\documentclass[reqno]{amsart} \usepackage{graphicx} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 59, pp. 1--13.\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 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/59\hfil Elliptic problems with small coefficients] {Asymptotic behavior of elliptic boundary-value problems with some small coefficients} \author[S. Guesmia\hfil EJDE-2008/59\hfilneg] {Senoussi Guesmia} \address{Senoussi Guesmia \newline Service de M\'{e}trologie Nucleaire\\ Universit\'{e} Libre de Bruxelles \\ C. P. 165, 50, Av. F. D. Roosevelt, B-1050 Brussels, Belgium \newline Laboratoire Math\'{e}matiques, Informatique et Applications (MIA)\\ 4, rue des Fr\`{e}res Lumi\`{e}re 68093 Mulhouse CEDEX France} \email{senoussi.guesmia@uha.fr, sguesmia@ulb.ac.be} \thanks{Submitted November 16, 2007. Published April 18, 2008.} \subjclass[2000]{35B25, 35B40, 35J25} \keywords{Elliptic problem; singular perturbations; asymptotic behavior} \begin{abstract} The aim of this paper is to analyze the asymptotic behavior of the solutions to elliptic boundary-value problems where some coefficients become negligible on a cylindrical part of the domain. We show that the dimension of the space can be reduced and find estimates of the rate of convergence. Some applications to elliptic boundary-value problems on domains becoming unbounded are also considered. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} We study the asymptotic behavior of the solutions of elliptic boundary-value problems, posed on bounded domains of $\mathbb{R}^{n}=\mathbb{R}^{p}\times\mathbb{R}^{n-p}$ with cylindrical part, where the coefficients and the domains depend on a parameter $\theta $. We show under certain conditions on the coefficients that the solution of such problems converges towards a solution of another elliptic problem in $\mathbb{R}^{n-p}$, faster than any power of $\theta $ on the cylindrical part. More specifically, we are interested in problems invariant by translations (cylindrical symmetry) arbitrary in $p$ directions, and we compare the solution of our problem with that of an ideal problem independent of the coordinates associated with these $p$ directions. This study was inspired to us, on one hand by the theory of "Singular Perturbation" of boundary problem, which is the framework of this paper, and on the other hand by the ideas and the tools given in some works of Chipot and Rougirel (see \cite{ch}, \cite{rougirel}) where another study of the asymptotic behavior of elliptic boundary-value problems on domains becoming unbounded is given. We would like to note that is difficult to locate similar studies in the literature, except some examples studied in \cite{LION 3} and recently some cases have been considered in \cite{sen1} and \cite{guesmia}. The paper is organized as follows: In the second section, we give some useful lemmas which will be used in the following sections. We show the main theorem in the third section where we investigate the rate of convergence estimates. Next, in the fourth section, we apply this result to the asymptotic behavior of the solutions of elliptic problems on domains becoming unbounded in one or several directions and we extend some results of \cite{ch} and \cite{rougirel} for more general domains. In the last section, we give the rate of convergence according to the size of the domain in all directions. Let $( \Omega _{\theta }) _{\theta >0}$ be a family of bounded Lipschitz domains of $\mathbb{R}^{n}$, satisfying \begin{equation} \Delta \times \omega \subset \Omega _{\theta },\quad \Delta \times \partial \omega \subset \partial \Omega _{\theta },\quad P_{X_{2}}\Omega _{\theta }\subset \omega _{0}, \label{fe01} \end{equation} where $\omega _{0}$ and $\omega $ are two bounded Lipschitz domains of $\mathbb{R}^{n-p}$, $\Delta $ is a bounded Lipschitz domain of $\mathbb{R}^{p}$, $n$ and $p$ two positive integers with $n>$ $p\geq 1$ and $P_{X_{i}}$ the projection on the $X_{i}$ axis, such that for $x=(x_{1},x_{2},\dots ,x_{n})\in \mathbb{R}^{n}$, we set $X_{1}=(x_{1},\dots ,x_{p})$ and $X_{2}=(x_{p+1},\dots ,x_{n})$. \begin{figure}[ht] \begin{center} \includegraphics[width=0.4\textwidth]{fig1} \end{center} \label{fig1} \caption{The domain $\Omega _{\theta }$.} \end{figure} We would like to consider the following three boundary-value problems \begin{equation} \begin{gathered} \sum_{i,j=1}^{n}-\partial _{i}(a_{ij}^{\theta }\partial _{j}u)+a_{0}u=f \quad\text{in }\Omega _{\theta } \\ u=0\quad \text{on }\partial \Omega _{\theta }, \end{gathered} \label{fe1} \end{equation} \begin{equation} \begin{gathered} \sum_{i,j=p+1}^{n}-\partial _{i}(a_{ij}\partial _{j}u) +a_{0}u=f \quad \text{in }\omega \\ u=0 \quad \text{on }\partial \omega , \end{gathered} \label{fe2} \end{equation} and \begin{equation} \begin{gathered} \sum_{i,j=p+1}^{n}-\partial _{i}(a_{ij}\partial _{j}u) +a_{0}u=h \quad \text{in }\omega _{0} \\ u=0 \quad \text{on }\partial \omega _{0} \end{gathered} \label{fe2b} \end{equation} where $\theta $ is a positive parameter. Since we are interested in $\theta $ close to $0$, we can take $\theta <1$. Assume that \begin{equation} f,h\in L^{2}(\omega _{0}). \label{fe3} \end{equation} Consider then \begin{equation} a_{ij}^{\theta }\in L^{\infty }(P_{X_{1}}\Omega _{\theta }\times \omega _{0}), \label{fe4} \end{equation} for all $i,j=1,\dots,n$. \begin{remark} \rm We can only suppose that \[ a_{ij}^{\theta }\in L^{\infty }(\Omega _{\theta }), \quad \text{for }j=1,\dots,n \] and we extend the coefficients on $P_{X_{1}}\Omega _{\theta }\times \omega _{0}$, keeping the assumptions below. \end{remark} Assume that the coefficients $a_{ij}^{\theta }$ are independent of $X_{1}$ for $j\geq p+1$, and independent of $\theta $ for $i\geq p+1$ and $j\geq p+1$, i.e. \begin{gather} a_{ij}^{\theta }(x)=a_{ij}^{\theta }(X_{2})\quad \text{for }j\geq p+1 \label{fe5} \\ a_{ij}^{\theta }(x)=a_{ij}(X_{2})\quad \text{for }i\geq p+1,\; j\geq p+1. \label{fe6} \end{gather} Furthermore, we assume the ellipticity condition; i.e., there exist a constant $\lambda >0$, such that \begin{equation} \sum_{i,j=1}^{n}a_{ij}^{\theta }(x)\xi _{i}\xi _{j}\geq \lambda \theta | \boldsymbol{\xi}^{1}| ^{2}+\lambda | \boldsymbol{\xi}^{2}| ^{2},\quad \text{ a.e. }x\in P_{X_{1}}\Omega _{\theta }\times \omega _{0}, \; \forall \xi \in \mathbb{R}^{n}, \label{felliptic} \end{equation} where $\boldsymbol{\xi}^{1}=( \xi _{1},\dots ,\xi _{p}) $ and $\boldsymbol{\xi}^{2}=( \xi _{p+1},\dots ,\xi _{n}) $. Consequently, \begin{equation} \sum_{i,j=p+1}^{n}a_{ij}(x)\xi _{i}\xi _{j}\geq \lambda | \xi | ^{2},\quad \text{a.e. }x\in \omega _{0},\;\forall \xi \in \mathbb{R}^{n-p}. \label{felliptic2} \end{equation} In addition, we suppose that there exist constants $\alpha $ $(0<\alpha \leq 1/2)$ and $C>0$, such that \begin{gather} | a_{ij}^{\theta }(x)| \leq C\theta ^{\frac{1}{2} +\alpha }\quad \text{for }i\leq p,\; j\leq p \label{fe11} \\ | a_{ij}^{\theta }(x)| \leq C\theta ^{\alpha } \quad \text{for }i\geq p+1,\; j\leq p\text{ or }i\leq p,\;j\geq p+1 \label{fe12} \end{gather} a.e. $x\in \Delta \times \omega $. The existence of the term $a_{0}$ does not have any influence on the final result, then we put $a_{0}=0$. \begin{remark} \label{rmk1.2} \rm As a model example, we consider the singularly perturbed Laplacian problem, defined on a cylindrical domain $\Omega _{\theta }=\Delta \times \omega $, \begin{gather*} -\theta \Delta _{X_{1}}u-\Delta _{X_{2}}u=f \quad \text{in }\Omega _{\theta } \\ u=0\quad \text{on }\partial \Omega _{\theta }. \end{gather*} \end{remark} The variational problems corresponding to (\ref{fe1}), (\ref{fe2}) and (\ref{fe2b}) are \begin{equation} \begin{gathered} a(u,v)=\int_{\Omega _{\theta }}\sum_{i,j=1}^{n}a_{ij}^{\theta }(x)\partial _{j}u_{\theta }\partial _{i}vdx=\int_{\Omega _{\theta }}fvdx, \\ u,v\in H_{0}^{1}(\Omega _{\theta }), \end{gathered} \label{fe7} \end{equation} \begin{equation} \begin{gathered} a_{\omega }(u,v)=\int_{\omega }\sum_{i,j=p+1}^{n}a_{ij}(X_{2})\partial _{j}u_{\infty }\partial _{i}vdX_{2}=\int_{\omega }fvdX_{2}, \\ u,v\in H_{0}^{1}(\omega ), \end{gathered} \label{fe8} \end{equation} and \begin{equation} \begin{gathered} a_{\omega _{0}}(u,v)=\int_{\omega _{0}}\sum_{i,j=p+1}^{n}a_{ij}(X_{2})\partial _{j}u_{h}\partial _{i}vdX_{2}=\int_{\omega _{0}}hvdX_{2}, \\ u,v\in H_{0}^{1}(\omega _{0}). \end{gathered} \label{fe8b} \end{equation} According to the Lax-Milgram theorem, the existence and the uniqueness of the solution $u_{\theta }$ in $H_{0}^{1}(\Omega _{\theta })$ of the problem (\ref{fe7}), the solution $u_{\infty }$ in $H_{0}^{1}(\omega )$ of the problem (\ref{fe8}) and the solution $u_{h}$ in $H_{0}^{1}(\omega _{0})$ of the problem (\ref{fe8}) are assured. First of all, we need to introduce some preliminary results. \section{Some estimates} We start with the following Lemmas which will be used frequently in this paper. \begin{lemma}\label{restriction} Let $v$ be an element of $H_{0}^{m}(\Omega _{\theta })$. Then \begin{gather} v(X_{1},.) \in H_{0}^{m}(\omega _{0})\quad \text{a.e. }X_{1}\in P_{X_{1}}\Omega _{\theta }, \label{fe41} \\ v(X_{1},.) \in H_{0}^{m}(\omega )\quad \text{a.e. }X_{1}\in \Delta . \label{fe041} \end{gather} \end{lemma} \begin{proof} By the density of $\mathcal{D}(\Omega _{\theta }) $ in $H_{0}^{m}(\Omega _{\theta })$, there exists a sequence $\phi _{n}$ of $\mathcal{D}(\Omega _{\theta }) $, such that \[ \int_{\Omega _{\theta }}\nabla (v-\phi _{n})dx \to 0 \quad \text{as $n\to \infty$}. \] We extend $v$ and $\phi _{n}$ by $0$ on $P_{X_{1}}\Omega _{\theta }\times \omega _{0}$ $( \Omega _{\theta }\subset P_{X_{1}}\Omega _{\theta }\times \omega _{0}) $, then we have $\phi _{n}\in \mathcal{D}( P_{X_{1}}\Omega _{\theta }\times \omega _{0})$, $v\in H_{0}^{m}(P_{X_{1}}\Omega _{\theta }\times \omega _{0})$ and \[ \int_{P_{X_{1}}\Omega _{\theta }}\int_{\omega _{0}}| \nabla (v-\phi _{n})| ^{2}dx \to 0 \quad \text{as $n\to\infty$}. \] We can extract a subsequence $\phi _{n_{k}}$, such that as $k\to\infty$: \begin{gather*} \int_{\omega _{0}}| \nabla _{X_{2}}(v-\phi _{n_{k}})| ^{2}dx \to 0\quad \text{ a.e. }X_{1}\; \in P_{X_{1}}\Omega _{\theta }, \\ \int_{\omega }| \nabla _{X_{2}}(v-\phi_{n_{k}})| ^{2}dx \to 0 \quad \text{a.e. }X_{1} \in \Delta , \end{gather*} which give (\ref{fe41}) and (\ref{fe041}). \end{proof} \begin{lemma}\label{cas positif} Under the preceding hypotheses, we assume that $h=f\geq 0$ (resp. $h=f\leq 0$). Then we have \[ 0\leq u_{\theta }\leq u_{h},\quad (\text{resp. }u_{h}\leq u_{\theta }\leq 0). \] \end{lemma} \begin{proof} We apply the weak maximal principle for elliptic problems (see \cite{ordre2}) to obtain the inequalities $u_{\theta }\geq 0$ and $u_{h}\geq 0$. For the second inequality, if we use (\ref{fe41}) we can take $v\in H_{0}^{1}(\Omega _{\theta })$ in \eqref{fe8b} and integrate on $P_{X_{1}}\Omega _{\theta }$, to get \[ \int_{\Omega _{\theta }}\sum_{i,j=p+1}^{n}a_{ij}(x)\partial _{j}u_{h}\partial _{i}vdx=\int_{\Omega _{\theta }}fvdx, \] because $v$ vanishes in the exterior of $\Omega _{\theta }$. By comparison with \eqref{fe7}, we deduce \[ \int_{\Omega _{\theta }}\sum_{i,j=1}^{n}a_{ij}^{\theta }(x)\partial _{j}u_{\theta }\partial _{i}vdx=\int_{\Omega _{\theta }}\sum_{i,j=p+1}^{n}a_{ij}(x)\partial _{j}u_{h}\partial _{i}vdx. \] Taking into account the independence of $u_{\infty }$ on $X_{1}$, we deduce \begin{align*} \int_{\Omega _{\theta }}\sum_{i,j=1}^{n}a_{ij}^{\theta }(x)\partial _{j}(u_{\theta }-u_{h})\partial _{i}vdx &= \int_{\Omega _{\theta }}\sum_{\substack{ 1\leq i\leq p \\ p+1\leq j\leq n}}^{n}a_{ij}^{\theta }(x)\partial _{j}u_{h}\partial _{i}vdx \\ &=\int_{\Omega _{\theta }}\sum_{\substack{ 1\leq i\leq p \\ p+1\leq j\leq n}}^{n}\partial _{i}(a_{ij}^{\theta }(x)\partial _{j}u_{h}v)dx\\ &=\int_{\partial \Omega _{\theta }}\sum_{\substack{ 1\leq i\leq p \\ p+1\leq j\leq n}} ^{n}a_{ij}^{\theta }(x)\partial _{j}u_{h}v\nu _{i}dx, \end{align*} because $a_{ij}^{\theta }$ is independent of $X_{1}$ for $1\leq i\leq p$ and $p+1\leq j\leq n$. Then, since $v$ vanishes on the boundary, we deduce that \begin{equation} \int_{\Omega _{\theta }}\sum_{i,j=1}^{n}a_{ij}^{\theta }(x)\partial _{j}(u_{\theta }-u_{h})\partial _{i}vdx=0, \label{fe42} \end{equation} for all $v\in H_{0}^{1}(\Omega _{\theta })$. On the other hand, Theorem 2.8 in \cite{CHI} shows that \[ \gamma [ ( u_{\theta }-u_{h}) ^{+}] =[ \gamma ( u_{\theta }-u_{h})] ^{+}. \] Then since $u_{\theta }\in H_{0}^{1}(\Omega _{\theta })$ and $u_{h}\geq 0$, we have \[ \gamma [( u_{\theta }-u_{h}) ^{+}] =0, \] which allows us to take $v=( u_{\theta }-u_{h}) ^{+}\in H_{0}^{1}(\Omega _{\theta })$ in \eqref{fe42}, then we get \begin{align*} &\int_{\Omega _{\theta }}\sum_{i,j=1}^{n}a_{ij}^{\theta }(x)\partial _{j}(u_{\theta }-u_{h})\partial _{i}( u_{\theta }-u_{\infty })^{+}vdx \\ &=\int_{u_{\theta }-u_{h}\geq 0}\sum_{i,j=1}^{n}a_{ij}^{\theta }(x)\partial _{j}(u_{\theta }-u_{h})\partial _{i}( u_{\theta }-u_{h})^{+}vdx=0. \end{align*} By the ellipticity assumption \eqref{felliptic}, it follows that \[ | \nabla (u_{\theta }-u_{h})^{+}| _{L^{2}(\Omega _{\theta }) }^{2}\leq 0. \] Therefore, $(u_{\theta }-u_{h})^{+}=$const and $(u_{\theta}-u_{h})^{+}\in H_{0}^{1}(\Omega _{\theta })$, then we have $(u_{\theta }-u_{h})^{+}=0$, which gives the second inequality $u_{\theta }\leq u_{\infty }$. For the second case when $f\leq 0$, it is enough to take $-f$ in place of $f$ above. \end{proof} Let $u_{+}$ (resp. $u_{-}$) be the solution of \eqref{fe8b} replacing $h$ by $f^{+}$ (resp. $-f^{-}$). \begin{lemma}\label{cas gen} Under the preceding assumptions, we have \[ u_{-}\leq u_{\theta }\leq u_{+}. \] \end{lemma} \begin{proof} Let $u_{\theta ,+}$ (resp. $u_{\theta ,-}$) be the solution of \eqref{fe7} replacing $f$ by $f^{+}$ (resp. $-f^{-}$). Let us notice that \[ -f^{-}\leq f\leq f^{+},\quad f^{+}\geq 0,\quad -f^{-}\leq 0 \] a.e. $x\in \omega _{0}$, then applying the weak maximal principle for elliptic problems, we get \[ u_{\theta ,-}\leq u_{\theta }\leq u_{\theta ,+}. \] If we use lemma \ref{cas positif}, we obtain $u_{-}\leq u_{\theta ,-}$, $u_{\theta ,+}\leq u_{+}$. This completes the proof. \end{proof} Next, we show the convergence of $u_{\theta}$ to $u_{\infty }$ and we estimate the rate of this convergence. \section{Asymptotic behavior} According to Lemma \ref{restriction}, testing (\ref{fe8}) with $v\in H_{0}^{1}(\Delta\times \omega )$ and integrating on $\Delta $ yields \[ \int_{\Omega _{\theta }}\sum_{i,j=p+1}^{n}a_{ij}^{\theta }(x)\partial _{j}u_{\infty }\partial _{i}vdx=\int_{\Omega _{\theta }}fvdx, \] because $v$ vanishes in the exterior of $\Delta \times \omega $. By \eqref{fe7}, we remark that \[ \int_{\Omega _{\theta }}\sum_{i,j=1}^{n}a_{ij}^{\theta }(x)\partial _{j}u_{\theta }\partial _{i}v\,dx=\int_{\Omega _{\theta }}\sum_{i,j=p+1}^{n}a_{ij}(x)\partial _{j}u_{\infty }\partial _{i}vdx. \] Using the independence of $u_{\infty }$ on $X_{1}$, it comes \begin{equation} \int_{\Omega _{\theta }}\sum_{i,j=1}^{n}a_{ij}^{\theta }(x)\partial _{j}(u_{\theta }-u_{\infty })\partial _{i}vdx=\int_{\Omega _{\theta }}\sum_{\substack{ 1\leq i\leq p \\ p+1\leq j\leq n}}^{n}a_{ij}^{\theta }(x)\partial _{j}u_{\infty }\partial _{i}vdx. \label{fe9} \end{equation} On the other hand, the independence of $u_{\infty }$ and of the coefficients $a_{ij}^{\theta }$ on $X_{1}$ for $1\leq i\leq p$ and $p+1\leq j\leq n$, gives \begin{align*} \int_{\Omega _{\theta }}\sum_{\substack{ 1\leq i\leq p \\ p+1\leq j\leq n}}^{n}a_{ij}^{\theta }(x)\partial _{j}u_{\infty }\partial _{i}vdx &=\sum_{\substack{ 1\leq i\leq p \\ p+1\leq j\leq n}} ^{n}\int_{\Omega _{\theta }}\partial _{i}(a_{ij}^{\theta }(x)\partial _{j}u_{\infty }v)dx \\ &=\sum_{\substack{ 1\leq i\leq p \\ p+1\leq j\leq n}} ^{n}\int_{\partial \Omega _{\theta }}a_{ij}^{\theta }(x)\partial _{j}u_{\infty }v\nu _{i}dx=0, \end{align*} because $v$ vanishes on the boundary. Consequently, (\ref{fe9}) becomes \begin{equation} \int_{\Omega _{\theta }}\sum_{i,j=1}^{n}a_{ij}^{\theta }(x)\partial _{j}(u_{\theta }-u_{\infty })\partial _{i}vdx=0 \quad \text{for all }v\in H_{0}^{1}(\Delta \times \omega )). \label{fe10} \end{equation} For $\epsilon >0$, we set \[ \Delta _{\epsilon }=\{ x\in \Delta : d(\partial \Delta ,x)>\epsilon\} . \] Let $(\rho _{\epsilon })_{\epsilon >0}$ be a family of smooth functions on $\mathbb{R}^{p}$, such that \[ \mathop{\rm supp}\rho _{\epsilon }\subset \Delta _{\frac{\epsilon }{2}}, \quad (\Delta _{\epsilon }\subset \Delta _{\frac{\epsilon }{2}}), \] $\rho _{\epsilon }(x)=1$ for all $x$ in $\Delta_{\epsilon }$ and for all $x$ in $\Delta $, $\rho _{\epsilon }$ satisfies \[ 0\leq \rho _{\epsilon }(x)\leq 1. \] If we take $v=\rho _{\epsilon }^{2}(u_{\theta }-u_{\infty })\in H_{0}^{m}(\Delta \times \omega ))$ in (\ref{fe10}), we deduce that \[ \int_{\Delta \times \omega }\sum_{i,j=1}^{n}a_{ij}^{\theta }(x)\partial _{j}( u_{\theta }-u_{\infty })\partial _{i}(\rho _{\epsilon }^{2}(u_{\theta }-u_{\infty }))dx=0, \] whence \begin{align*} &\int_{\Delta \times \omega }\sum_{i,j=1}^{n}\rho _{\epsilon }^{2}a_{ij}^{\theta }(x)\partial _{j}(u_{\theta }-u_{\infty }) \partial _{i}(u_{\theta }-u_{\infty })dx\\ &= -2\int_{\Delta \times \omega }\sum_{\substack{ 1\leq i\leq p \\ 1\leq j\leq n}}a_{ij}^{\theta }(x)\rho _{\epsilon }\partial _{j}( u_{\theta }-u_{\infty })(u_{\theta }-u_{\infty }) \partial _{i}\rho _{\epsilon }dx. \end{align*} Using (\ref{felliptic}) and noting that $\rho _{\epsilon }$ vanishes in the exterior of $\Delta _{\frac{\epsilon }{2}}$ and depends only on $X_{1}$, it follows that \begin{align*} &\int_{\Delta _{\frac{\epsilon }{2}}\times \omega }\lambda \theta \sum_{i=1}^{p}\rho _{\epsilon }^{2}(\partial _{i}( u_{\theta }-u_{\infty }))^{2}dx+\int_{\Delta _{\frac{ \epsilon }{2}}\times \omega }\lambda '\sum_{i=p+1}^{n}\rho _{\epsilon }^{2}(\partial _{i}(u_{\theta }-u_{\infty }) )^{2}dx \\ &\leq -2\int_{\Delta _{\frac{\epsilon }{2}}\times \omega }\sum _{\substack{ 1\leq i\leq p \\ 1\leq j\leq p}}a_{ij}^{\theta }(x)\rho _{\epsilon }\partial _{j}(u_{\theta }-u_{\infty })( u_{\theta }-u_{\infty })\partial _{i}\rho _{\epsilon }dx \\ &\quad -2\int_{\Delta _{\frac{\epsilon }{2}}\times \omega }\sum _{\substack{ 1\leq i\leq p \\ p+1\leq j\leq n}}a_{ij}^{\theta }(x)\rho _{\epsilon }\partial _{j}(u_{\theta }-u_{\infty })( u_{\theta }-u_{\infty })\partial _{i}\rho _{\epsilon }dx. \end{align*} We estimate the second member using (\ref{fe11}), (\ref{fe12}) and the fact that the derivative of $\rho _{\epsilon }$ is bounded, we get \begin{align*} &\int_{\Delta _{\frac{\epsilon }{2}}\times \omega }\theta \sum_{i=1}^{p}(\rho _{\epsilon }\partial _{i}(u_{\theta }-u_{\infty }))^{2}dx+\int_{\Delta _{\frac{\epsilon }{2} }\times \omega }\sum_{i=p+1}^{n}(\rho _{\epsilon }\partial _{i}(u_{\theta }-u_{\infty }))^{2}dx \\ &\leq C\theta ^{\frac{1}{2}+\alpha }\Big[ \int_{\Delta _{\frac{ \epsilon }{2}}\times \omega }\sum_{1\leq j\leq p}(\rho _{\epsilon }\partial _{j}(u_{\theta }-u_{\infty }))^{2}dx \Big] ^{1/2}\Big[ \int_{\Delta _{\frac{\epsilon }{2}}\times \omega }(u_{\theta }-u_{\infty })^{2}dx\Big] ^{1/2} \\ &\quad +C\theta ^{\alpha }\Big[ \int_{\Delta _{\frac{\epsilon }{2}}\times \omega }\sum_{1\leq j\leq p}(\rho _{\epsilon }\partial _{j}(u_{\theta }-u_{\infty }))^{2}dx\Big] ^{1/2}\Big[ \int_{\Delta _{\frac{\epsilon }{2}}\times \omega }(u_{\theta }-u_{\infty })^{2}dx\Big] ^{1/2}. \end{align*} According to the Young inequality $ab\leq \varepsilon a^{2}+\frac{b^{2}}{\varepsilon }$ with $\varepsilon =\frac{1}{2C} \theta ^{\frac{1}{2}-\alpha }$ in the first term of the right hand side, and $\varepsilon =\frac{1}{2C}\theta ^{-\alpha }$ in the second term of the right hind side, we deduce \begin{equation} \label{fe13} \begin{aligned} &\frac{1}{2}\int_{\Delta _{\frac{\epsilon }{2}}\times \omega }\theta \sum_{i=1}^{p}(\rho _{\epsilon }\partial _{i}(u_{\theta }-u_{\infty }))^{2}dx+\frac{1}{2}\int_{\Delta _{\frac{ \epsilon }{2}}\times \omega }\sum_{i=p+1}^{n}(\rho _{\epsilon }\partial _{i}(u_{\theta }-u_{\infty }))^{2}dx\\ &\leq C\theta ^{2\alpha }\int_{\Delta _{\frac{\epsilon }{2}}\times \omega }(u_{\theta }-u_{\infty })^{2}dx. \end{aligned} \end{equation} Using Poincar\'{e}'s inequality and since $u_{\theta }-u_{\infty }$ vanishes on $\partial \omega $ for a.e. $X_{1}$, \[ \frac{1}{| \omega | ^{2}}\int_{\omega }( u_{\theta }-u_{\infty })^{2}dX_{2}\leq \frac{1}{2}\int_{\omega }\sum_{p+1\leq i\leq n}(\partial _{i}(u_{\theta }-u_{\infty }))^{2}dX_{2}\text{ a.e. }X_{1}\text{ in }\Delta _{ \frac{\epsilon }{2}}, \] where $| \omega | $ is the diameter of $\omega $, then (\ref{fe13}) becomes \begin{align*} &\frac{1}{| \omega | ^{2}}\int_{\Delta _{\frac{ \epsilon }{2}}\times \omega }(\rho _{\epsilon }(u_{\theta }-u_{\infty }))^{2}dx+\int_{\Delta _{\frac{\epsilon }{2} }\times \omega }\theta \sum_{i=1}^{p}(\rho _{\epsilon }\partial _{i}(u_{\theta }-u_{\infty }))^{2}dx\\ &+ \int_{\Delta _{\frac{\epsilon }{2}}\times \omega }\sum_{i=p+1}^{n}(\rho _{\epsilon }\partial _{i}( u_{\theta }-u_{\infty }))^{2}dx\\ &\leq C\theta ^{2\alpha }\int_{\Delta _{\frac{\epsilon }{2}}\times \omega }(u_{\theta}-u_{\infty })^{2}dx. \end{align*} According to the definition of $\rho _{\epsilon }$, we obtain \begin{equation} \label{fe14} \begin{aligned} &\frac{1}{| \omega | ^{2}}\int_{\Delta _{\epsilon }\times \omega }(u_{\theta }-u_{\infty }) ^{2}dx+\int_{\Delta _{\epsilon }\times \omega }\theta \sum_{i=1}^{p}(\partial _{i}(u_{\theta }-u_{\infty }))^{2}dx \\ &\quad + \int_{\Delta _{\epsilon }\times \omega }\sum_{i=p+1}^{n}( \partial _{i}(u_{\theta }-u_{\infty }))^{2}dx\\ &\leq C\theta ^{2\alpha }\int_{\Delta _{\frac{\epsilon }{2}}\times \omega }(u_{\theta }-u_{\infty })^{2}dx, \end{aligned} \end{equation} in particular \begin{equation} \int_{\Delta _{\epsilon }\times \omega }(u_{\theta }-u_{\infty })^{2}dx\leq C(\theta ^{\alpha }| \omega | )^{2}\int_{\Delta _{\frac{\epsilon }{2}}\times \omega }( u_{\theta }-u_{\infty })^{2}dx. \label{fe14b} \end{equation} Choosing $\epsilon =\frac{\varepsilon }{2^{k}}$ for $k=0,\dots,\tau -1$ and $\varepsilon >0$, we get \[ \int_{\Delta _{\frac{\varepsilon }{2^{k}}}\times \omega }( u_{\theta }-u_{\infty })^{2}dx\leq C(\theta ^{\alpha }| \omega | )^{2}\int_{\Delta _{\frac{ \varepsilon }{2^{k+1}}}\times \omega }(u_{\theta }-u_{\infty }) ^{2}dx. \] Iterating the above formula, leads to \[ \int_{\Delta _{\frac{\varepsilon }{2}}\times \omega }(u_{\theta }-u_{\infty })^{2}dx\leq C(\theta ^{\alpha }| \omega | )^{2(\tau -1)}\int_{\Delta _{\frac{ \varepsilon }{2^{\tau }}}\times \omega }(u_{\theta }-u_{\infty })^{2}dx. \] Applying Lemma \ref{cas gen}, we obtain \[ \int_{\Delta _{\frac{\varepsilon }{2}}\times \omega }(u_{\theta }-u_{\infty })^{2}dx\leq C(\theta ^{\alpha }| \omega | )^{2(\tau -1)}\int_{\omega }( | u_{+}| +| u_{-}| +| u_{\infty }| )^{2}dx, \] whence \[ \int_{\Delta _{\frac{\varepsilon }{2}}\times \omega }(u_{\theta }-u_{\infty })^{2}dx\leq C_{\omega }\theta ^{2\alpha (\tau -1)}, \] with \begin{equation} C_{\omega }=C| \omega | ^{2(\tau -1) }\int_{\omega }(| u_{+}| +| u_{-}| +| u_{\infty }| )^{2}dx. \label{fe46} \end{equation} Using (\ref{fe14}) with $\epsilon =\varepsilon $, we get the estimates \begin{gather} \int_{\Delta _{\varepsilon }\times \omega }\sum_{i=1}^{p}(\partial _{i}(u_{\theta }-u_{\infty }))^{2}dx \leq C_{\omega }\theta ^{2\alpha \tau -1}, \label{fe18} \\ \int_{\Delta _{\varepsilon }\times \omega }\sum_{i=p+1}^{n}(\partial _{i}(u\theta -u_{\infty }))^{2}dx \leq C_{\omega }\theta ^{2\alpha \tau }. \label{fe19} \end{gather} Finally, for any constant $r>0$, choosing $\tau $ such that $\tau \alpha >r$. Hence, we can state the following theorem. \begin{theorem} \label{feh1convergence}Under conditions \eqref{fe3}--\eqref{felliptic}, \eqref{fe11} and \eqref{fe12}, for any open subset $\Phi $ of $\Delta \times \omega $ with boundary disjoint of $\partial \Delta \times \omega $, it holds that \[ u_{\theta }\to u_{\infty }\text{ \ in }H^{1}(\Phi ), \] and for any $r>0$, \begin{gather} \int_{\Phi }| \nabla _{X_{1}}u_{\theta }| ^{2}dx \leq C_{\omega }\theta ^{2r-1}, \label{fe15} \\ \int_{\Phi }| \nabla _{X_{2}}(u_{\theta }-u_{\infty })| ^{2}dx \leq C_{\omega }\theta ^{2r}, \label{fe16} \end{gather} where $C_{\omega }$ is a constant given above and independent of $\theta $. \end{theorem} \begin{proof} It is sufficient to take $\varepsilon =d(\partial \Delta ,P_{X_{1}}\Phi )$. \end{proof} \begin{remark} \rm We can take $f\in H^{-1}(\omega _{0})$ to show the same results. In this case we can consider $f$ as an element of $H^{-1}(\Omega _{\theta }) $ by \[ <\widetilde{f}(t),v>_{H^{-1}(\Omega _{\theta }) }=\int_{P_{X_{1}}\Omega _{\theta }}_{H^{-1} (\omega _{0})}dX_{1},\text{ \ }v\in H_{0}^{1}(\Omega _{\theta }) , \] where $\widetilde{v}$ is the extension of $v$ by $0$ on $P_{X_{1}}\Omega _{\theta }\times \omega _{0}$. \end{remark} \section{Application to the case of large size domains \label{feappliquation elliptique}} We will see in this paragraph that the asymptotic behavior of the solution of linear elliptic problems of order two on domain $\overline{\Omega }_{\ell }$ satisfied for $\ell '\geq \ell $ \begin{equation} \overline{\Omega }_{\ell }=(-\ell ,\ell )^{p}\times \omega \text{ or }(-\ell ,\ell )^{p}\times \omega \subset \overline{\Omega }_{\ell }\subset (-\ell ',\ell ')^{p}\times \omega \label{fe38} \end{equation} which is studied in the book of Chipot \cite[Chapter 2 and 3]{ch}, can be casted in the preceding study without supposing any assumption on $\ell '$ (considering domains more general than (\ref{fe38})), by giving a particular form to the coefficients $a_{ij}^{\theta }$. Indeed, Let $( \overline{\Omega }_{\ell })_{\ell >0}$ be a family of bounded Lipschitz domains of $\mathbb{R}^{p}\times \omega _{0}$ (see Figures 2 and 3), such that for any $\ell >0$, $\overline{\Omega }_{\ell }$ contains the cylinder $(-\ell ,\ell )^{p}\times \omega $ and $(-\ell ,\ell )^{p}\times \partial \omega $ is a part of the boundary of $\overline{\Omega }_{\ell }$, where $\omega _{0}$ and $\omega $ are defined in the first section. We consider the two boundary-value problems defined by \begin{equation} \begin{gathered} \sum_{i,j=1}^{n}-\partial _{i}(a_{ij}\partial _{j}u) +a_{0}u=f \quad \text{in }\overline{\Omega }_{\ell } \\ u=0\quad \text{on }\partial \overline{\Omega }_{\ell }, \end{gathered} \label{fe28} \end{equation} and \begin{equation} \begin{gathered} \sum_{i,j=p+1}^{n}-\partial _{i}(a_{ij}\partial _{j}u) +a_{0}u=f \quad \text{in }\omega \\ u=0 \quad \text{on }\partial \omega. \end{gathered} \label{fe29} \end{equation} We suppose that $f\in L^{2}(\omega )$, \begin{equation} a_{0},\; a_{ij}\in L^{\infty }(\mathbb{R}^{p}\times \omega _{0}), \label{fe30} \end{equation} and \begin{equation} a_{0}(x)=a_{0}(X_{2})\geq 0,\quad a_{ij}(x)=a_{ij}(X_{2})\quad\text{for }j\geq p+1. \label{fe31} \end{equation} Moreover, we assume that there exists a constant $\lambda >0$, such that \begin{equation} \sum_{i,j=1}^{n}a_{ij}(x)\xi _{i}\xi _{j} \geq \lambda | \boldsymbol{\xi}| ^{2},\quad \text{a.e. }x\in \mathbb{R}^{p}\times \omega _{0},\;\forall \xi \in \mathbb{R}^{n}. \label{fe32} \end{equation} \begin{figure}[ht] \begin{center} \includegraphics[width=6cm, height=6cm]{fig2} \end{center} \label{fig2} \caption{The domain $\overline{\Omega }_{\ell }.$} \end{figure} \begin{figure}[ht] \begin{center} \includegraphics[width=6cm, height=6cm]{fig3} \end{center} \label{fig3} \caption{The domain $\overline{\Omega }_{\ell _{1}}$ has another form for $\ell _{1}>\ell$.} \end{figure} Then the solutions $\overline{u}_{\ell }$ and $u_{\infty }$ of (\ref{fe28}) and (\ref{fe29}) respectively satisfy \begin{equation} \int_{\overline{\Omega }_{\ell }}\sum_{i,j=1}^{n}a_{ij}(x)\partial _{j}\overline{u}_{\ell }\partial _{i}v+a_{0}(x)\overline{u}_{\ell }vdx=\int_{\overline{\Omega }_{\ell }}fvdx, \quad \text{a.e. }v\in H_{0}^{1}(\overline{\Omega }_{\ell }), \label{fe33} \end{equation} and \begin{equation} \int_{\omega }\sum_{i,j=p+1}^{n}a_{ij}(X_{2})\partial _{j}u_{\infty }\partial _{i}v+a_{0}(X_{2})u_{\infty }vdX_{2}=\int_{\omega }fvdX_{2},\quad \text{a.e. }v\in H_{0}^{1}(\omega ). \label{fe34} \end{equation} We take $\theta =\frac{1}{\ell ^{2}}$ and use the change of variable given by \begin{equation} \psi :(X_{1},X_{2})\mapsto y=\Big(Y_{1}=\frac{X_{1}}{\ell },Y_{2}=X_{2}\Big), \label{fe35} \end{equation} in $(\ref{fe33})$, and we set $\psi (\overline{\Omega } _{\ell })=\Omega _{\theta }$, thus we obtain \begin{align*} &\int_{\Omega _{\theta }}\sum_{i,j=1}^{p}\frac{1}{\ell ^{2}} a_{ij}(\ell Y_{1},Y_{2})\partial _{j}\overline{u}_{\ell }(\ell Y_{1},Y_{2})\partial _{i}v(\ell Y_{1},Y_{2})\ell ^{p}dy\\ &+ \int_{\Omega _{\theta }}\sum_{\substack{ 1\leq i\leq p,\text{ } p+1\leq j\leq n \\ or \\ 1\leq j\leq p,\text{ }p+1\leq i\leq n}}\frac{1}{ \ell }a_{ij}(\ell Y_{1},Y_{2})\partial _{j}\overline{u}_{\ell }(\ell Y_{1},Y_{2})\partial _{i}v(\ell Y_{1},Y_{2})\ell ^{p}dy\\ &+ \int_{\Omega _{\theta }}\sum_{i,j=p+1}^{p}a_{ij}(Y_{2})\partial _{j}\overline{u}_{\ell }(\ell Y_{1},Y_{2})\partial _{i}v(\ell Y_{1},Y_{2})\ell ^{p}dy\\ &=\int_{\Omega _{\theta }}f(Y_{2})v(\ell Y_{1},Y_{2})\ell ^{p}dy. \end{align*} Setting \begin{gather*} u_{\theta }(Y_{1},Y_{2}) =\overline{u}_{\ell }(\ell Y_{1},Y_{2}), \\ a_{ij}^{\theta }(Y_{1},Y_{2}) =\frac{1}{\ell ^{2}}a_{ij}(\ell Y_{1},Y_{2}) \text{ \ \ for }i,j=1,\dots,p, \\ a_{ij}^{\theta }(Y_{1},Y_{2}) =\frac{1}{\ell }a_{ij}(\ell Y_{1},Y_{2}) \text{ \ \ \ for }1\leq i\leq p0$ and for $\Phi =(-\sigma,\sigma)^{p}\times \omega $ with $0<\sigma<1$, that there exists $C>0$ independent of $\ell $, such that \begin{gather*} \int_{(-\sigma,\sigma)^{p}\times \omega }| \nabla _{X_{1}}u_{\theta }| ^{2}dy \leq C\theta ^{r+p-2}, \\ \int_{(-\sigma,\sigma)^{p}\times \omega }| \nabla _{X_{2}}(u_{\theta }-u_{\infty }) | ^{2}dy \leq C\theta ^{r+p}. \end{gather*} Again, we use the change of variable \eqref{fe35} to obtain \[ \| \overline{u}_{\ell }-u_{\infty }\| _{H^{1}((-\sigma\ell,\sigma\ell)^{p}\times \omega)}\leq \frac{C}{\ell ^{r}}. \] \section{Estimate according to all directions} In the applications, we say that the size of the domain is large specifically in some directions if we take into account the size ratio between all the directions, for instance in the domain $(0,1)\times (0,\varepsilon)$, the size of $(0,1)$ is considered large when $\varepsilon$ become negligible. However all the estimates of $\overline{u}_{\ell}-u_{\infty }$ given in \cite{sen}, \cite{ch} and \cite{rougirel}, only show an estimate of the error of convergence with respect to $\ell $. In the following, we investigate this estimate with respect to the size ratio between $\ell $ and $|\omega |$. Then, we suppose in this section that $\omega =\omega _{0}$ and a bounded domain $\overline{\Omega } _{\ell }$ satisfies \begin{equation} (-\ell ,\ell )^{p}\times \omega \subset \overline{\Omega }_{\ell }\subset \mathbb{R}^{p}\times \omega ; \label{fe43} \end{equation} in addition, we assume that \begin{equation} f\in L^{\infty }(\omega ). \label{fe44} \end{equation} First, we show the following estimate. \begin{figure}[ht] \begin{center} \includegraphics[width=6cm, height=4cm]{fig4} \end{center} \label{fig4} \caption{The domain $\overline{\Omega } _{\ell }$.} \end{figure} \begin{lemma} Let $u_{+}$ (resp. $u_{-}$) be the solution of \eqref{fe8} replacing $h$ by $f^{+}$ (resp. $-f^{-}$). It holds that \begin{equation} | u_{+}| _{L^{2}(\omega )},\text{ }| u_{-}| _{L^{2}(\omega )},\text{ }| u_{\infty }| _{L^{2}(\omega )}\leq C\left[ \mathop{\rm meas}\omega )\right] ^{1/2}| \omega | ^{2} \label{fe45} \end{equation} where $C$ is a constant independent of $\omega $ and $\mathop{\rm meas}\omega)$ denotes the measure of $\omega $. \end{lemma} \begin{proof} We give the proof for $u_{+}$, the proof for $u_{-}$ and $u_{\infty }$ are similar. Taking $v=u_{+}$ in $(\ref{fe8})$ and using the ellipticity condition $(\ref{felliptic2})$, we obtain \[ \lambda '\int_{\omega }| \nabla u_{+}| ^{2}dX_{2}\leq | f^{+}| _{L^{2}(\omega )}| u_{+}| _{L^{2}(\omega )}. \] Using \eqref{fe44} and applying Poincar\'{e}'s inequality, then there exists a constant $C$ independent of $\omega $, such that \[ \frac{1}{| \omega | ^{2}}| u_{+}| _{L^{2}(\omega )}^{2}\leq C[ \mathop{\rm meas}\omega )] ^{1/2}| u_{+}| _{L^{2}(\omega )}, \] which gives $(\ref{fe45})$. \end{proof} This enables us to state the following corollary. \begin{corollary} Let $\overline{u}_{\ell }$ be the solution of $( \ref{fe33})$ where $\overline{\Omega }_{\ell }$ is given by \eqref{fe43}. If we suppose that \eqref{fe30}, \eqref{fe31}, \eqref{fe32} and \eqref{fe44} hold, then for any $\tau >0$ and any $0<\sigma<1$ there exists a constant $C_{\sigma}>0$ independent of $\ell $ and $\omega $, such that \begin{equation} | \nabla (\overline{u}_{\ell }-u_{\infty }) | _{L^{2}\left(( -\sigma\ell,\sigma\ell\right)^{p}\times \omega)} \leq C_{\sigma}\ell ^{p}\mathop{\rm meas}(\omega )| \omega | ^{2} \big(\frac{| \omega | }{\ell }\big)^{2\tau}. \label{fe47} \end{equation} \end{corollary} \begin{proof} If we use the change of variable $(\ref{fe35})$ in $(\ref {fe18})$ and $(\ref{fe19})$, and we apply the lemma above to estimate the constant $C_{\omega }$ defined in \eqref{fe46}, then we deduce \eqref{fe47}. \end{proof} \subsection*{Acknowledgements} We would like to thank Professor M. Kirane for his useful comments that helped improving this article. We would also like to thank Professor R. 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