\documentstyle[twoside]{article}
\pagestyle{myheadings}
\markboth{\hfil Behaviour near the boundary \hfil EJDE--1997/12}%
{EJDE--1997/12\hfil V. N. Domingos Cavalcanti \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.\ {\bf 1997}(1997), No.\ 12, pp. 1--18. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
 \vspace{\bigskipamount} \\
 Behaviour near the boundary for solutions of elasticity systems
\thanks{ {\em 1991 Mathematics Subject Classifications:}
93B05, 93C20, 35B37.\hfil\break\noindent
{\em Key words and phrases:} Behaviour near the boundary, controllability, 
elasticity system
\hfil\break\noindent
\copyright 1997 Southwest Texas State University  and University of
North Texas. \hfil\break\noindent
Submitted April 1, 1997. Published July 31, 1997.} }
\date{}
\author{V. N. Domingos Cavalcanti }
\maketitle

\def\diver{\mathop{\rm div}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\def\det{\mathop{\rm det}}

\begin{abstract}
In this article we study the behaviour near the boundary for
weak solutions of the system
$$
u''-\mu\Delta u-(\lambda +\mu )\nabla (\alpha (x)\,\diver u)=h\,,
$$
with $u(x,t)=0$ on the boundary of a domain $\Omega\in {\bf R}^n$, and
$u(x,0)=u^0$, $u'(x,0)=u^1$ in $\Omega$. 
We show that the Sobolev norm of the solution in an 
$\varepsilon$-neighbourhood of the boundary can be estimated 
independently of $\varepsilon$. 
\end{abstract}

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\section{Introduction}
Let $\Omega$ be a bounded domain in ${\bf R}^n$ with a $C^3$-boundary
$\Gamma$, and let $\nu (x)$ be the unit exterior normal of $\Gamma$ 
at a point $x$. For $T>0$, we denote by $Q$ the finite cylinder
$\Omega\times ]0,T[$,
and by $\Sigma$ its lateral boundary $\Gamma\times ]0,T[$. 
For an open subset $\Gamma_0$ of $\Gamma$, $\Sigma_0$ denotes $\Gamma_0\times ]0,T[$.
For each $\varepsilon >0$, $\omega_{\varepsilon}$ denotes the
$\varepsilon$-neighbourhood of $\Gamma_0$ in $\Omega$ defined by
$$
\omega_{\varepsilon}=\bigcup_{x\in\Gamma_0}B(x,\varepsilon )\cap
\Omega\, ,$$
where $B(x,\varepsilon )$ is the open ball with center $x$ and radius
$\varepsilon$.
Functional spaces and their inner products are denoted as follows
\begin{eqnarray*}
H=[L^2(\Omega )]^n,&& (u,v)_H=\sum^{_n}_{i=1}(u_i,v_i)_{L^2(
\Omega )}\quad\forall u,v\in H\\
V=[H^1_0(\Omega )]^n,&& ((u,v))_V=\sum_{i=1}^n(\nabla u_i,\nabla
v_i)_{L^2(\Omega )}\quad\forall u,v\in V\,.
\end{eqnarray*}

This article is concerned with the behaviour near the boundary for
weak solutions of the system
\begin{eqnarray}\label{eq:1.1}
&&u^{\prime\prime}-\mu\Delta u-(\lambda +\mu )\nabla (\alpha (x)\diver
u)=h\quad\hbox{in}\quad Q\nonumber\\
&&u=0\quad\hbox{on}\quad\Sigma\\
&&u(0)=u^0,\quad u'(0)=u^1\quad\hbox{in}\quad\Omega \,,\nonumber
\end{eqnarray}
with
\begin{equation}\label{eq:1.2}\{u^0,u^1,h\}\in V\times H\times L^
1(0,T;H)\,,\end{equation}
where $\lambda , \mu >0$ are the Lam\'e constants, and
$\alpha\in C^1(\overline {\Omega})$ is a real function such that for
all $x$ in $\overline\Omega$ and some $\alpha_0$, 
$\alpha (x)\geq\alpha_0>0$.

Recall that if $u$ is the unique solution of the above system, for
the energy function
$$E(t)=\frac 12|u'(t)|^2_H+\frac {\mu}2||u(t)||^2_V+\frac {(\lambda
+\mu )}2|\alpha^{\frac 12}(x)\diver u(t)|^2_{L^2(\Omega )}$$
there exists a positive constant $c$ such that
$$\sup_{t\in [0,T]}E(t)\leq cE_0\,,$$
where
\begin{equation}\label{eq:1.3}E_0=||u_0||^2_V+|u^1|^2_H+||h||^2_{
L^1(0,T;H)}.\end{equation}

We shall prove that  the $H$ norm of the solution $u$ of (\ref{eq:1.1}) in an 
$\varepsilon$-neighbour-hood of
the boundary can be estimated independently of $\varepsilon$. 
It will be done by studying first the behaviour of  $\nabla u$ in the same 
neighbourhood.
We use the method developed by J.P. Puel and C. Fabre [7] for solutions
of the wave equation, which is an extension of the multipliers
method introduced by F. Rellich for elliptic equations and used
by C. Morawetz in hyperbolic equations.
Their method was also applied to Schrodinger equations, and to equations 
for vibrating beams (cf. C. Fabre [4] and C. Fabre - J. P. Puel [6]).

The interest in the above result lies in the fact that this estimate, 
combined with other results, allow us to obtain the boundary control 
as the limit of internal controls.  Then, provided that the internal 
controls exist we obtain the boundary control passing to the limit as 
$\varepsilon \rightarrow 0$.  This is a way of getting boundary control 
when we only have the guarantee that the system is internal controllable. 

To use this kind of argument is a powerful tool since we act where it is 
convenient and then pass to the limit.  In this direction we can cite the 
work by M. E. Bradley and M. A. Horn [1] where they control the Von K\'arm\'an 
system $w''-\gamma^2\Delta w' + \Delta^2 w + b(x)w' = [w,\chi(w)]$, by 
showing that the boundary control which stabilizes its solution when 
$\gamma \neq 0 $ also stabilizes the solution to the system obtained 
in the limit as $\gamma \rightarrow 0 $.  Here $\gamma $ is a parameter 
which is proportional to the thickness of the Von K\'arm\'an plate.
 
\section{Geometrical Properties of $\Omega$}

Using the fact that $\Omega$ is a bounded domain of ${\bf R}^n$ with a $
C^3-$boundary
$\Gamma$, we obtain the following lemma.

\begin{lemma} \label{Lemma 2.1} There exist open sets $U_1,\dots U_m$ 
and a positive constant
$\varepsilon_0$ which satisfy 
\begin{itemize}
\item For each $\varepsilon$ in $]0,\varepsilon_0[$,
 $\overline {\omega_{\varepsilon}}\subset\bigcup_{i=1}^m U_i$, where
 $\overline {\omega_{\varepsilon}}$ denotes the closure of
$\omega_{\varepsilon}$ in ${\bf R}^n$.
\item For each $x\in\omega_{\varepsilon}\cap U_i$, there exists a unique 
$(y,z)\in (\Gamma\cap U_i)\times ]0,\varepsilon [$  such that
$x=y-z\nu (y)$; $i=1,...,m$.  
\item There are functions $F_i^{-1}:x\to (w,z)$, $C^2$-diffeomorphisms
 defined from $\omega_{\varepsilon}\cap U_i$ onto their image; $i=1,\dots,m$.
\end{itemize}\end{lemma}

\paragraph{Proof.}
Due to the regularity of $\Gamma$, which is the
$C^3$-boundary of $\Omega$, by M. Do Carmo [2] (section 2.7,
proposition 1) for each point $p\in\Gamma$ and $X_p:U\rightarrow
\Gamma$, a
$C^3$-parametrization of a neighbourhood of $p$, there exist a
neighbourhood $W_p\subset X_p(U)$ of $p$ in $\Gamma$ and $\varepsilon_
p>0$ such that the
segments of the normal lines through points $q\in W_p$, with center
at $q$ and length $2\varepsilon_p$, are disjoint; that is, $W_p$ has a tubular
neighbourhood.

Considering $0<2r_p<\varepsilon_p$ where $\overline {B_{2r_p}}$$(
p)$$\subset B_p$, $W_p=B_p\cap\Gamma$ and $B_p$ is an open set of ${\bf R}^n$ we obtain:
$$\overline {\Omega}=\Omega\cup\Gamma\subset\Omega\cup\left(\bigcup_{
p\in\Gamma}B_{r_p}(p)\right).$$
Using the compactness of $\overline {\Omega}$, there exist $p_1,\dots
,p_m$ such that
$$\Gamma\subset\bigcup_{i=1}^mB_{r_{p_i}}(p_i).$$
Defining
$$U_i=B_{2r_{p_i}}(p_i);\quad i=1,\dots ,m\quad\mbox{and}\quad
\varepsilon_0=\frac 12\min\left\{r_{p_1},\dots ,r_{p_m}\right\}\,,$$
we conclude that for every $\varepsilon\in ]0,\varepsilon_0]$, 
$\overline {\omega_{\varepsilon}}\subset\bigcup_{i=1}^mU_i$.

On the other hand, we observe that for each $p_i\in\Gamma$ the related
$C^3$- parametrization $X_i$ implies that the function
\begin{eqnarray*}
&F_i:X_i^{-1}(\Gamma\cap U_i)\times ]0,\varepsilon [\rightarrow\omega_{
\varepsilon}\cap U_i& \\
&(w,z)\mapsto F_i(w,z)=X_i(w)-zN_i(w)\,,&
\end{eqnarray*}
where $\varepsilon\in ]0,\varepsilon_0[$ and $N_i(w)$ is the normal 
vector at point $X_i(w),$ is a $C^2$-diffeomorphism.

Moreover, according to the choice of the open sets $U_i$ we
conclude that for every $x\in\omega_{\varepsilon}\cap U_i$ where $
\varepsilon\in ]0,\varepsilon_0]$ there exist
a unique normal projection $y$ of $x$ on $\Gamma$ and unique $z\in ]0,\varepsilon
[$ such that $x=y-z\nu (y).$ \hfil      $\Box$

\paragraph{Remark.} It follows from the above lemma that if
$x\in\omega_{\varepsilon}\cap U_i$ for sufficiently small $\varepsilon$, 
the normal projection of $x$ onto $\Gamma$ is uniquely defined.  
Furthermore,  
$$p(x)=X_i(\omega )=F_i(\omega ,0)\,,$$
where $F_i(w,z)=X_i(w)-zN_i(w)$. 

\begin{lemma} Let $\det JF_i(w,z)$ denote the determinant of the
Jacobian matrix of $F_i$ at $(w,z)$. Then
\begin{description}
\item{(i)} there exist positive constants $\varepsilon_0, m, M$, such 
that $\forall (w,z)\in X_i^{-1}(\Gamma\cap U_i)\times [0,\varepsilon_0]$,
$$ m\leq |\det JF_i(w,z)|\leq M$$
\item{(ii)}  $|\det JF_i(w,0)|=1$, $\forall w\in X_i^{-1}(\Gamma\cap U_i)$
\item{(iii)} the function $(w,z)\to |\det JF_i(w,z)|$ is a
$C_{}^1$-function.
\item{(iv)} for function  $v$ on $\omega_{\varepsilon}\cap U_i$,
define $\hat {v}(w,z)=(v\circ F_i)(w,z)$. If
$v\in H^1(\omega_{\varepsilon}\cap U_i)$ then
\begin{equation}\label{eq:2.1}\frac {\partial}{\partial z}\hat {v}
(w,z)=-\nabla v(F_i(w,z))\cdot\nu (F_i(w,0)\,.\end{equation}
\end{description}
\end{lemma}

\paragraph{Proof.} Since $F_i(w,z)=X_i(w)-zN_i(w)$ it follows that
$|\det JF_i(\omega ,0)|=||N_i(\omega )||$ and consequently $|\det
JF_i(\omega ,0)|=1$,
$\forall\omega\in X_i^{-1}(\Gamma\cap U_i)$. Besides, taking into account 
the regularity of
the boundary of $\Omega$, we get that $\det JF_i(\omega ,z)$ is a 
$C^1-$function. For a $\varepsilon_i>0$ small enough we obtain that 
$\det JF_i(\omega ,z)$ does not change
its sign on $]0,\varepsilon_i[$, which allow us to conclude (iii). Combining the
two results obtained in (ii) and (iii) we get (i). Finally,  from the
regularity of the functions $F_i(w,z)$, observing that
$\frac {\partial}{\partial z}F_{i_k}(w,z)=-N_{i_k}(w)$ and from the identity
$$
\frac {\partial}{\partial z}(v\circ F_i)(w,z)=\sum_{k=1}^n\frac {
\partial v}{\partial x_k}(F_i(w,z))\frac {\partial}{\partial z}F_{
i_k}(w,z)\,,$$
which holds for regular functions v, we obtain (iv). \hfil   $\Box$


\section{Fundamental Identity}
In this section we prove an identity that is essential in the proof of our main result.

\begin{lemma}  If
$u=(u_1,...,u_n)$ is the solution of (\ref{eq:1.1})-(\ref{eq:1.2}), for
every vector valued function
$g\in W^{2,\infty}(\Omega ,{\bf R}^n)$ collinear to the normal vector on $
\Gamma$, we have
\begin{eqnarray}\label{eq:3.1}
&&2\mu\sum_{i,j,k=1}^n\int_Q\frac {\partial
u_i}{\partial x_j}\frac {\partial u_i}{\partial x_k}\frac {\partial
g_k}{\partial x_j}\,dx\,dt+\mu\sum_{i=1}^n\int_Q(\nabla u_i\cdot\nabla
(\diver g))u_i\,dx\,dt\nonumber \\
&&+2(\lambda +\mu )\sum_{i,j=1}^n{}\int_Q\alpha (x)\diver u\frac {
\partial u_i}{\partial x_j}\frac {\partial g_j}{\partial x_i}\,dx\,
dt\nonumber \\
&&+(\lambda +\mu )\int_Q\alpha (x)\diver u(u\cdot\nabla (\diver g))\,dx\,dt \\
&=&2\sum_{i=1}^n\int_Qh_i(\nabla u^{}_i\cdot g)\,dx\,dt\,+\,\sum_{i
=1}^n\int_Qh_iu^{}_idivg\,dx\,dt\nonumber \\
&&+\mu\sum_{i=1}^n\int_{\Sigma}|\frac {\partial u_i}{\partial\nu}
|^2g\cdot\nu d\Gamma dt\,+\,(\lambda +\mu )\int_{\Sigma}\alpha (x
)(divu)^2g\cdot\nu d\Gamma \,dt\nonumber \\
&&+(\lambda +\mu )\int_Q\nabla\alpha\cdot g(\diver u)^2\,dx\,dt\,-\,2\sum_{
i=1}^n\int_{\Omega}u_i'(T)(\nabla u_i(T)\cdot g)\,dx\nonumber \\
&&+2\sum_{i=1}^n\int_{\Omega}u_i'(0)(\nabla u_i(0)\cdot g)\,dx
-\sum_{i=1}^n\int_{\Omega}u_i'(T)u_i(T)\diver g\,dx\nonumber \\
&&+\sum_{i=1}^n\int_{\Omega}u_i'(0)u_i(0)\diver g\,dx\,.\nonumber
\end{eqnarray}
\end{lemma}

\paragraph{Proof.}
For initial data $\left\{u^0,u^1,h\right\}\in\left[H_0^1
(\Omega )\times H^2(\Omega )\right]^n\times V\times L^1(0,T;V)$ let
$$
2\nabla u\cdot g=\left(2\nabla u_1\cdot g,\dots ,
2\nabla u_n\cdot g\right)\,,$$
where
$$2\nabla u_i\cdot g=2\sum_{k=1}^n\frac {\partial u_i}{\partial x_k}g_k\,.
$$
Multiplying (\ref{eq:1.1}) by $2\nabla u\cdot g$ we obtain
\begin{eqnarray}\label{eq:3.2}
&&\sum_{i=1}^n\int_Qu_i^{\prime\prime}
(2\nabla u_i\cdot g)\,dx\,dt-\mu\sum_{i=1}^n\int_Q\Delta u_i(2\nabla
u_i\cdot g)\,dx\,dt \nonumber \\
&&-(\lambda +\mu )\sum_{i=1}^n\int_Q\frac {\partial}{\partial x_i}\left
(\alpha (x)\diver u\right)\left(2\nabla u_i\cdot g\right)\,dx\,dt\\
&&=\sum_{i=1}^n\int_Qh_i\left(2\nabla u_i\cdot g\right)\,dx\,dt\,.\nonumber
\end{eqnarray}
Integrating by parts with respect to $t$, by Gauss Theorem,
\begin{eqnarray}\label{eq:3.3}
\lefteqn{2\sum_{i=1}^n\int_Qu_i^{\prime\prime}\left
(\nabla u_i\cdot g\right)\,dx\,dt} && \nonumber \\
&=&2\sum_{i=1}^n\int_{\Omega}u_i'(T)\left
(\nabla u_i(T)\cdot g\right)\,dx
-2\sum_{i=1}^n\int_{\Omega}u_i'(0)\left(\nabla u_i(0)\cdot g\right
)dx \\
&&+\sum_{i=1}^n\int_Q\left(u_i'\right)^2\diver g\,dx\,dt\,.\nonumber
\end{eqnarray}
Using the fact that
\begin{equation}\label{eq:3.4}
\frac {\partial u_i}{\partial
x_k}=\nu_k\frac {\partial u_i}{\partial \nu}\quad\mbox{on}\quad \Gamma\,,\end{equation}
 by Green and Gauss Theorems we have
\begin{eqnarray}\label{eq:3.5}
\lefteqn{-\mu\sum_{i=1}^n\int_Q\Delta u_i(2 \nabla u_i\cdot g)\,dx\,dt} && \\
&=&-\mu\sum_{i=1}^n\int_Q\left|\nabla u_i\right |^2 \diver g\,dx\,dt
+ 2\mu\sum_{i,j,k=1}^n\int_Q\frac {\partial u_i}{\partial x_j}\frac {
\partial u_i}{\partial x_k}\frac {\partial g_k}{\partial x_j}\,dx\,dt
\nonumber \\
&&-\mu\sum_{i=1}^n\int_{\Sigma}\left(\frac {\partial u_i}{\partial
\nu}\right)^2g\cdot\nu\,d\Gamma dt\,.\nonumber
\end{eqnarray}
Finally, by (\ref{eq:3.4}) and by Gauss theorem we have
\begin{eqnarray}\label{eq:3.6}
\lefteqn{-(\lambda +\mu )\sum_{i=1}^n\int_Q\frac {
\partial}{\partial x_i}\left(\alpha (x)\diver u\right)\left(2\nabla
u_i\cdot g\right)\,dx\,dt} && \\
&=&2(\lambda +\mu )\sum_{k=1}^n\int_Q\alpha (x)\diver u\frac {\partial}{
\partial x_k}(\diver u)g_k\,dx\,dt\,\nonumber \\
&&+2(\lambda +\mu )\sum_{i,k=1}^n\int_Q\alpha (x)\diver u\frac {\partial 
u_i}{\partial x_k}\frac {\partial g_k}{\partial x_i}\,dx\,dt\,.\nonumber \\
&&-2(\lambda
+\mu )\int_{\Sigma}\alpha (x)\left(\diver u\right)^2g\cdot\nu\,d\Gamma \,dt
\,.\nonumber \\
&=&-(\lambda +\mu )\int_Q\left(\diver u\right)^2\left(\nabla\alpha
(x)\cdot g\right)dx\,dt-(\lambda +\mu )\int_Q\left(\diver u\right)^2\alpha
(x)\diver g\,dx\,dt\,\nonumber \\
&&+(\lambda +\mu )\int_{\Sigma}\alpha (x)\left(\diver u\right)^2g\cdot
\nu\,d\Gamma dt\,.\nonumber \\
&&+2(\lambda +\mu )\sum_{i,k=1}^n\int_Q\alpha (x)\diver u
\frac {\partial u_i}{\partial x_k}\frac {\partial g_k}{\partial
x_i}\,dx\,dt\,.\nonumber \\
&&-2(\lambda +\mu )\int_{\Sigma}\alpha (x)\left(\diver u\right)^2g\cdot
\nu\,d\Gamma dt\,.\nonumber 
\end{eqnarray}
Replacing (\ref{eq:3.3}), (\ref{eq:3.5}) and (\ref{eq:3.6}) in (\ref{eq:3.2}) 
it follows that
\begin{eqnarray}\label{eq:3.7}
&&2\sum_{i=1}^n\int_{\Omega}u_i'(T)\left
(\nabla u_i(T)\cdot g\right)\,dx-2\sum_{i=1}^n\int_{\Omega}u_i'(0
)\left(\nabla u_i(0)\cdot g\right)dx \\
&&+\sum_{i=1}^n\int_Q\left(u_i'\right)^2\diver g\,dx\,dt-\mu\sum_{i=1}^
n\int_Q\left|\nabla u_i\right|^2 \diver g\,dx\,dt\nonumber \\
&&+2\mu\sum_{i,j,k=1}^n\int_Q\frac {
\partial u_i}{\partial x_j}\frac {\partial u_i}{\partial x_k}\frac {
\partial g_k}{\partial x_j}\,dx\,dt
-\mu\sum_{i=1}^n\int_{\Sigma}\left(\frac {\partial u_i}{\partial
\nu}\right)^2g\cdot\nu\,d\Gamma dt\nonumber \\
&&-(\lambda +\mu )\int_Q\left(\diver 
u\right)^2\left(\nabla\alpha (x)\cdot g\right)dx\,dt
-(\lambda +\mu )\int_Q\left(\diver u\right)^2\alpha (x)\diver g\,
dx\,dt\nonumber \\
&&+2(\lambda +\mu )\sum_{i,k=1}^n\int_Q\alpha (x)\diver u\frac {
\partial u_i}{\partial x_k}\frac {\partial g_k}{\partial x_i}\,dx
dt
-(\lambda +\mu )\int_{\Sigma}\alpha (x)\left(\diver u\right)^2g\cdot
\nu\,d\Gamma dt\nonumber \\
&=&\sum_{i=1}^n\int_Qh_i\left(2\nabla u_i\cdot g\right
)\,dx\,dt\,.\nonumber
\end{eqnarray}
Let $u\diver g=\left(u_1\diver g,\dots ,u_n\diver g\right)$,
where
$u_i\diver g=\sum_{j=1}^nu_i\frac {\partial g_j}{\partial x_j}$.
Now, multiplying (\ref{eq:1.1}) by $u\diver g$ we get
\begin{eqnarray}\label{eq:3.8}
&&\sum_{i=1}^n\int_Qu_i^{\prime\prime}
u_i\diver g\,dx\,dt-\mu\sum_{i=1}^n\int_Q\Delta u_iu_i\diver g\,dx\,
dt\nonumber \\
&&-(\lambda +\mu )\sum_{i=1}^n\int_Q\frac {\partial}{\partial x_i}\left
(\alpha (x) \diver u\right)u_i\diver g\,dx\,dt\\
&=&\sum_{i=1}^n\int_Qh_iu_i\diver g\,dx\,dt\,.\nonumber
\end{eqnarray}
Using integration by parts we obtain
\begin{eqnarray}\label{eq:3.9}
\lefteqn{\sum_{i=1}^n\int_Qu_i^{\prime\prime}u_i\diver g\,dx\,dt} \\
&=&\sum_{i=1}^
n\int_{\Omega}u_i'(T)u_i(T)\diver g\,dx
-\sum_{i=1}^n\int_{\Omega}u_i'(0)u_i(0)\diver g\,dx\,.\nonumber \\
&&-\sum_{i=1}^n\int_Q\left|u_i'\right|^2\diver g\,dx\,dt\,.\nonumber
\end{eqnarray}
and by Green's Theorem
\begin{eqnarray}\label{eq:3.10}
\lefteqn{-\mu\sum_{i=1}^n\int_Q\Delta u_iu_i\diver g\,dx\,dt} \\
&=&\mu\sum_{i=1}^n\int_Q\left|\nabla u_i\right|^2\diver g\,dx\,dt
+\mu\sum_{i=1}^n\int_Q\nabla u_i\cdot\nabla\left(\diver g\right)u_
i\,dx\,dt.\nonumber
\end{eqnarray}
Since $u=0$ on $\Gamma$, by Gauss Theorem we have
\begin{eqnarray}\label{eq:3.11}
\lefteqn{-(\lambda +\mu )\sum_{i=1}^n\int_Q\frac {\partial}{\partial x_i}\left
(\alpha (x)\diver u\right)u_i\diver g\,dx\,dt}\\
&=&(\lambda +\mu )\int_Q\alpha (x)\left(\diver u\right)^2\diver g\,dx
dt\nonumber \\
&&+(\lambda +\mu )\int_Q\alpha (x)\,(\diver u)u\cdot\nabla\left(
\diver g\right)dx\,dt\,.\nonumber
\end{eqnarray}
Then, from (\ref{eq:3.8})-(\ref{eq:3.11}), it follows that
\begin{eqnarray}\label{eq:3.12}
&&\sum_{i=1}^n\int_{\Omega}u_i'(T)u_i(T)\,div \,g \,dx-\sum_{i=1}^n
\int_{\Omega}u_i'(0)u_i(0)\diver g\,dx\nonumber \\
&&-\sum_{i=1}^n\int_Q\left|u_i'\right|^2\diver g\,dx\,dt+\mu\sum_{i=1}^
n\int_Q\left|\nabla u_i\right|^2\diver g\,dx\,dt \\
&&+\mu\sum_{i=1}^n\int_Q\nabla u_i\cdot\nabla\left(\diver g\right)u_
i\,dx\,dt+(\lambda +\mu )\int_Q\alpha (x)\left(\diver u\right)^2\diver 
g\,dx\,dt\nonumber \\
&&+(\lambda +\mu )\int_Q\alpha (x)\,(\diver u)u\cdot\nabla\left(\diver
g\right)dx\,dt\nonumber \\
&&=\sum_{i=1}^n\int_Qh_iu_i\diver g\,dx\,dt\nonumber
\end{eqnarray}
Adding (\ref{eq:3.7}) and (\ref{eq:3.12}), and using that
$\left[H_0^1(\Omega )\cap H^2(\Omega )\right]^n\times V\times
L^1(0,T;V)$ is dense in $V\times H\times L^1(0,T;V)$,
we obtain (\ref{eq:3.1}). \hfil $\Box$

\begin{lemma} Let $u$ be the solution of (\ref{eq:1.1}) with
initial data satisfying (\ref{eq:1.2}), and let $v_r=\theta_ru$,
where $\{\theta_r\}_{1\leq r\leq m}$ is a $C^{\infty}$ partition of the 
unity relative to the open sets $U_1,\dots ,U_m$. Then  $v_r=(v_{r_1},\dots
,v_{r_n})$ is the solution to
\begin{eqnarray}\label{eq:3.13}
&&v_r^{\prime\prime}-\mu\Delta v_r-(\lambda +\mu )\nabla (\alpha (x
)\diver v_r)=h_r\quad\hbox{in } Q\nonumber \\
&&v_r=0\quad on\quad\Sigma\\
&&v_r(0)=\theta_ru^0,\,v_r'(0)=\theta_ru^1\quad on\quad\Omega\,,\nonumber
\end{eqnarray}
where 
$$h_r=\theta_rh-2\mu\nabla\theta_r\cdot\nabla u-\mu u\Delta\theta_
r-(\lambda +\mu )\alpha (x)divu\nabla\theta_r
-(\lambda +\mu )\nabla (\alpha (x)u\cdot\nabla\theta_r)$$
and $\mbox{supp}\,v_r\subset U_r\times [0,T]$.
Furthermore, if $\varepsilon_0$ is the minimum between the two 
epsilons found in Lemmas 2.1 and 2.2, the function
$$
G(\varepsilon )=\left\{\begin{array}{ll}
\frac 1\varepsilon \int_0^T\int_{\omega_{\varepsilon}
\cap U_r} \sum_{i=1}^n|\nabla v_{r_i}(x,t)\cdot\nu (p(x))|^2\,dx\,dt
,\quad &\varepsilon\in ]0,\varepsilon_0]\\
\int_0^T\int_{\Gamma\cap U_r}\sum_{i=1}^n\left|\frac {\partial
v_{r_i}}{\partial\nu}\right|^2d\Gamma \,dt &\varepsilon =0\,,\end{array}
\right.$$
where $F_r(W_r\times ]0,\varepsilon [)=\omega_{\varepsilon}
\cap U_r$, is continuous on $[0,\varepsilon_0]$.
\end{lemma}

\paragraph{Proof.} The continuity of $G$ on $[0,\varepsilon_0]$ 
follows from Lemma 2.2. \hfil $\Box$

\begin{lemma} Let $\delta$ be a positive number. Then there exists a
real number $\gamma$$\in ]0,\delta [$, and there exist positive decreasing
functions $\rho_{\varepsilon}\in W^{2,\infty}(0,\varepsilon )$,  where 
$\varepsilon =\delta +\gamma$,  such that
\begin{equation}\label{eq:3.14}
\rho_{\varepsilon}(\varepsilon )=0
\quad\rho_{\varepsilon}'(\varepsilon )=0\quad\mbox{and}\quad
\rho_{\varepsilon}'=-\frac 1{\delta}\,,\quad \mbox{in}\quad [0,\delta ]\,.\end{equation}
Furthermore,
\begin{equation}\label{eq:3.15}||\rho_{\varepsilon}||_{L^{\infty}
(0,\varepsilon )}\leq C_1,\,\,\,||\rho_{\varepsilon}'||_{L^{\infty}
(0,\varepsilon )}\leq\frac {C_2}{\varepsilon},\,\,||\rho_{\varepsilon}^{
\prime\prime}||_{L^{\infty}(0,\varepsilon )}\leq\frac {C_3}{\varepsilon^
2}\end{equation}
for positive constants $C_1,C_2,C_3$,  and 
\begin{equation}\label{eq:3.16}
\frac {\gamma}2\int_0^{\varepsilon}
|\rho_{\varepsilon}^{\prime\prime}(z)|^2z^2dz=\frac 9{16}\,.\end{equation}
\end{lemma}


\paragraph{Proof.} Let $\gamma$ be the positive value when solving for
$x$ in 
\begin{equation}\label{eq:3.17}1+\frac x{\delta}+\frac 13(\frac x{
\delta})^2=\frac 98\,.
\end{equation}
Then $\gamma =\left(\sqrt {\frac 76}-1\right)\frac 32\delta$ which 
belongs to the interval $]0,\delta [$. 
Put $\varepsilon =\gamma +\delta$, and define 
$\rho_{\varepsilon}:[0,\varepsilon ]\rightarrow R$, by
$$\rho_{\varepsilon}(z)=\left\{\begin{array}{ll}
1-\frac {\gamma}{2\delta}-\frac z{\delta}\,, &z\in [0,\delta]\\
\frac 1{2\gamma\delta}(\delta +\gamma -z)^2\,,&z\in [\delta
,\delta +\gamma ]\,.\end{array}
\right.$$
Then $\rho_{\varepsilon}\in W^{2,\infty}(0,\varepsilon )$ and satisfies 
(\ref{eq:3.14}),(\ref{eq:3.15}).
To show that $\rho_\varepsilon$ satisfies (\ref{eq:3.16}), we use 
(\ref{eq:3.17}) as follows
\begin{eqnarray*}
\frac {\gamma}2\int_0^{\varepsilon}|\rho_{\varepsilon}^{\prime\prime}
(z)|^2z^2\,dz&=&\frac {\gamma}2\int_{\gamma}^{\delta +\gamma}\frac 1{
\gamma^2\delta^2}z^2\,dz\\
&=&\frac 12\left(1+\frac {\gamma}{\delta}+
\frac {\gamma^2}{3\delta^2}\right) \\
&=&\frac9{16}\,.
\end{eqnarray*}

\section{Behaviour of the Solution $u$ in
$\omega_{\varepsilon}\times ]0,T[$}
Our goal in this section, which contains the main result of this work,
is to study the behaviour of the solution $u$ of the elasticity system
given in (\ref{eq:1.1}) in $\omega_{\varepsilon}\times ]0,T[$.

\begin{theorem} There exist positive constants $C$ and $\varepsilon_0$
such that every solution $u$ of (\ref{eq:1.1}) where (\ref{eq:1.2}) 
holds with $\alpha_1<\mu/(3n(\lambda +\mu ))$ satisfies
$$\frac 1{\varepsilon}
\sum_{i=1}^n\int_0^T\int_{\omega_{\varepsilon}}|\nabla u_i(x,t)\cdot
\nu (p(x))|^2\,dx\,dt\leq cE_0\quad \forall\varepsilon\in ]0,\varepsilon_0]
\,.$$
Furthermore, $C$ and $\varepsilon_0$ depend only on the positive number T, 
the function $\alpha (x)$, the geometry of $\Omega$, and the 
Lam\'e constants.  
\end{theorem}  

\paragraph{Proof.} Without loss of generality assume that $\Gamma_0=\Gamma$.
Let $u$ be the solution of (\ref{eq:1.1}) and define $v_r=\theta_
ru$ where $\{\theta_r\}_{1\leq r\leq m}$
is a $C^{\infty}$ partition of the unity relative to the open sets $
U_1,\dots ,U_m$ given in lemma 2.1.  According to Lemma 3.2, $v_r$ is
the solution of (\ref{eq:3.13}) and the function $G$ is continuous
on $[0,\varepsilon_0]$.
Let $\delta_0\in [0,\varepsilon_0]$ be a value such that
$$G(\delta_0)=\max_{\varepsilon\in [0,\varepsilon_0]}G(\varepsilon
)\,.$$
If $\delta_0=0$, we obtain
$$\max_{\varepsilon\in [0,\varepsilon_0]}G(\varepsilon )=\sum_{i=
1}^n\int_0^T\int_{\Gamma\cap U_r}\left|\frac {\partial v_{r_i}}{\partial
\nu}\right|^2d\Gamma \,dt\,.$$
But, considering the trace theory developed by M. Milla
Miranda [9], we get
$$\frac {\partial v_{r_i}}{\partial\nu}=\theta |_{\Gamma}\frac {\partial
u_i}{\partial\nu}$$
and consequently,
$$\max_{\varepsilon\in [0,\varepsilon_0]}G(\varepsilon )\leq c^{*}
\sum_{i=1}^n\int_0^T\int_{\Gamma}\left|\frac {\partial u_i}{\partial\nu}
\right|^2d\Gamma \,dt\,.$$
On the other hand, proceeding as in J. L. Lions [8] (chapter
IV - pages 224, 225), that is, using the multiplier method, we
obtain that $\frac{\partial u_i}{\partial\nu}\in L^2(\Sigma )$, $
i=1,2,...,n$ and
$$\sum_{i=1}^n\int_0^T\int_{\Sigma}\left|\frac {\partial u_i}{\partial
\nu}\right|^2d\Gamma dt\leq cE_0\,.$$
Then, we conclude that
$$\max_{\varepsilon\in [0,\varepsilon_0]}G(\varepsilon )\leq cE_0\,.
$$
If $\delta_0\in ]\frac {\varepsilon_0}2,\varepsilon_0[$, we have
$$\max_{\varepsilon\in [0,\varepsilon_0]}G(\varepsilon )\leq\frac
2{\varepsilon_0}\sum_{i=1}^n\int_0^T\int_{\omega_{\delta_0}\cap U_
r}|\nabla v_{r_i}(x,t)|^2\,dx\,dt\leq cE_0$$
where c, in both cases, is the desired constant.
We can then restrict ourselves to the case $0<\delta_0<\frac {\varepsilon_
0}2$.

As $\delta_0>0$, according to lemma 3.3, there exists a real number
 $\gamma\in ]0,\delta_0[$. Put $\varepsilon =\gamma +\delta_
0$, then there exists a decreasing function
$\rho_{\varepsilon}\in W^{2,\infty}(0,\varepsilon )$ such that
\begin{equation}\label{eq:4.1}
\rho_{\varepsilon}(\varepsilon )=0,
\quad\rho_{\varepsilon}'(\varepsilon )=0,\quad\mbox{and}\quad
\rho_{\varepsilon}'=-\frac 1{\delta_0}\,\,\,in\,\,[0,\delta_0]
\end{equation}
and (\ref{eq:3.15}) and (\ref{eq:3.16}) are satisfied.
Let us consider now the following vector valued function
\begin{equation}\label{eq:4.2}g_{\varepsilon}(x)=\rho_{\varepsilon}
(z)\nu (y),\quad x\in\omega_{\varepsilon}\cap U_r,\quad \mbox{where}\quad
x=y-z\nu (y).\end{equation}
Noting that $\frac {\partial z}{\partial x_k}=-\nu_k$ we have
$$\frac {\partial g_{\varepsilon_j}}{\partial x_k}(x)=-\rho_{\varepsilon}'
(z)\nu_k(p(x))\nu_j(p(x))+\rho_{\varepsilon}(z)\frac {\partial\nu_
j}{\partial x_k}(p(x)),\quad x\in\omega_{\varepsilon}\,.$$
So, we obtain
$$\diver g_{\varepsilon}(x)=-\rho_{\varepsilon}'(z)+\rho_{\varepsilon}
(z)\diver\nu (p(x)),\quad x\in\omega_{\varepsilon}$$
and
$$\nabla (\diver g_{\varepsilon})(x)=\rho_{\varepsilon}^{\prime\prime}
(z)\nu (p(x))-\rho_{\varepsilon}'(z)\diver(\nu (p(x))+\rho_{\varepsilon}
(z)\nabla (\diver\nu (p(x)))\,.$$

Next, we use identity (\ref{eq:3.1}) taking as function g the family of functions
$g_{\varepsilon}$ defined in (4.2), replacing $u$ by $v_r$ the solution of
(\ref{eq:3.13}) and
observing that ${\rm supp}\,v_r\subset U_r\times [0,T].$

From the above equalities, Lemma 2.2, and taking into account that
$F_r(W_r\times ]0,\varepsilon [)=\omega_{\varepsilon}\times U_r$, we get by 
lemma 3.1 the following expression
\begin{eqnarray}\label{eq:4.3}
&&-2\mu\sum_{i=1}^n\int_0^T\int_{W_r}
\int_0^{\varepsilon}\rho_{\varepsilon}'(z)\left|\frac {\partial\hat {v}_{
r_i}}{\partial z}(w,z,t)\right|^2|\det JF_r(w,z)|dz\,dw\,dt\\
&&-\mu\sum_{i=1}^n\int_0^T\int_{W_r}\int_0^{\varepsilon}\rho_{\varepsilon}^{
\prime\prime}(z)\frac {\partial\hat {v}_{r_i}}{\partial z}(w,z,t)
\hat {v}_{r_i}(w,z,t)|\det JF_r(w,z)|dz\,dw\,dt\nonumber \\
&&+(\lambda +\mu )\sum_{i=1}^n\int_0^T\int_{\omega_{\varepsilon}\cap
U_r}\alpha (x)\rho_{\varepsilon}^{\prime\prime}(z)\diver v_r(v_r\cdot
\nu (p(x)))\,dx\,dt\nonumber \\
&&=R_1+R_2\,.\nonumber
\end{eqnarray}
where $R_1$ is given by
\begin{eqnarray*}
&&-2\mu\sum_{i,j,k=1}^n\int_0^T\int_{\omega_{\varepsilon}\cap
U_r}\rho_{\varepsilon}(z)\frac {\partial v_{r_i}}{\partial x_j}\frac {
\partial v_{r_i}}{\partial x_k}\frac {\partial\nu_j}{\partial x_k}
(p(x))\,dx\,dt\\
&&-\mu\sum_{i=1}^n\int_0^T\int_{\omega_{\varepsilon}\cap U_r}\rho_{
\varepsilon}(z)(\nabla v_{r_i}\cdot\nabla (\diver \nu (p(x))))v_{r_i}\,
dx\,dt\\
&&-2(\lambda +\mu )\sum_{i,j=1}^n\int_0^T\int_{\omega_{\varepsilon}
\cap U_r}\alpha (x)\rho_{\varepsilon}(z)\diver v_r\frac {\partial v_{
r_i}}{\partial x_j}\frac {\partial\nu_j}{\partial x_i}(p(x))\,dx\,dt\\
&&-(\lambda +\mu )\int_0^T\int_{\omega_{\varepsilon}\cap U_r}\alpha
(x)\rho_{\varepsilon}(z)\diver v_r(v_r\cdot\nabla (\diver \nu (p(x))))\,dx\,dt\\
&&-2\sum_{i=1}^n\int_0^T\int_{W_r}\int_0^{\varepsilon}\rho_{\varepsilon}
(z)\hat {h}_{r_i}(w,z,t)\frac {\partial\hat {v}_{r_i}}{\partial z}
(w,z,t)\hat {v}_{r_i}(w,z,t)|\det JF_r(w,z)|dz\,dw\,dt\\
&&+\sum_{i=1}^n\int_0^T\int_{W_r}\int_0^{\varepsilon}\rho_{\varepsilon}
(z)\diver \nu (F_r(w,0))\hat {h}_{r_i}(w,z,t)|\det JF_r(w,z)|dz\,dw\,dt\\
&&+\int_{\Sigma}\{\mu\sum_{i=1}^n|\frac {\partial v_{r_i}}{\partial
\nu}|^2+(\lambda +\mu )\alpha (x)|\diver v_r|^2\}\rho_{\varepsilon}(0
)d\Gamma \,dt\\
&&+(\lambda +\mu )\int_0^T\int_{\omega_{\varepsilon\cap U_r}}\rho_{
\varepsilon}(z)(\nabla\alpha (x)\cdot\nu (p(x)))|\diver v_r|^2\,dx\,dt\\
&&+2\sum_{i=1}^n\int_{W_r}\int_0^{\varepsilon}\rho_{\varepsilon}(
z)\hat {v}_{r_i}(w,z,T)\frac {\partial\hat {v}_{r_i}}{\partial z}
(w,z,T)|\det JF_r(w,z)|dz\,dw\\
&&-2\sum_{i=1}^n\int_{W_r}\int_0^{\varepsilon}\rho_{\varepsilon}(
z)\hat {v}_{r_i}(w,z,0)\frac {\partial\hat {v}_{r_i}}{\partial z}
(w,z,0)|\det JF_r(w,z)|dz\,dw\\
&&-\sum_{i=1}^n\int_{\omega_{\varepsilon\cap U_r}}\rho_{\varepsilon}
(z)\diver \nu (p(x))v_{r_i}'(T)v_{r_i}(T)\,dx\\
&&+\sum_{i=1}^n\int_{\omega_{\varepsilon\cap U_r}}\rho_{\varepsilon}
(z)\diver \nu (p(x))v_{r_i}'(0)v_{r_i}(0)\,dx
\end{eqnarray*}
and $R_2$ is given by
\begin{eqnarray*}
&&-\mu\sum_{i=1}^n\int_0^T\int_{W_r}\int_0^{\varepsilon}\rho_{
\varepsilon}'(z)\diver \nu (F_r(w,0))\frac {\partial\hat {v}_{r_i}}{\partial 
z}(w,z,t)\hat {v}_{r_i}(w,z,t,)\times \\
&&\hskip 2.7cm |\det JF_r(w,z)|dz\,dw\,dt\\
&&-2(\lambda +\mu )\sum_{i=1}^n\int_0^T\int_{W_{{\rm r}}}\int_0^{
\varepsilon}\alpha (F_r(w,z))\rho_{\varepsilon}'(z)\diver v_r(F_r(w,z
),t)\frac {\partial\hat {v}_{r_i}}{\partial z}(w,z,t)\times \\
&&\hskip 2.7cm \nu_i(F_r(w,0))|\det JF_r(w,z)|dz\,dw\,dt\\
&&+(\lambda +\mu )\int_0^T\int_{\omega_{\varepsilon}\cap U_r}\alpha
(x)\rho_{\varepsilon}'(z)\diver \nu (p(x))\diver v_r(v_r\cdot\nu (p(x)))\,
dx\,dt\\
&&-\sum_{i=1}^n\int_0^T\int_{W_r}\int_0^{\varepsilon}\rho_{\varepsilon}'
(z)\hat {h}_{r_i}(w,z,t)\hat {v}_{r_i}(w,z,t)|\det JF_r(w,z)|dz\,dw\,dt\\
&&+\sum_{i=1}^n\int_{\omega_{\varepsilon}\cap U_r}\rho_{\varepsilon}'
(z)v_{r_i}'(T)v_{r_i}(T)\,dx\,\,-\,\,\sum_{i=1}^n\int_{\omega_{\varepsilon
\cap U_r}}\rho_{\varepsilon}'(z)v_{r_i}'(0)v_{r_i}(0)\,dx\,.
\end{eqnarray*}
Note that the expression for $R_1$ collects the terms which involves 
$\rho_{\varepsilon}(z)$ and the expression $R_2$ the terms related to 
$\rho_{\varepsilon}^{\prime}(z)$.

Taking into account the regularity of the functions $\alpha$, $\rho_{
\varepsilon}$ and  of the
normal vector $\nu$, and considering that
\begin{equation}\label{eq:4.4}\frac {\partial v_{r_i}}{\partial\nu}
=\theta_r\vert_{\Gamma}\frac {\partial u_i}{\partial\nu}\quad\hbox{
and}\quad \diver v_r\vert_{\Gamma}=\theta_r\vert_{\Gamma}\diver u
\vert_{\Gamma}\end{equation}
we have a positive constant $C_1$ such that $|R_1|\leq C_1E_0$.

On the other hand, observing that:
$$\frac {\partial v_{r_i}}{\partial x_j}(x)=-\frac {\partial\hat {
v}_{r_i}}{\partial z}(w,z)\nu_j(F_r(w,0))+D_w\hat {v}_{r_i}D_{x_j}
w$$
where
$$D_w\hat {v}_{r_i}D_{x_j}w=\sum_{k=1}^{n-1}\frac {\partial\hat {
v}_{r_i}}{\partial w_k}(w,z)\frac {\partial w_k}{\partial x_j},$$
we can write
\begin{equation}\label{eq:4.5}\diver v_r(x)=\sum_{j=1}^n\{-\frac {\partial
\hat {v}_{r_j}}{\partial z}(w,z)\nu_j(F_r(w,0))+D_w\hat {v}_{r_j}
D_{x_j}w\}.\end{equation}
Using the H\"older inequality, we have
$$|\hat {v}_{r_i}(w,z,t)|^2\leq z\int_0^z\left|\frac {\partial\hat {v}_{
r_i}}{\partial z}(w,s,t)\right|^2ds.$$
From this inequality, and using in (\ref{eq:4.5}) an analogous
argument to the one used by C. Fabre and J. P. Puel in [6] (page 196) 
we conclude that there exists $C_2>0$ such that
\begin{equation}\label{eq:4.6}|R_2|\leq C_2E_0.\end{equation}

From (\ref{eq:4.3}), (\ref{eq:4.4}) and (\ref{eq:4.6}) we find a positive 
constant $C_3$ independent
of $\varepsilon$, $\delta_0$ and $u$ for which, 
\begin{eqnarray}\label{eq:4.7}
\lefteqn{-2\mu\sum_{i=1}^n\int_0^T\int_{W_r}
\int_0^{\varepsilon}\rho_{\varepsilon}'(z)\left|\frac {\partial\hat {v}_{
r_i}}{\partial z}(w,z,t)\right|^2|\det JF_r(w,z)|dz\,dw\,dt}&& \\
&\leq&\mu\sum_{i=1}^n\int_0^T\int_{W_r}\int_0^{\varepsilon}\rho_{
\varepsilon}^{\prime\prime}(z)\frac {\partial\hat {v}_{r_i}}{\partial
z}(w,z,t)\hat {v}_{r_i}(w,z,t)|\det JF_r(w,z)|\,dz\,dw\,dt\nonumber \\
&&-(\lambda +\mu )\int_0^T\int_{\omega_{\varepsilon}\cap U_r}\alpha
(x)\rho_{\varepsilon}^{\prime\prime}(z)\diver v_r(v_r\cdot\nu (p(x)))\,
dx\,dt\,+\,C_3E_0\,.\nonumber
\end{eqnarray}
But using (\ref{eq:4.5}) we have
\begin{eqnarray}\label{eq:4.8}
\lefteqn{-(\lambda +\mu )\int_0^T\int_{\omega_{
\varepsilon}\cap U_r}\alpha (x)\rho_{\varepsilon}^{\prime\prime}(
z)\diver v_r(v_r\cdot\nu (p(x)))\,dx\,dt} && \\
&=&(\lambda +\mu )\sum_{i,j=1}^n\int_0^T\int_{W_r}\int_0^{\varepsilon}
\alpha (F_r(w,z))\rho_{\varepsilon}^{\prime\prime}(z)\frac {\partial
\hat {v}_{r_i}}{\partial z}(w,z,t)\nu_i(F_r(w,0))\hat {v}_{r_j}(w
,z,t)\times \nonumber \\
&&\hskip 3.8cm \nu_j(F_r(w,0))|\det JF_r(w,z)|\,dz\,dw\,dt\nonumber \\
&&-(\lambda +\mu )_{}\int_0^T\int_{W_r}\int_0^{\varepsilon}\alpha
(F_r(w,z))\rho_{\varepsilon}^{\prime\prime}(z)\sum_{i=1}^nD_w\hat {
v}_{r_i}D_{x_i}w\times \nonumber \\
&&\hskip 3.8cm  (\hat {v}_{r_{}}(w,z,t)\cdot\nu (F_r(w,0)))|\det JF_{{\rm r}}
(w,z)|\,dz\,dw\,dt\,.\nonumber
\end{eqnarray}
Using the Cauchy-Schwarz inequality and (\ref{eq:4.1}) we prove that
\begin{eqnarray}\label{eq:4.9}
&&(\lambda +\mu )\sum_{i,j=1}^n\int_0^
T\int_{W_r}\int_0^{\varepsilon}\alpha (F_r(w,z))\rho_{\varepsilon}^{
\prime\prime}(z)\frac {\partial\hat {v}_{r_i}}{\partial z}(w,z,t)
\nu_i(F_r(w,0))\times \nonumber \\
&&\hskip 3cm  \hat {v}_{r_j}(w,z,t)\nu_j(F_r(w,0))|\det JF_r(w,z)|\,dz\,dw\,dt
\\
&\leq &(\lambda +\mu )\alpha_1n\{\frac 1{\gamma}\int_0^T\int_{W_r}
\int_{\delta_0}^{\varepsilon}\sum_{i=1}^n\left|\frac {\partial\hat {v}_{
r_i}}{\partial z}(w,z,t)\right|^2|\det JF_r(w,z)|\,dz\,dw\,dt\}^{1/2}
\times \nonumber \\
&& \hskip 1cm \{\gamma\int_0^T\int_{W_r}\int_{\delta_0}^{\varepsilon}\sum_{
i=1}^n|\rho_{\varepsilon}^{\prime\prime}(z)|^2|\hat {v}_{r_i}(w,z
,t)|^2|\det JF_r(w,z)|\,dz\,dw\,dt\}^{1/2}\nonumber \\
&\leq& (\lambda +\mu )\alpha_1nG(\delta_0)^{1/2}(\frac 94G(\delta_
0)+C_4E_0)^{1/2}\,,\nonumber
\end{eqnarray}
where the last inequality comes from (\ref{eq:3.16}) and the identity
\begin{eqnarray*}
\lefteqn{\hat {v}_{r_j}(w,z,t)|\det JF_r(w,z)|^{1/2} }\\
&=&\int_0^z\frac {\partial
\hat {v}_{r_j}}{\partial z}(w,s,t)|\det JF_r(w,z)|^{1/2}ds
+\int_0^z\hat {v}_{r_j}(w,s,t)\frac {\partial}{\partial z}(|\det 
JF_r(w,s)|^{1/2})ds\,.
\end{eqnarray*}
Now, using an analogous argument to the one used in C. Fabre and J.P. Puel
[6](page 197) we get
\begin{eqnarray} \label{eq:4.10}
&(\lambda +\mu )|\sum_{i=1}^n\int_0^T\int_{W_r}\int_0^{\varepsilon}
\alpha (F_r(w,z))\rho_{\varepsilon}^{\prime\prime}(z)[D_w\hat {
v}_{r_i}D_{x_i}w]\times& \nonumber \\
& (\hat {v}_r(w,z,t)\cdot\nu (
F_r(w,0)))|\det JF_r(w,z)|\,dz\,dw\,dt\;|\leq C_5E_0\,.&
\end{eqnarray}
So, by (\ref{eq:4.8})-(\ref{eq:4.10}) we have
\begin{eqnarray}\label{eq:4.11}
\lefteqn{-(\lambda +\mu )\int_0^T\int_{\omega_{\varepsilon}\cap U_r}\alpha
(x)\rho_{\varepsilon}^{\prime\prime}(z)\diver v_r(v_r\cdot\nu (p(x)))\,
dx\,dt}&& \\
&\leq&\frac 32(\lambda +\mu )\alpha_
1nG(\delta_0)+C_6(\lambda +\mu )\alpha_1nG(\delta_0)^{\frac 12}\,
E_0^{\frac 12}+C_7E_0\,.\nonumber
\end{eqnarray}
Proceeding in the same way we did to obtain (\ref{eq:4.9}) we get
\begin{eqnarray}\label{eq:4.12}
&&\mu\sum_{i=1}^n\int_0^T\int_{W_r}\int_
0^{\varepsilon}\rho_{\varepsilon}^{\prime\prime}(z)\frac {\partial
\hat {v}_{r_i}}{\partial z}(w,z,t)\hat {v}_{r_i}(w,z,t)|\det JF_r(w
,z)|\,dz\,dw\,dt \nonumber \\
&&\leq\frac 32\mu G(\delta_0)+C_8\mu G(\delta_0)^{1/2}E_0^{1/2}\,. 
\end{eqnarray}
Finally, using the fact that $\rho_{\varepsilon}'(z)\leq 0$ we have
\begin{eqnarray}\label{eq:4.13}
&&-2\mu\sum_{i=1}^n\int_0^T\int_{W_r}
\int_0^{\varepsilon}\rho_{\varepsilon}'(z)\left|\frac {\partial\hat {v}_{
r_i}}{\partial z}(w,z,t)\right|^2|\det JF_r(w,z)|\,dz\,dw\,dt \nonumber \\
&&\geq 2\mu G(\delta_0)\,.
\end{eqnarray}
Replacing the inequalities (\ref{eq:4.11})-(\ref{eq:4.13}) in (\ref{eq:4.7}) we obtain
$$2\mu G(\delta_0)\left[\frac 14-\frac 34n\frac {(\lambda +\mu )}{\mu}
\alpha_1\right]\leq C_9G(\delta_0)^{1/2}\,E_0^{1/2}+C_{10}E_0\,
.$$
Since $\alpha_1<\frac {\mu}{3n(\lambda +\mu )}$, there exists $
\zeta_0>0$ such that
$$0<\zeta_0<\frac 14-\frac 34n\frac {(\lambda +\mu )}{\mu}\alpha_
1$$
and consequently there exists $C>0$, independent of $\varepsilon$, $
u$ and $\delta_0$ such that
$$\max_{\varepsilon\in [0,\varepsilon_0]}G(\varepsilon )\leq CE_0\,
.$$
So, in any case, there exists $C>0$ independent of $\varepsilon$, $
u$ and $\delta_0$ such that
\begin{eqnarray*}
\lefteqn{\frac 1{\varepsilon}\sum_{i=1}^n\int_0^T\int_{\omega_{\varepsilon}
\cap U_r}|\nabla v_{r_i}(x,t)\cdot\nu (p(x))|^2\,dx\,dt}&& \\
&\leq& C\{||u_0||^2_V+|u_1|_H^2+||h||^2_{L_{}^1(0,T,H)}\},\quad\forall
\varepsilon\in ]0,\varepsilon_0]\,.
\end{eqnarray*}
But since $u=\sum_{r=1}^mv_r$, Theorem 4.1 is proved. \hfil$\Box$

From the above theorem we obtain the following result. 
  
\begin{theorem} There exist positive constants $C$ and $\varepsilon_0$
such that every solution $u$ of (\ref{eq:1.1}) where
(\ref{eq:1.2}) holds with $\alpha_1\leq\frac {\mu}{3n(\lambda +\mu
)}$ satisfies
$$\frac 1{\varepsilon^
3}\sum_{i=1}^n\int_0^T\int_{\omega_{\varepsilon}}|u_i(x,t)|^2\,dx\,
dt\leq CE_0\,, \forall\varepsilon\in ]0,\varepsilon_0]\,.$$
Furthermore,  $C$ and $\varepsilon_0$
depend only  on the real positive number $T$, the function
$\alpha (x)$, the geometry of $\Omega$, and the Lam\'e constants.
\end{theorem}   

\paragraph{Proof.} Let $u=(u_1,\dots,u_n)$ be the solution of (\ref{eq:1.1})
with (\ref{eq:1.2}) and let $\varepsilon_0$ be the minimum between the two 
epsilons found in Lemmas 2.1 and 2.2.  Considering ${{\{\theta_r\}}_{1 \leq r \leq m}}$
 a $C^\infty$ partition of the unity relative to the open sets $U_1,\dots,U_m$ 
given in Lemma 2.1 we obtain $u=\sum_{r=1}^m{u \theta_r}$.
Then, for every $\varepsilon\in ]0,\varepsilon_0]$ we obtain
\begin{eqnarray}\label{eq:4.14}
\int_0^T\int_{\omega_{\varepsilon}}|u_i(x,t)|^2\,dx\,dt
&=&\int_0^T\int_{\omega_{\varepsilon}}|\sum_{r=1}^mu_i\theta_r|^2\,dx\,dt
\nonumber \\
&\leq& C_{11}\sum_{r=1}^m\int_0^T\int_{\omega_{\varepsilon}\cap U_
r}|u_i|^2|\theta_r|^2\,dx\,dt \nonumber \\
&\leq& C_{11}\sum_{r=1}^m\int_0^T\int_{\omega_{
\varepsilon}\cap U_r}|u_i|^2\,dx\,dt\nonumber \\
&=&C_{11}\sum_{r=1}^m\int_0^T\int_{W_r}\int_0^{\varepsilon}|\hat {
u}_i(w,z,t)|^2|\det JF_r(w,z)|\,dz\,dw\,dt\nonumber \\
&\leq& C_{12}\sum_{r=1}^m\int_0^T\int_{W_r}\int_0^{\varepsilon}|\hat {u}_i(w,z,t)|^2
\,dz\,dw\,dt\,,
\end{eqnarray}
where $\hat {u}_i(w,z,t)= u_i(F_r(w,z),t)$.
Since
$$\hat {u}_i(w,z,t)=\int_0^z\frac {\partial\hat {u}_i}{\partial s}
(w,s,t)\,ds\,,$$
it follows that
$$|\hat {u}_i(w,z,t)|^2\leq z\int_0^z\left|\frac {\partial\hat {u}_i}{
\partial s}(w,s,t)\right|^2ds\,.$$
Consequently from (\ref{eq:4.14}) we have 
\begin{eqnarray*}
\lefteqn{\int_0^T\int_{\omega_{\varepsilon}}|u_i(x,t)|^2\,dx\,dt }&&\\
&\leq& C_{12}
\sum_{r=1}^m\int_0^T\int_{W_r}\int_0^{\varepsilon}(z\int_0^z\left|\frac {
\partial\hat {u}_i}{\partial s}(w,s,t)\right|^2\,ds)dz\,dw\,dt  \\
&\leq& C_{12}\sum_{r=1}^m\int_0^T\int_{W_r}\int_0^{\varepsilon}z\int_
0^{\varepsilon}\left|\frac {\partial\hat {u}_i}{\partial s}(w,s,t)\right|^2\,
ds\,dz\,dw\,dt  \\
&=&C_{12}(\sum_{r=1}^m\int_0^T\int_{W_r}\int_0^{\varepsilon}\left|\frac {
\partial\hat {u}_i}{\partial s}(w,s,t)\right|^2\,ds\,dw\,dt)(\int_0^{\varepsilon}
z\,dz) \\
&=&C_{13}\varepsilon^2\sum_{r=1}^m\int_0^T\int_{W_r}\int_0^{\varepsilon}
|\nabla u_i(F_r(w,s),t)\cdot\nu (F_r(w,0))|^2\,ds\,dw\,dt
\end{eqnarray*}
where the last equality comes from (\ref{eq:2.1}).
Therefore,
\begin{eqnarray*}
\int_0^T\int_{\omega_{\varepsilon}}|u_i(x,t)|^2\,dx\,dt 
&\leq& C_{14} \varepsilon^2\sum_{r=1}^m\int_0^T\int_{\omega_{\varepsilon}\cap U_
r}|\nabla u_i(x,t)\cdot\nu (p(x))|^2\,dx\,dt \\
&\leq &C_{14}\varepsilon^2\int_0^T\int_{\omega_{\varepsilon}}|\nabla 
u_i(x,t)\cdot\nu (p(x))|^2\,dx\,dt\,. 
\end{eqnarray*}
Then, we obtain the desired result from Theorem 4.1, in view 
of 
$$\frac 1{\varepsilon^3}\int_0^T\int_{\omega_{\varepsilon}}|u_i(x
,t)|^2\,dx\,dt\leq C_{14}\frac 1{\varepsilon}\int_0^T\int_{\omega_{
\varepsilon}}|\nabla u_i(x,t)\cdot\nu (p(x))|^2\,dx\,dt.
$$

\paragraph{Acknowledgments.} I want to express my sincere gratitude
to my adviser Professor Manuel Milla Miranda, to Luiz Adauto Medeiros
for his stimulating discussions, and to  Jean
Pierre Puel and Caroline Fabre for their helpful comments. I should also thank
God, my family and my friends for the strength received.


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\end{thebibliography}
\bigskip
{\sc Val\'eria Neves Domingos Cavalcanti\\
Universidade Estadual de Maring\'a \\
87020-900 Maring\'a-PR, Brazil} \\
Email address: valeria@gauss.dma.uem.br 
\end{document}