}e(u^{\varepsilon_{m}}_{\delta})-
A^{\varepsilon_{m}}\widetilde{e(u^{\varepsilon_{m}}}))\rightharpoonup
(A_{\delta} e(u_{\delta})-A^{0} e(u^{0}))\quad
\text{weakly in } L^{2}(\Omega)^{n\times n}.
$$
In view of the weak lower
semi-continuity of the $L^{2}$-norm, these two last relations give
$$
\|(A_{\delta} e(u_{\delta})-A^{0} e(u^{0}))\|_{L^{2}(\Omega)^{n\times n}} \leq
c\delta^{1/2},
$$
for every $u_{\delta}\in U_{\delta}$.
Hence,
$$
\sup_{u_{\delta}\in U_{\delta}} \|(A_{\delta} e(u_{\delta})-A^{0}
e(u^{0}))\|_{L^{2}(\Omega)^{n\times n}} \leq
c\delta^{1/2},
$$
which, together with Lemma \ref{li7}, gives the claimed result.
\end{proof}


\section{Main results}

\begin{theorem}\label{li10}
Let $A_{\delta}\in W_{\delta}$. Then, the solution $u_{\delta}$ of
\eqref{e10} satisfies  as $\delta\to 0$
\begin{equation}\label{e20}
\begin{gathered}
u_{\delta}\to u^{0}\quad\text{strongly in } H^{1}_{0}(\Omega)^{n},\\
A_{\delta}e(u_{\delta})\to A^{0} e(u^{0})\quad\text{strongly in }
L^{2}(\Omega)^{n\times n},
\end{gathered}
\end{equation}
where  $u^{0}$ is the solution of \eqref{e4}. Moreover,   one has
the convergence:
\begin{equation}\label{e21}
 \forall p\in[1,\infty[,\quad A_{\delta}\to A^{0},
\end{equation}
strongly in $L^{p}(\Omega)$ and weakly $\star$ in
$L^{\infty}(\Omega)$.
\end{theorem}

\begin{theorem} \label{li11}
Let $f^{\varepsilon},\;f$ be in $H^{-1}(\Omega)^{n}$  satisfying
\eqref{e8}. Suppose that
\begin{equation}\label{e22}
\forall\varepsilon>0,\;\;\langle f_{\varepsilon},v\rangle=0,\quad
\forall v\in H_{0}^{1}(\Omega)^{n},\;v=0\text{ on }\Omega_{\varepsilon}.
\end{equation}
Then, as $\delta\to0$,
\begin{equation}\label{e23}
\begin{gathered}
u^{\varepsilon}_{\delta}\to u^{\varepsilon}\quad
\text{strongly in }H^{1}_{0}(\Omega_{\varepsilon})^{n},\\
 A^{\varepsilon}_{\delta}e(u^{\varepsilon}_{\delta})\to
A^{\varepsilon} \widetilde{e(u^{\varepsilon})}\quad\text{strongly in }
L^{2}(\Omega)^{n\times n},
\end{gathered}
\end{equation}
where  $u^{\varepsilon}$ and $u^{\varepsilon}_{\delta}$ are the
solutions of \eqref{e2} and \eqref{e9} respectively.
\end{theorem}

\begin{theorem}\label{li12}
Let $f^{\varepsilon},\;f\in H^{-1}(\Omega)^{n}$  satisfying
\eqref{e8} and  \eqref{e22}. Then, as $(\varepsilon,\delta)\to(0,0)$
\begin{equation}\label{e24}
\begin{gathered}
u^{\varepsilon}_{\delta}\to u^{0}\quad\text{weakly in }
H^{1}_{0}(\Omega)^{n},\\
A^{\varepsilon}_{\delta}e(u^{\varepsilon}_{\delta})\to
A^{0}e(u^{0})\quad\text{weakly in }L^{2}(\Omega)^{n\times n},
\end{gathered}
\end{equation}
where  $u^{0}$ and $u^{\varepsilon}_{\delta}$ are the solutions of
\eqref{e4} and \eqref{e9} respectively.
\end{theorem}

To prove these results we  use  similar arguments as those used
in \cite{ciodam}. Before giving these proofs, we recall the
following lemma:

\begin{lemma}[\cite{ciodam}] \label{li3'}
 Let $\{\psi_{m}\}$ be a sequence of $L^{2}(\Omega)$.
Suppose that there exists $\psi,\;\phi\in L^{2}(\Omega)$ such that
\begin{gather*}
\psi_{m}\to \psi\quad\text{strongly  in }L^{2}_{loc}(\Omega),\\
\forall m\in\mathbb{N},\;|\psi_{m}|\leq\phi\quad\text{a.e. in }\Omega.
\end{gather*}
Then, $\psi_{m}\to \psi$ strongly  in $L^{2}(\Omega)$.
\end{lemma}

\begin{proof}[Proof of Theorem \ref{li10}]
Observe first that (\ref{e20}) follows immediately from
Proposition \ref{li9}.
On the other hand, taking in \eqref{e2} and \eqref{e9},
$f^{\varepsilon}=f\doteq-\mathop{\rm div}(A^{0}e(\varphi\Lambda x))$ with
$\varphi\in\mathcal{D}(\Omega)$ and
$\Lambda\in\mathbb{R}_{S}^{n\times n}$, then \eqref{e4} reads
\begin{gather*}
-\mathop{\rm div}\;(A^{0}e( u^{0}) )
=-\mathop{\rm div}(A^{0}e(\varphi\Lambda x))\quad\text{in }\Omega ,\\
u^{0}=0\quad\text{on }\partial \Omega.
\end{gather*}
This implies, in view of the fact that
$A^{0}\in\mathcal{M}_{e}(\frac{\alpha}{C^{2}},\beta,\Omega)$, that
$u^{0}=\varphi\Lambda x$.
This, together with (\ref{e20}), gives
\begin{gather*}
u_{\delta}\to \varphi\Lambda x\quad\text{strongly in }
H^{1}_{0}(\Omega)^{n},\\
A_{\delta}e(u_{\delta})\to A^{0} e(\varphi\Lambda x)\quad
\text{strongly in }L^{2}(\Omega)^{n\times n}.
\end{gather*}
Taking now $\varphi\in\mathcal{D}(\Omega)$ such that $\varphi=1$ on
$\omega$ and where $\omega\subset\subset\Omega$, one obtains
\begin{equation}\label{e24bis}
\begin{gathered}
u_{\delta}\to \Lambda \quad\text{strongly in }H^{1}(\omega)^{n},\\
A_{\delta}e(u_{\delta})\to A^{0} \Lambda\quad\text{strongly in }
L^{2}(\omega)^{n\times n}.
\end{gathered}
\end{equation}
On the other hand,  one has almost everywhere in $\omega$,
\begin{align*}
|A_{\delta}\Lambda-A^{0}\Lambda|
&\leq |A_{\delta}\Lambda-A_{\delta}e(u_{\delta})|+
|A_{\delta}e(u_{\delta})-A^{0}\Lambda|\\
&\leq \beta|\Lambda-e(u_{\delta})|+
|A_{\delta}e(u_{\delta})-A^{0}\Lambda|.
\end{align*}
Then, using (\ref{e24bis}), one gets
\begin{equation*}
A_{\delta}\Lambda\to A^{0} \Lambda \quad\text{strongly in }
L^{2}(\omega)^{n\times n},
\end{equation*}
for every $\omega\subset\subset\Omega$.
Since $|A_{\delta_{m}}\Lambda|\leq\beta|\Lambda|$, this gives by
Lemma \ref{li3'} written for $\psi_{m}=A_{\delta_{m}}\Lambda$ (with
$\delta_{m}\to0$),
\begin{equation*}
A_{\delta}\Lambda\to A^{0} \Lambda \quad\text{strongly in }
L^{2}(\Omega)^{n\times n}.
\end{equation*}
By  the symmetric properties of $A_{\delta}$ and $A^{0}$,
this convergence is still valid for every matrix
$\Lambda\in\mathbb{R}^{n\times n}$. Thus
\begin{equation*}
A_{\delta}\to A^{0} \quad\text{strongly in }L^{2}(\Omega)^{n\times n}.
\end{equation*}
 From this convergence and  the fact that
$\|A_{\delta}\|_{L^{\infty}(\Omega)}\leq \beta$, one obtains
convergence (\ref{e21}).
\end{proof}

\begin{proof}[Proof of Theorem \ref{li11}]
 From  hypothesis \eqref{e22}, Proposition \ref{li8} and the fact
that $P^{\varepsilon}({u^{\varepsilon}_{\delta}}_
{|_{\Omega_{\varepsilon}}})-u^{\varepsilon}_{\delta}=0$
in $\Omega_{\varepsilon}$, it follows that
\begin{gather*}
\|{u^{\varepsilon}_{\delta}}_{|_{\Omega_{\varepsilon}}}-u^{\varepsilon}
\|_{H^{1}_{0}(\Omega_{\varepsilon})^{n}}\leq
\|P^{\varepsilon}({u^{\varepsilon}_{\delta}}_{|_{\Omega_{\varepsilon}}})-P^{\varepsilon}u^{\varepsilon}
\|_{H^{1}_{0}(\Omega)^{n}} \leq
c\delta^{1/2},\\
\|A^{\varepsilon}_{\delta}e(u^{\varepsilon}_{\delta})-A^{{\varepsilon}
}\widetilde{e( u^{{\varepsilon} }) }\ \|_{L^{2}(\Omega)^{n\times n}}
\leq c\delta^{1/2}.
\end{gather*}
Passing to the limit as $\delta\to0$, one obtains
(\ref{e23}).
\end{proof}

\begin{proof}[Proof of Theorem \ref{li12}]
\textbf{(i)} Under hypothesis \eqref{e22}, Proposition \ref{li8}
gives
\begin{equation*}
\lim_{(\varepsilon,\delta)\to(0,0)} \|P^{\varepsilon}
({u^{\varepsilon}_{\delta}}_{|_{\Omega_{\varepsilon}}})
-P^{\varepsilon}u^{\varepsilon} \|_{H^{1}_{0}(\Omega)^{n}}=0
\end{equation*}
and the fact that
$A^{\varepsilon}\overset{H_{e}}{\rightharpoonup}A^{0}$ implies
$$
P_{\varepsilon}u^{\varepsilon}-u^{0}\rightharpoonup0
\quad\text{weakly in } H^{1}_{0}(\Omega).
$$
Hence, by passing to the weak limit in $H^{1}_{0}(\Omega)$ as
$(\varepsilon,\delta)\to(0,0)$ in the following equality:
\begin{equation*}
P_{\varepsilon}({u^{\varepsilon}_{\delta}}_{|_{\Omega_{\varepsilon}}})-u^{0}=
(P_{\varepsilon}({u^{\varepsilon}_{\delta}}_{|_{\Omega_{\varepsilon}}})-
P_{\varepsilon}u^{\varepsilon})+(P_{\varepsilon}u^{\varepsilon}-u^{0}),
\end{equation*}
one deduces
\begin{equation}\label{e25}
P_{\varepsilon}({u^{\varepsilon}_{\delta}}_{|_{\Omega_{\varepsilon}}})
\rightharpoonup u^{0} \quad\text{weakly in }
H^{1}_{0}(\Omega)^{n}.
\end{equation}
On the other hand, using
 \eqref{e22} and Proposition \ref{li8}, one gets
\begin{gather*}
\|P^{\varepsilon}({u^{\varepsilon}_{\delta}}_{|_{\Omega_{\varepsilon}}})-
P^{\varepsilon}u^{\varepsilon} \|_{H^{1}_{0}(\Omega)^{n}} \leq
c\delta^{1/2},\\
\|e(u^{\varepsilon}_{\delta})\|_{L^{2}(T_{\varepsilon})^{n\times n}}
\leq c.
\end{gather*}
This implies
\begin{gather*}
\|u^{\varepsilon}_{\delta}\|_{H^{1}_{0}(\Omega_{\varepsilon})^{n}}\leq
\|P^{\varepsilon}({u^{\varepsilon}_{\delta}}_{|_{\Omega_{\varepsilon}}})
\|_{H^{1}_{0}(\Omega)^{n}}\leq c\delta^{1/2}+
\|P^{\varepsilon}u^{\varepsilon}
\|_{H^{1}_{0}(\Omega)^{n}},\\
\|e(u^{\varepsilon}_{\delta})\|_{L^{2}(T_{\varepsilon})^{n\times n}}
\leq c.
\end{gather*}
Since $P^{\varepsilon}u^{\varepsilon}$ is bounded independently of
$\varepsilon$ in $H^{1}_{0}(\Omega)^{n}$, one deduces that
$u^{\varepsilon}_{\delta}$ is bounded independently of
$\varepsilon$ and $\delta$ in $H^{1}_{0}(\Omega)^{n}$.
This, together with (\ref{e25}) and Proposition \ref{li6},
 gives (\ref{e24})i).

\noindent\textbf{(ii)} Using Proposition \ref{li8}, the fact that
$A^{\varepsilon}\overset{H_{e}}{\rightharpoonup}A^{0}$ and
\begin{equation*}
A^{\varepsilon}_{\delta}e(u^{\varepsilon}_{\delta})-A^{0}e(u^{0})=
(A^{\varepsilon}_{\delta}e(u^{\varepsilon}_{\delta})-A^{\varepsilon}
\widetilde{e(u^{\varepsilon})})+
(A^{\varepsilon}\widetilde{e(u^{\varepsilon})}-A^{0}e(u^{0})),
\end{equation*}
one obtains the convergence \ref{e24}(ii).
\end{proof}

\subsection*{Acknowledgments}
This work  was  finalized when the author  was visiting the
Laboratoire Jacques-Louis Lions at the Paris University Pierre
et Marie Curie, supported by a grant of the Algerian Ministry
for the higher education and scientific research. The author  is
very thankful for the hospitality during this visit. The author
is also  grateful to  P. Donato  for the useful discussions and
advice.

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