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\markboth{\hfil Diffusion equation for composite materials \hfil EJDE--2000/15}
{EJDE--2000/15\hfil Mohamed El Hajji \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.~{\bf 2000}(2000), No.~15, pp.~1--11. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
 A diffusion equation for composite materials 
\thanks{ {\em 1991 Mathematics Subject Classifications:} 31C40, 31C45, 
60J50, 31C35, 31B35.
\hfil\break\indent
{\em Key words and phrases:} Diffusion equation, composite material, 
\hfil\break\indent asymptotic behavior, $H^0$-convergence.
\hfil\break\indent
\copyright 2000 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Submitted October 14, 1999. Published February 22, 2000.} }
\date{}

%
\author{Mohamed El Hajji}
\maketitle

\begin{abstract} 
 In this article, we study the asymptotic behavior of solutions to 
 the diffusion equation with non-homogeneous Neumann boundary conditions.
 This equation models a composite material that occupies a perforated 
 domain, in ${\mathbb R}^N$, with small holes  whose sizes are measured
 by a number $r_\varepsilon$.  
 We examine the case when $r_\varepsilon < \varepsilon^{N/(N-2)}$ with
 zero-average data around the holes, and the case when
 $\lim_{\varepsilon\to 0}{r_\varepsilon/\varepsilon}=0$ with
 nonzero-average data.
\end{abstract}

\newtheorem{theorem}{Theorem}


\section{Introduction}

As a model for a composite material occupying a perforated domain 
in ${\mathbb R}^N$, the diffusion equation with non-homogeneous 
boundary conditions has been the object of many studies. In particular,
we are interested in the properties of the solution to the equation
\begin{eqnarray}
 &- \mathop{\rm div} \left(A^{\varepsilon} \nabla u_{\varepsilon}
\right) = f\quad \hbox {in } \Omega_{\varepsilon},& \nonumber \\
&\left(A^{\varepsilon}(x) \nabla u_{\varepsilon} \right)\cdot {n} 
= h_\varepsilon\; \hbox { on }
\partial S_{\varepsilon},& \label{I} \\
 &u_{\varepsilon} = 0\;\hbox { on } \partial \Omega ,& \nonumber
\end{eqnarray}
where $\Omega_\varepsilon$ is the perforated domain 
obtained by extracting a  set of holes $S_\varepsilon$  from $\Omega$,
 $f$ and $h_\varepsilon$ are given functions, and $A^\varepsilon$ is an 
operator in the space 
\begin{eqnarray*}
M(\alpha, \beta ; \Omega)&=&\big\{
A\in [L^\infty ({\mathbb  R}^N)]^{N^2} :
 (A(x)\lambda, \lambda)\geq \alpha |\lambda|^2,\\
&&\quad|A(x)\lambda|\leq \beta |\lambda|\, 
\forall \lambda\in {\mathbb  R}^N, p.p\cdot x\in \Omega\big\}
\end{eqnarray*}
which is defined for all real numbers $\alpha$ and $\beta$.  

When $h_\varepsilon\in L^2(\partial S_\varepsilon)$ and 
the domain has  holes of size $r_\varepsilon$, solutions to (\ref{I}) have been 
studied  by D. Cioranescu and P. Donato \cite{Cio-Do} for 
$r_\varepsilon=\varepsilon$ 
and $A^\varepsilon(x)=A({x\over\varepsilon})$ with 
$A\in M(\alpha,\beta;\Omega)$, and by
C. Conca and P. Donato \cite{Conca-Do} for $A=I$ and 
$r_\varepsilon\ll \varepsilon$.
Using the concept of $H^0$-convergence introduced by 
M. Briane et al. \cite{H-conv},  
P. Donato and M. El Hajji \cite{Hajji1} showed convergence of 
solutions in not-necessarily periodic domains. The $H^0$-convergence is proven
by showing strong convergence in $H^{-1}(\Omega)$ of the distribution, 
concentrated on the boundary of $S_\varepsilon$, given by   
\begin{equation}\label{II}
\langle\nu^\varepsilon_h,\varphi\rangle _{H^{-{1}}(\Omega),\;H^1_0(\Omega)}
\  =\  \langle h_\varepsilon, \varphi\rangle_{H^{-{1/ 2}} 
(\partial S_\varepsilon),\; H^{1/ 2}(\partial
S_\varepsilon)}, \quad \forall\varphi\in H^1_0(\Omega).
\end{equation}
This method allows the study the asymptotic behavior of solutions
to (\ref{I}) for 
$h_\varepsilon\in L^2(\partial S_\varepsilon)$ with
$r_\varepsilon >\varepsilon^{N/{N-2}}$ (see \cite{Hajji1}),
and for perforated domains with double periodicity with 
$h_\varepsilon\in H^{-{1/2}}(\partial S_\varepsilon)$ and 
${r_\varepsilon /\varepsilon}\to 0$ as $\varepsilon\to 0$
(see T. Levy \cite{Levy}). 

In this article, we study the perforated domains with 
$r_\varepsilon < \varepsilon^{N/{N-2}}$, and  perforated domains such 
that $r_\varepsilon/ \varepsilon \to 0$ as $\varepsilon\to 0$.    
In these situation the distribution given by (\ref{II}) does not converge 
strongly in $H^{-1}(\Omega)$, and so the method described above can not 
be applied. In spite of this, we describe the asymptotic behavior of 
solutions to (\ref{I}) using oscillating test functions. 
This method was introduced by L. Tartar \cite{Tartar} and has been used 
by many authors. 

\section{Statement of the main result}

Let $\Omega$ be a bounded open set of ${\mathbb  R}^N$, 
$Y=[0,l_1[\times..\times[0,l_N[$ be a representative cell, and $S$ be an 
open set of $Y$ with smooth boundary $\partial S$ such that 
$\overline S\subset Y$. Let $\varepsilon$ and $r_\varepsilon$ be 
terms of positive sequences such that $r_\varepsilon\leq\varepsilon$. 
Let $c$ denote positive constants independent of $\varepsilon$. 
We denote by $\tau(r_\varepsilon\overline S)$ the set of 
translations of $r_\varepsilon\overline S$ of the form 
$\varepsilon k_1+r_\varepsilon\overline S$ with $k\in{\mathbb  Z}^N$. 
Let $k_l=(k_1l_1,..,k_Nl_N)$ represent the holes in ${\mathbb  R}^N$.

We assume that the holes $\tau(r_\varepsilon\overline S)$ do not 
intersect the boundary $\partial\Omega$. If $S_\varepsilon$ is the set 
of the holes enclosed in $\Omega$, it follows that 
there exists a finite set ${\mathcal K}\in{\mathbb  Z}^N$
such that 
$$S_\varepsilon={\bigcup_{k\in {\cal K}}}r_\varepsilon(k_l+\overline S).$$
We set
\begin{equation}\label{lot1}
\Omega_\varepsilon=\Omega\setminus\overline {S_\varepsilon},
\end{equation}
and denote by $\chi_{\Omega_\varepsilon}$ the characteristic function 
of $\Omega_\varepsilon$. 
Let $V_\varepsilon$ denote the Hilbert space 
\[ 
V_\varepsilon=\big\{v\in H^1(\Omega_\varepsilon), 
v_{|_{\partial\Omega}}=0\big\}
\] 
equipped with the $H^1$-norm. Let $A(y)=(a_{ij}(y))_{ij}$ be a matrix 
such that
$$
A\in\left(L^\infty\left({\mathbb  R}^N\right)\right)^{N^2},
$$
$A$ is Y-periodic, and there there exist $\alpha >0$  such that 
\begin{equation} \label{lot2}
\sum_{i,j=1}^{N}a_{i j}\left(y\right)\lambda_i\lambda_j\geq\alpha
\left|\lambda\right|^2, \quad\mbox{ a.e. $y$ in }
{\mathbb  R}^{N},\quad\forall\,\lambda\in{\mathbb  R}^{N}. 
\end{equation}
We note that for every $\varepsilon>0$,
\begin{equation}\label{lot3}
A^\varepsilon(x)=A(\displaystyle{x\over \varepsilon})
\quad\mbox{ a.e. $x$ in }{\mathbb  R}^N.
\end{equation}
In this paper, we study the system
\begin{eqnarray}
&-\mathop{\rm div }(A^\varepsilon \nabla u_\varepsilon)=0\quad\mbox{in }
 \Omega_\varepsilon,\nonumber\\
&(A^\varepsilon \nabla u_\varepsilon).n=h_\varepsilon\quad\mbox{on }
\partial S_\varepsilon,\label{lot4}\\
&u_\varepsilon =0,\quad\mbox{on }\partial\Omega,\nonumber
\end{eqnarray}
where $\Omega_\varepsilon$ is given by (\ref{lot1}), 
$A^\varepsilon$ is given by  (\ref{lot3}), and $h_\varepsilon$ is given by
\begin{equation}\label{lot5}
h^\varepsilon(x)=h({x\over{r_\varepsilon}}),
\end{equation}
where $h\in L^2(\partial S)$ is $Y$-periodic function.  
Set
\begin{equation}\label{lot6}
I_h={1\over|Y|}\int_{\partial S}h \,d\sigma\,.
\end{equation}
We examine the case where $r_\varepsilon < \varepsilon^{N/(N-2)}$ 
with $I_h\neq 0$, and the case where 
$\lim_{\varepsilon\to o} r_\varepsilon/ \varepsilon = 0$ with $I_h=0$. 
The following result describes the asymptotic behavior of the solution 
to (\ref{lot4}) in the two cases.

\begin{theorem} Let $u_\varepsilon$ be the solution of (\ref{lot4}). 
Suppose that one of the following hypotheses is satisfied
\begin{equation}\label{lot7}
I_h\neq 0\quad\mbox{and}\quad
\left\{
\begin{array}{ll}
\displaystyle\lim_{\varepsilon\to 0}{{r_\varepsilon}\over{\varepsilon
^{N/(N-2)}}}=0& \mbox{if $N>2$},\\[3pt]
\displaystyle\lim_{\varepsilon\to 0}\varepsilon^{-2}
(\ln (\varepsilon/r_\varepsilon))^{-1}=0 & \mbox{if $N=2$},
\end{array}\right.
\end{equation}  
or
\begin{equation}\label{lot8}
I_h=0\quad\mbox{and}\quad 
\displaystyle\lim_{\varepsilon\to 0}{{r_\varepsilon}\over\varepsilon}=0\,.
\end{equation}
Then, for every $\varepsilon>0$, there exists an extension operator $P_\varepsilon$ defined from $V_\varepsilon$ to $H_0^1(\Omega)$ satisfying
\begin{eqnarray}
& P_\varepsilon\in{\cal L}\left(V_\varepsilon , H^1_0(\Omega)\right),&
   \label{lot9i}\\
& \left(P_\varepsilon v\right)_{\mid_{\Omega_\varepsilon}}=v\;
   \forall v\in V_\varepsilon,& \label{lot9ii}\\
& \left\Vert \nabla \left(P_\varepsilon v\right)\right\Vert_{\left(L^2
\left(\Omega\right)\right)^N}\leq C\,
\left\Vert \nabla v\right\Vert_{\left(L^2\left(\Omega_\varepsilon\right)
\right)^N},\;\forall v\in V_\varepsilon\,.& \label{lot9iii}
\end{eqnarray}
such that
$$({r_\varepsilon\over \varepsilon})^{-{N/2}}P_\varepsilon u_\varepsilon
\rightharpoonup u\quad\mbox{weakly in }H_0^1(\Omega),\quad\mbox{for }N>2
$$
and
$$P_\varepsilon[({r_\varepsilon\over\varepsilon})^{-{1/2}}
(\log {\varepsilon\over{r_\varepsilon}})^{-{1/2}}u_\varepsilon]
\rightharpoonup u\quad\mbox{weakly in }H_0^1(\Omega),\quad\mbox{for }N=2,
$$
where $u$ is the solution of the problem
\begin{eqnarray*}
&-\mathop{\rm div}(A^0.\nabla u)=0\quad\mbox{in }\Omega,&\\
&u =0\quad\mbox{on }\partial \Omega,&
\end{eqnarray*}
and the matrix $A^0=(a^0_{ij})_{ij}$ has entries 
\begin{equation}\label{lot10}
a^0_{ij}={1\over{|Y|}}\int_Y(a_{ji}
-\sum_{k=1}^Na_{ki}{{\partial{\chi}_j}\over{\partial y_k}} dy),
\end{equation}
and $\chi_j$ is a $Y$-periodic function that satisfies 
\begin{equation}\label{lot11}
-\mathop{\rm div}(A^t\nabla(y_j-{\chi}_j))=0\quad\mbox{in }Y
\end{equation}
\end{theorem}

\paragraph{Remark.}  One can replace the first equation of system (\ref{lot4}) 
by
$$-\mathop{\rm div}(A^\varepsilon\nabla u_\varepsilon)
=f_\varepsilon\quad\mbox{in }\Omega_\varepsilon,$$
with
\begin{eqnarray*}
&({{r_\varepsilon}\over\varepsilon})^{-{N/2}}f_\varepsilon\rightharpoonup 
f\quad\mbox{weakly in }L^2(\Omega)\quad\mbox{if }N>2,&\\
&({{r_\varepsilon}\over\varepsilon})^{-1}(\ln(\varepsilon/r_\varepsilon)
)^{-{1/2}}f_\varepsilon\rightharpoonup f\quad\mbox{weakly in }L^2(\Omega)
\quad\mbox{if }N=2.&
\end{eqnarray*}
Then $u$ will be the solution of
\begin{eqnarray*}
&-\mathop{\rm div}(A^0.\nabla u)=f\quad\mbox{in }\Omega,\\
&u =0\quad\mbox{on }\partial \Omega.
\end{eqnarray*}
This approach has been used in \cite{Conca-Do} for the case $A=I$, 
in \cite{Cio-Do} when $A=I$ and $r_\varepsilon=\varepsilon$, and 
in \cite{Hajji1} for the case where $r_\varepsilon> \varepsilon^{N/(N-2)}$ 
using the $H^0$-convergence and some arguments given by 
S. Kaizu in \cite{Kaizu}.


\section{Proof of the main result}

Observe first that $S_\varepsilon$ is admissible in $\Omega$,
in the sense of the $H^0$-convergence (\cite{Hajji1,Cio-SJ,Conca-Do,Tartar}).
Then there exists an extension operator $P_\varepsilon$ satisfying 
(\ref{lot10}).

On the other hand, the matrix $A^0$ can be defined by 
$$A^0\lambda={{\cal M}}_{Y}({}^tA\nabla w_\lambda)={1\over|Y|}
\int_Y{}^tA\nabla w-\lambda \,dy,\quad\forall\lambda\in{\mathbb  R}^N,$$
where for every $\lambda\in{\mathbb  R}$, $w_\lambda$ is the solution of 
the problem
\begin{eqnarray}\label{lot12}
&-\mathop{\rm div}({}^tA\nabla w_\lambda)=0\quad\mbox{in }Y,&\\
&\mbox{with } w_\lambda-\lambda y\quad\mbox{$Y$-periodic}.&\nonumber
\end{eqnarray}
For $x\in {\mathbb  R}^N$, let 
\begin{equation}\label{lot13}
w_\lambda^\varepsilon (x)=\varepsilon w_\lambda ({x\over\varepsilon}).
\end{equation}
To simplify notation, let
\begin{eqnarray}\label{lot14}
\delta^\varepsilon=
\left\{ \begin{array}{ll}
(r_\varepsilon/\varepsilon)^{-{N/2}}& \mbox{if $N>2$},\\
(r_\varepsilon/\varepsilon)^{-1}(\ln(\varepsilon/r_\varepsilon))^{-1/2}
& \mbox{if $N=2$}.
\end{array}\right.
\end{eqnarray}
Taking $u_\varepsilon$ as a test function in the variational formulation of 
(\ref{lot4}), and using the classical techniques of {\em a priori} estimates, one 
can easily show the existence of two constants c and c' independent of 
$\varepsilon$ such that
$$c'\leq \| \nabla (\delta^\varepsilon u_\varepsilon) \|_{L^2(\Omega_\varepsilon)
}\leq c.
$$
Hence, from (\ref{lot9iii}), up to a subsequence,
\begin{equation}\label{lot15}
P_\varepsilon(\delta^\varepsilon u_\varepsilon)\rightharpoonup 
u\quad\mbox{weakly in }H_0^1(\Omega).
\end{equation}
Set now $\xi^\varepsilon=A^\varepsilon\nabla[P_\varepsilon(\delta^\varepsilon 
u_\varepsilon)]$. Then using (\ref{lot15}), (\ref{lot9i})-(\ref{lot9iii}) and 
(\ref{lot2})-(\ref{lot3}), one shows that $\xi^\varepsilon$ is bounded in 
$L^2(\Omega)$, and so up to a subsequence
\begin{equation}\label{lot16}
\xi^\varepsilon\rightharpoonup \xi\quad\mbox{weakly in }L^2(\Omega).
\end{equation}

\paragraph{Case where (\ref{lot7}) is satisfied.}
Let $\phi\in D(\Omega)$. Then from the variational formulation of (\ref{lot4}),
 one has
\begin{equation}\label{lot17}
\int_\Omega \chi_{\Omega_\varepsilon}\xi^\varepsilon.\nabla\phi \,dx
=\delta^\varepsilon\int_{\partial S_\varepsilon}h_\varepsilon \phi \,d\sigma\,.
\end{equation}
If $N(\varepsilon)$ denotes the number of the holes included in $\Omega$, 
one has then
\begin{eqnarray}
|\delta^\varepsilon\int_{\partial S_\varepsilon}h_\varepsilon\phi \,d\sigma| 
&\leq &\| \phi\|_{L^\infty(\Omega)}\delta^\varepsilon\sum_{k\in{{\cal K}}}
\int_{\partial(r_\varepsilon(S+k))}|h({x\over{r_\varepsilon}})| \,d\sigma(x)
\nonumber\\
&\leq &c \delta^\varepsilon N(\varepsilon)r_\varepsilon^{N-1}
\int_{\partial S}|h| \,d\sigma  \label{lot18} \\
&\leq &c \delta^\varepsilon{{r_\varepsilon^{N-1}}\over{\varepsilon^N}}|
\partial S|^{1/2}\| h\|_{L^2(\partial S)}. \nonumber
\end{eqnarray}
 From (\ref{lot14}), one can write
$$
\delta^\varepsilon{{r_\varepsilon^{N-1}}\over{\varepsilon^N}}=
\left\{
\begin{array}{ll}
({{{r_\varepsilon}^{N-2}}\over{\varepsilon^N}})^{1/2} &\mbox{if $N>2$},\\[2pt]
[\varepsilon^{-2}(\ln(\varepsilon/r_\varepsilon))^{-1}]^{1/2}
&\mbox{if $N=2$},
\end{array}\right.
$$
and so in virtue of (\ref{lot7}), 
\begin{equation}\label{lot19}
\displaystyle\lim_{\varepsilon\to 0}\delta^\varepsilon{{{r_\varepsilon}^{N-1}}
\over{\varepsilon^N}}=0,
\end{equation}
hence
$$\lim_{\varepsilon\to 0}\delta^\varepsilon
\int_{\partial S_\varepsilon}h_\varepsilon\phi \,d\sigma =0\,.
$$
On the other hand, it is easy to show that 
$$\chi_{\Omega_\varepsilon}\to 1\quad\mbox{strongly in }L^p(\Omega)
\quad\forall p\in[1,\infty[,$$
hence
\begin{equation}\label{lot20}
\int_\Omega\chi_\varepsilon\xi^\varepsilon.\nabla\phi \,dx\to \int_\Omega\xi.
\nabla\phi \,dx.
\end{equation}
One can deduce that, as $\varepsilon\to 0$ in (\ref{lot17}), 
$$\int_\Omega\xi\nabla\phi \,dx=0,\quad\forall\phi\in D(\Omega).$$
Consequently
\begin{equation}\label{lot21}
-\mathop{\rm div}\xi=0\quad\mbox{in }\Omega.
\end{equation}
It remains to identify the function $\xi$. Let 
$w_\lambda^\varepsilon$ be the function defined by (\ref{lot12})-(\ref{lot13}).
Then 
$$w_\lambda^\varepsilon\rightharpoonup \lambda x\quad\mbox{weakly in }
H^1(\Omega), \mbox{ and so }L^p(\Omega)\mbox{ strong }\forall p<2^*
$$
where ${1/{2^*}}={1/2}-{1/N}$,  with $N\geq 2$.
Let $\phi\in D(\Omega)$, by choosing $\phi w_\lambda^\varepsilon$ as a test 
function in the variational formulation of (\ref{lot4}), one has
\begin{equation}\label{lot22}
\int_{\Omega_\varepsilon}\xi^\varepsilon \nabla(\phi w_\lambda^\varepsilon) 
\,dx=\delta^\varepsilon\int_{\partial {S_\varepsilon}}h_\varepsilon
\phi w_\lambda^\varepsilon \,dx\,. 
\end{equation}
To pass to the limit as $\varepsilon\to 0$ in (\ref{lot22}), we set 
\begin{equation}\label{lot23}
\int_{\Omega_\varepsilon}\xi^\varepsilon \nabla(\phi w_\lambda^\varepsilon) 
\,dx=J_1^\varepsilon+J_2^\varepsilon\,,
\end{equation}
where 
$$J_1^\varepsilon=\int_{\Omega_\varepsilon}\xi^\varepsilon \nabla \phi. 
w_\lambda^\varepsilon \,dx\quad\mbox{and}\quad
J_2^\varepsilon=\int_{\Omega_\varepsilon}\xi^\varepsilon 
\nabla w_\lambda^\varepsilon.\phi \,dx.
$$

Using the results given by G. Stampacchia in \cite{Stamp} 
(see also \cite{Brez}), one can deduce that $w_\lambda\in L^\infty(Y)$, so
\begin{equation}\label{lot24}
\chi_{\Omega_\varepsilon}w_\lambda^\varepsilon\to \lambda x\quad
\mbox{strongly in }L^2(\Omega).
\end{equation}
This with convergence (\ref{lot16}), gives
\begin{equation}\label{lot25}
J_1^\varepsilon =\int_\Omega \chi_{\Omega_\varepsilon}w_\lambda^\varepsilon 
\xi^\varepsilon \nabla\phi \,dx\to \int_\Omega \lambda x\xi\nabla\phi \,dx
\quad\mbox{as }\varepsilon\to 0\,.
\end{equation}
Now, we may write 
\begin{equation}\label{e33"}
J_2^\varepsilon=\displaystyle\int_\Omega \xi^\varepsilon \nabla 
w_\lambda^\varepsilon\phi \,dx-\displaystyle\int_{S_\varepsilon}\xi^\varepsilon
 \nabla w_\lambda^\varepsilon\phi \,dx\,.
\end{equation}
One the one hand,
\begin{eqnarray*}
\int_\Omega \xi^\varepsilon \nabla w_\lambda^\varepsilon\phi \,dx 
&=& \int_\Omega {}^tA^\varepsilon \nabla w_\lambda^\varepsilon
\nabla[P_\varepsilon(\delta^\varepsilon u_\varepsilon)\phi] \,dx
-\int_\Omega {}^tA^\varepsilon\nabla w_\lambda^\varepsilon
\nabla\phi P_\varepsilon[(\delta^\varepsilon u_\varepsilon] \,dx\\
&=&-\int_\Omega {}^tA^\varepsilon\nabla w_\lambda^\varepsilon
\nabla\phi P_\varepsilon[(\delta^\varepsilon u_\varepsilon] \,dx
\end{eqnarray*}
because, from the definition of $w_\lambda^\varepsilon$,
$$ \int_\Omega {}^tA^\varepsilon \nabla w_\lambda^\varepsilon
\nabla[P_\varepsilon(\delta^\varepsilon u_\varepsilon)\phi] \,dx=0\,.
$$
 From the definition of $A^0$,
${}^tA^\varepsilon\nabla w_\lambda^\varepsilon\rightharpoonup A^0\lambda$
weakly in $L^2(\Omega)$.
 From (\ref{lot15}), up to a subsequence,
$$P_\varepsilon(\delta^\varepsilon u_\varepsilon)\to u\quad\mbox{strongly in}L^2(\Omega),$$
which implies 
$$\int_\Omega {}^tA^\varepsilon\nabla w_\lambda^\varepsilon
\nabla\phi P_\varepsilon(\delta^\varepsilon u_\varepsilon) \,dx
\to \int_\Omega A^0\lambda u \nabla \phi \,dx\,.
$$
Hence
\begin{equation}\label{lot26}
\int_\Omega \xi^\varepsilon \nabla w_\lambda^\varepsilon \phi \,dx\to 
-\int_\Omega A^0\lambda \nabla \phi u \,dx\,.
\end{equation}
On the other hand,
\begin{equation}\label{lot27}
|\int_{S_\varepsilon}\xi^\varepsilon \nabla w_\lambda^\varepsilon\phi \,dx|
\leq c\| \xi^\varepsilon\|_{L^2(\Omega)}\| \nabla w_\lambda^\varepsilon
\|_{L^2(S_\varepsilon)}.
\end{equation}
Since $\| \xi^\varepsilon\|_{L^2(\Omega)}$ is bounded, 
\begin{equation}\label{lot28}
|\int_{S_\varepsilon}\xi^\varepsilon
\nabla w_\lambda^\varepsilon
\phi \,dx|\leq c \| \nabla w_\lambda^\varepsilon
\|_{L^2(S_\varepsilon)}.
\end{equation}
Note that
\begin{eqnarray*}
\| \nabla w_\lambda^\varepsilon\|_{L^2(S_\varepsilon)}^2 
&=&\int_{S_\varepsilon}|(\nabla w_\lambda^\varepsilon)(x)|^2 \,dx\\
&=&\sum_{k\in{{\cal K}}}\int_{r_\varepsilon (S+k)}|(\nabla 
w_\lambda^\varepsilon)(x)|^2 \,dx\\
&=&\sum_{k\in{{\cal K}}}\int_{r_\varepsilon (S+k)}|({\nabla}_y 
w_\lambda)({x\over\varepsilon})|^2 \,dy\\
&=& N(\varepsilon)\varepsilon^N\int_{{r_\varepsilon\over\varepsilon}S}|
({\nabla}_y w_\lambda)(y)|^2 \,dy\\
&\leq& c\int_{{r_\varepsilon\over\varepsilon}S}|\nabla w_\lambda|^2 \,dy\,.
\end{eqnarray*}
Since
$r_\varepsilon/\varepsilon\to 0$ and $w_\lambda\in H^1(Y)$, it follows that
$$\int_{r_\varepsilon S/\varepsilon} |\nabla w_\lambda|^2 \,dy\to 0\,.$$
Using (\ref{lot28}), one has
$$\int_{S_\varepsilon}\xi^\varepsilon
\nabla w_\lambda^\varepsilon \phi \,dx\to 0\quad\mbox{as }\varepsilon\to 0.
$$
This, with (\ref{e33"}) and convergence (\ref{lot26}) imply that
\begin{equation}\label{lot29}
J_2^\varepsilon\to -\int_\Omega A^0\lambda \nabla\phi u \,dx.
\end{equation}

Next we pass to the limit in the right hand of (\ref{lot22}).
With the same argument as in (\ref{lot18}), 
\begin{eqnarray*}
|\delta^\varepsilon\int_{\partial S_\varepsilon} h_\varepsilon\phi 
w_\lambda^\varepsilon \,d\sigma |&\leq & c\delta^\varepsilon N(\varepsilon)
r_\varepsilon^{N-1}\int_{\partial S}|h w_\lambda| \,d\sigma\\
&\leq & c\delta^\varepsilon{{r_\varepsilon^{N-1}}\over{\varepsilon^N}}
\| w_\lambda\|_{L^2(\partial S)}|\partial S|^{1/2}\| h\|_{L^2(\partial S)}.
\end{eqnarray*}
Since we have shown that
$$\displaystyle\lim_{\varepsilon\to 0}\delta^\varepsilon{{r_\varepsilon^{N-1}}
\over{\varepsilon^N}}=0,$$
from (\ref{lot7}) one deduces that
$$\delta^\varepsilon\int_{\partial S_\varepsilon} h_\varepsilon\phi 
w_\lambda^\varepsilon \,d\sigma \to 0.
$$
Finally, by passing to the limit as $\varepsilon\to 0$ in (\ref{lot22}), 
and using (\ref{lot27}) and (\ref{lot29}) one obtains
$$\int_\Omega\lambda x \xi\nabla \phi \,dx-\int_\Omega A^0\lambda 
\nabla\phi u \,dx=0,
$$
hence, from (\ref{lot21}) it follows that
$$\int_\Omega \xi\lambda \phi \,dx=\int_\Omega A^0 \lambda \nabla u 
\phi \,dx\forall\phi\in D(\Omega),\forall\lambda\in{\mathbb  R}^N,
$$
i.e., $\xi=A^0 \nabla u$.

\paragraph{Case where (\ref{lot8}) is satisfied.}
Let $\phi\in D(\Omega)$. Then from the variational formulation of (\ref{lot4}),
\begin{equation}\label{lot30}
\int_\Omega\chi_{\Omega_\varepsilon}\xi^\varepsilon.\nabla\phi \,dx
=\delta^\varepsilon\int_{\partial S_\varepsilon}h_\varepsilon \phi \,d\sigma.
\end{equation} 
The arguments used the proof of (\ref{lot20}) can be applied here to obtain
$$\int_\Omega\chi_{\Omega_\varepsilon}\xi^\varepsilon.\nabla\phi \,dx
\to\int_\Omega\xi.\nabla\phi \,dx\,.$$

To pass to the limit in the right-hand side of (\ref{lot30}), we introduce
$N$ as the solution to 
\begin{eqnarray*}
&-\mathop{\rm div}N=0\quad\mbox{in }S,&\\
& N.n=-h\quad\mbox{on }\partial S\,.&
\end{eqnarray*}
The existence of $N$ is assured by the hypothesis $I_h=0$. Set 
$\displaystyle{N_\varepsilon(x)=N({{x-\varepsilon k}\over{r_\varepsilon}})}$, 
for $x$ in $(\varepsilon Y\setminus r_\varepsilon S)_k$. Then
$$
\int_{\partial S_\varepsilon}h_\varepsilon\phi \,d\sigma
=\int_{S_\varepsilon}\nabla\phi.N_\varepsilon \,dx,$$
hence
$$\delta^\varepsilon|\int_{\partial S_\varepsilon}h_\varepsilon\phi \,d\sigma|
\leq \delta^\varepsilon\| \nabla\phi\|_{L^2(S_\varepsilon)}\| 
N_\varepsilon\|_{L^2(S_\varepsilon)}.$$
Note that
$\| N_\varepsilon\|_{L^2(S_\varepsilon)}\leq c({r_\varepsilon\over\varepsilon})
^{N/2}\| N\|_{L^2(S)}$, so
\begin{equation}\label{lot31}
\delta^\varepsilon|\int_{\partial S_\varepsilon}h_\varepsilon\phi \,d\sigma|
\leq c\delta^\varepsilon({{r_\varepsilon}\over\varepsilon})^{N/2}
\| \nabla\phi\|_{L^1(S_\varepsilon)}.
\end{equation}
Since
$$
\delta^\varepsilon({{r_\varepsilon}\over\varepsilon})^{N/2}=
\left\{
\begin{array}{ll}
1& \mbox{if $N>2$} \\[2pt]
(\ln(\varepsilon/ r_\varepsilon))^{-1/2} & \mbox{if $N=2$},
\end{array} \right.
$$
it follows from (\ref{lot31}), when $N=2$, that
$$\lim_{\varepsilon\to 0}\delta^\varepsilon|
\int_{\partial S_\varepsilon}h_\varepsilon\phi \,d\sigma|=0\,.$$
For $N>2$, one has
$$\delta^\varepsilon|\int_{\partial S_\varepsilon}h_\varepsilon\phi 
\,d\sigma|\leq c\| \nabla\phi\|_{L^2(S_\varepsilon)}.
$$
Since $\chi_{\Omega_\varepsilon}\to 1$ strongly in $L^p(\Omega)$, for all 
$p\in[1, \infty [$ and $\phi\in D(\Omega)$, one deduces that
\[ 
\int_{\Omega}(1-\chi_\varepsilon)|\nabla\phi|^2 \,dx\to 0\,.
\] 
Hence, by passing to the limit as $\varepsilon\to 0$ in (\ref{lot30}), 
one obtains
$$\int_\Omega \xi.\nabla\phi \,dx=0\,,$$
then
$-\mathop{\rm div}\xi=0\quad\mbox{in }\Omega$.
 
Let $w_\lambda^\varepsilon$ be the function defined by 
(\ref{lot12})-(\ref{lot13}) and $\phi\in D(\Omega)$. As in the previous case, 
by using $\phi w_\lambda^\varepsilon$ as a test function in the variational 
formulation of (\ref{lot4}), one has
$$\int_{\Omega_\varepsilon}\xi^\varepsilon.\nabla(\phi w_\lambda^\varepsilon) 
\,dx=\delta^\varepsilon\int_{\partial S_\varepsilon}h_\varepsilon \phi 
w_\lambda^\varepsilon \,d\sigma\,.
$$
 From (\ref{lot23}), (\ref{lot25}) and (\ref{lot29}), one has
\begin{equation}\label{lot32}
\displaystyle{\int_{\Omega_\varepsilon}\xi^\varepsilon.
\nabla(\phi w_\lambda^\varepsilon) \,dx\to\int_\Omega\lambda x.\nabla\phi 
\,dx-\int_\Omega A^0\lambda.\nabla\phi u \,dx}.
\end{equation}
Now we show that
\begin{equation}\label{lot33}
\delta^\varepsilon|\int_{\partial S_\varepsilon}h_\varepsilon \phi 
w_\lambda^\varepsilon \,d\sigma|\to 0.
\end{equation}
One has
\begin{eqnarray*}
\lefteqn{\delta^\varepsilon|\int_{\partial S_\varepsilon}h_\varepsilon \phi 
w_\lambda^\varepsilon \,d\sigma|}\\
&=& \delta^\varepsilon|
\int_{S_\varepsilon}\nabla(\phi w_\lambda^\varepsilon).N_\varepsilon \,dx|\\
&\leq &\delta^\varepsilon|\int_{S_\varepsilon}\nabla\phi.w_\lambda
^\varepsilon.N_\varepsilon \,d\sigma|+\delta^\varepsilon|
\int_{S_\varepsilon}\phi.\nabla w_\lambda^\varepsilon.N_\varepsilon \,d\sigma|\\
&\leq &\delta^\varepsilon\| N_\varepsilon\|_{L^2(S_\varepsilon)}
\| \nabla\phi w_\lambda^\varepsilon\|_{L^2(S_\varepsilon)}
+ \delta^\varepsilon\| N_\varepsilon\|_{L^2(S_\varepsilon)}\| 
\phi\nabla w_\lambda^\varepsilon\|_{L^2(S_\varepsilon)}\\
&\leq &c\delta^\varepsilon({{r_\varepsilon}\over\varepsilon})^{N/2}
\left\{\| \nabla\phi w_\lambda^\varepsilon\|_{L^2(S_\varepsilon)}
+\| \phi\nabla w_\lambda^\varepsilon\|_{L^2(S_\varepsilon)}\right\}\\
&\leq &c\delta^\varepsilon({{r_\varepsilon}\over\varepsilon})^{N/2}
\left\{\| \nabla\phi\|_{L^\infty(\Omega)}\| w_\lambda^\varepsilon\|
_{L^2(S_\varepsilon)}+\| \phi\|_{L^\infty(\Omega)}\| \nabla 
w_\lambda^\varepsilon\|_{L^2(S_\varepsilon)}\right\}.\\
&\leq &c\delta^\varepsilon({{r_\varepsilon}\over\varepsilon})^{N/2}
\left\{\| w_\lambda^\varepsilon\|_{L^2(S_\varepsilon)}+\| 
\nabla w_\lambda^\varepsilon\|_{L^2(S_\varepsilon)}\right\}.
\end{eqnarray*}
Note that
\begin{eqnarray*}
\| \nabla w_\lambda^\varepsilon\|_{L^2(S_\varepsilon)}^2 
&=& \int_{S_\varepsilon}|\nabla w_\lambda^\varepsilon|^2 \,dx
=\sum_{k\in{{\cal K}}}\int_{r_\varepsilon(S+k)}|\nabla 
w_\lambda^\varepsilon|^2 \,dx\\
&\leq & N(\varepsilon)\varepsilon^N\int_{{{r_\varepsilon}\over\varepsilon}S}
|\nabla w_\lambda|^2 \,dx\leq c\int_{{{r-\varepsilon}\over\varepsilon}S}
|\nabla w_\lambda|^2 \,dx\,.
\end{eqnarray*}
Since $w_\lambda\in H^1(S)$ and $r_\varepsilon/\varepsilon \to 0$, 
$$\lim_{\varepsilon\to 0}\int_{{{r_\varepsilon}\over\varepsilon}S}|\nabla 
w_\lambda|^2 \,dx =0\,.$$
Hence $\| \nabla w_\lambda^\varepsilon\|_{L^2(S_\varepsilon)}\to 0$. 
On the other hand, one has
$\| w_\lambda^\varepsilon\|_{L^2(S_\varepsilon)}\leq c$. 
Finally, as 
$$\lim_{\varepsilon\to 0}\delta^\varepsilon({{r_\varepsilon}\over\varepsilon})
^{N/2}=0\,,$$
one deduces (\ref{lot33}). This and (\ref{lot32}) completes the proof, using 
the same arguments as in the previous case.


\paragraph{Acknowledgments}
The author would like to thank Professor Patrizia Donato for her help on this 
work.

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\noindent{\sc Mohamed El Hajji } \\
Universit\'e de  Rouen, UFR des Sciences \\
UPRES-A 60 85 (Labo de Math.) \\ 
76821 Mont Saint Aignan, France \\
e-mail: Mohamed.Elhajji@univ-rouen.fr

\end{document}