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To display a typeset version of a journal * %_ * article easily, we suggest that you retrieve the article in DVI, * %_ * PostScript, or PDF format. * %_ ************************************************************************** % Author Package file for use with AMS-LaTeX 1.2 \controldates{24-SEP-2003,24-SEP-2003,24-SEP-2003,24-SEP-2003} \RequirePackage[warning,log]{snapshot} \documentclass{era-l} \issueinfo{9}{11}{}{2003} \dateposted{September 29, 2003} \pagespan{88}{93} \PII{S 1079-6762(03)00115-X} \copyrightinfo{2003}{American Mathematical Society} \newcommand{\ds}{\displaystyle} \newcommand{\defeq}{\> \lower 2pt \hbox{$\buildrel{\rm def} \over = $}\>} \newcommand{\new}{\rightharpoonup} \newcommand{\newt}{\Longrightarrow} \newcommand{\ol}{\overline} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{corollary}{Corollary} \theoremstyle{remark} \newtheorem{remark}{Remark} \begin{document} \title[DIRICHLET PROBLEM IN HALF-SPACE]{On asymptotic behavior of solutions \linebreak[1] of the Dirichlet problem in half-space\linebreak[1] for linear and quasi-linear elliptic equations} \author{Vasily Denisov} \address{Moscow State University, Faculty of Computational Mathematics and Cybernetics, Leninskie gory, Moscow 119899, Russia} \email{V.Denisov@g23.relcom.ru} \thanks{The second author was supported by INTAS, grant 00-136 and RFBR, grant 02-01-00312.} \author{Andrey Muravnik} \address{Department of Differential Equations, Moscow State Aviation Institute, Volokolamskoe shosse 4, Moscow, A-80, GSP-3, 125993, Russia} \email{abm@mailru.com} \date{March 6, 2002} \subjclass[2000]{Primary 35J25; Secondary 35B40, 35J60} \keywords{Asymptotic behaviour of solutions, BKPZ-type non-linearities} \commby{Michael E. Taylor} \begin{abstract} We study the Dirichlet problem in half-space for the equation $\nobreak{\Delta u+g(u)|\nabla u|^2=0,}$ where $g$ is continuous or has a power singularity (in the~latter case positive solutions are considered). The results presented give necessary and sufficient conditions for the existence of (pointwise or uniform) limit of the solution as $y\to\infty,$ where $y$ denotes the spatial variable, orthogonal to the hyperplane of boundary-value data. These conditions are given in terms of integral means of the boundary-value function. \end{abstract} \maketitle \section*{Introduction} The phenomenon called \textit{stabilization} is well known for \textit{parabolic} equations both in linear (see e.g. \cite{DR} and references therein) and non-linear (see e.g. \cite{DM} and references therein) cases; it means the existence of a finite limit of the solution as $t\to\infty.$ However, there are well-posed non-isotropic \textit{elliptic} boundary-value problems in unbounded domains (see e.g. \cite{GT}) for which we can talk about stabilization in the following sense: the solution has a finite limit as a selected \textit{spatial} variable tends to infinity. This paper is devoted to the Dirichlet problem in half-space for elliptic equations. We present necessary and sufficient conditions for the stabilization of its solution; here the spatial variable, orthogonal to the hyperplane of boundary-value data, plays the role of time. In Section~\ref{s1}, the linear case is presented; Sections \ref{s2} and \ref{s3} are devoted to quasi-linear equations with the so-called Burgers-Kardar-Parisi-Zhang non-linearity type (see e.g. \cite{KPZ}, \cite{MHKZ}). Equations with such non-linearities arise, for example, in modeling of directed polymers and interface growth. They also present an independent theoretical interest because they contain second powers of the first derivatives (see e.g. \cite{P} and references therein). Note that we deal with the stabilization problem in cylindrical domains with an \textit{unbounded} base (in particular, here the base of the cylinder is the whole $E^N$). As in the parabolic case, this problem is principally different (this refers both to the results and to the methods of research) from the stabilization problem in cylindrical domains with a \textit{bounded} base. The latter problem has been investigated for a~rather broad class of non-linear elliptic equations by means of methods of dynamical systems (see e.g. \cite{M} and references therein). \section{Linear equations}\label{s1} In the half-space $\{y\ge 0\}$ we consider the Dirichlet problem \begin{equation}\label{eq1} p(x)\frac{\partial^2 u}{\partial y^2} + \sum^{N}_{i,j=1}\frac{\partial}{\partial x_i} \Bigl(a_{ij}(x,y)\frac{\partial u}{\partial x_j}\Bigr)=0, \quad (x,y)\in \{y>0\}, \end{equation} \begin{equation}\label{eq2} u(x,0)=\varphi(x), \quad x\in E^N, \end{equation} where the coefficients of equation~\eqref{eq1} are measurable and bounded and the uniform ellipticity condition is satisfied, i.e., there exists a positive constant $\lambda$ such that for any $x,y\in\{y>0\}$, \begin{equation}\label{eq3} \lambda^{-1}\le p(x)\le \lambda, \quad \lambda^{-1}|\xi|^2\le \sum^{N}_{i,j=1}a_{ij}(x,y)\xi_i\xi_j \le \lambda|\xi|^2; \end{equation} the boundary-value function $\varphi(x)$ is supposed to be continuous and bounded. We say that $u(x,y)$ is a solution of \eqref{eq1} if $u$ is bounded on $\{y>0\},$ belongs to $W^1_2$ in any strictly interior subdomain of $\{y\ge 0\}$, and satisfies the integral identity \[ \int_{0}^{+\infty} \int_{E^N} \Bigl[ \sum_{i,j=1}^{N}a_{ij}(x,y) \frac{\partial u}{\partial x_j} \frac{\partial v}{\partial x_i} + p(x) \frac{\partial u}{\partial y} \frac{\partial v}{\partial y} \Bigr]dx dy=0 \] for any $v(x,y)\in C^{\infty}_{0}\{y>0\}$. It is known (see e.g. \cite{GT}) that, under the above assumptions, a solution $u(x,y)$ of \eqref{eq1}, \eqref{eq2} exists and is unique; it is bounded and continuous on $\{y>0\}$ and coincides at $y=0$ with a continuous boundary-value function $\varphi(x)$ on $E^N$. Moreover, it follows from \cite{G} and \cite{KS} that $u(x,y)$ satisfies H\"older's condition with a constant $\alpha>0$ depending merely on $\lambda$ and $N$. We discuss the question of finding necessary and sufficient conditions for the existence of \begin{equation}\label{eq4} \lim_{y\to +\infty}u(x,y)=l \end{equation} for any $x\in E^N$ (or uniformly with respect to $x\in E^N$). First of all we introduce a few definitions related to the means of multi-variable functions. Let $g(x)$ be bounded on $E^N$, let $K$ be a compact subset of $E^N$, and let $R$ be a large positive parameter. We consider the following family of functions: \[\{g(Rx)\}=\{g(Rx_1, Rx_2,\dots, Rx_N)\},\,R>0.\] Suppose $g(x)$ has a weak limit at infinity in $L_{\rm loc}^2$, i.e., \begin{equation}\label{eq5} g(Rx)\new{\ol g}(x), \end{equation} where the symbol $\new$ denotes weak convergence in $L^2(K).$ It follows from the boundedness of $g(x)$ that \eqref{eq5} is equivalent to the following two assertions: \begin{equation}\label{eq61} \lim_{R\to\infty} \int_{E^N} g(Rx)\psi(x)\, dx= \int_{E^N} {\ol g}(x)\psi(x)\, dx\textrm{ for any }\psi\in C^{\infty}_{0}(E^N), \end{equation} \begin{equation}\label{eq62} \lim_{R\to\infty} \int_{K} g(Rx)\, dx= \int_{K} {\ol g}(x)\, dx. \end{equation} We say that $g(x)$ has a mean value, if it has a weak limit~\eqref{eq5} at infinity in $L^2_{\rm loc}$ and that limit is a constant. Relation \eqref{eq62} yields that the above definition of the mean value is equivalent to the following one: \begin{equation}\label{eq63} {\ol g}=\lim_{R\to\infty} \frac{1}{|K_R|} \int_{K_R} g(y)\, dy, \end{equation} where $K_R\defeq\{y\in E^N\big\vert y=Rx,\, x\in K\},$ $|K_R|$ denotes the Lebesgue measure of~$K_R$. We say that $g(x)$ has a uniform mean value ${\ol g}$ on $E^N$ if for any $\psi\in C^{\infty}_{0}(E^N)$ there exists \begin{equation}\label{eq64} \lim_{R\to\infty} \int_{E^N} g(\beta+Rx)\psi(x)\, dx= {\ol g} \int_{E^N} \psi(x)\, dx, \end{equation} and this limit is uniform with respect to $\beta\in E^N$. The latter definition is equivalent to the following one: \begin{equation}\label{eq65} \lim_{R\to\infty} \frac{1}{|K_R|} \int_{K_R} g(\beta+y)\, dy={\ol g} \end{equation} uniformly with respect to $\beta\in E^N$. Let $f(x,y)\defeq f(x_1,\dots, x_N; y)$ be bounded on the half-space $\{y\ge 0\}$. We say that it satisfies condition~$A$ if there exists a constant ${\ol f}$ such that \begin{equation}\label{eq66} \lim_{R\to\infty} \int_{|x|\le 1} \int_{0}^{1} \bigl[f(Rx,Ry)-{\ol f}\bigr]^2 \, dy\, dx = 0. \end{equation} Note that an $A$-type condition was introduced in \cite{GM} and \cite{K}. It was developed further in \cite{Z}. Now we are ready to formulate our main results concerning linear equations. \begin{theorem}\label{thm1} Suppose $p(x)$ has a mean value and the elliptic operator in \eqref{eq1} is the Laplacian. Then limit \eqref{eq4} exists at any point ${x\!\in\! E^N}$ if and only if \begin{equation}\label{eq7} \lim_{R\to\infty} {\frac {\int_{|x|\le R}p(x)\varphi(x)\, dx} {\int_{|x|\le R}p(x)\, dx}} =l. \end{equation} \end{theorem} \begin{theorem}\label{thm2} Assume $p(x)$ has a mean value and the coefficients $a_{ij}(x,y)$ satisfy condition~$A$ with a positive definite matrix $\{{\ol a}_{ij}\}_{i,j=1}^n.$ Then limit \eqref{eq4} exists at any point ${x\!\in\! E^N}$ if and only if \begin{equation}\label{eq8} \lim_{R\to\infty} {\frac {\int_{(Bx,x)\le R^2}\varphi(x)p(x)\, dx} {\int_{(Bx,x)\le R^2}p(x)\, dx}} =l, \end{equation} where $B$ denotes the inverse matrix of $\{{\ol a}_{ij}\}_{i,j=1}^n.$ \end{theorem} \begin{theorem}\label{thm3} Limit \eqref{eq4} exists uniformly with respect to ${x\!\in\! E^N}$ if and only if \begin{equation}\label{eq9} \lim_{R\to\infty} {\frac {\int_{|y|\le R}p(x+y)\varphi(x+y)\, dy} {\int_{|y|\le R}p(x+y)\, dy}} =l \end{equation} uniformly with respect to ${x\!\in\! E^N}.$ \end{theorem} \begin{remark} In the case of Cauchy problem for \textit{parabolic} equations, assertions similar to Theorems~\ref{thm1}--\ref{thm3} were proved in \cite{GM}--\cite{Z}. \end{remark} \section{Quasi-linear equations with regular coefficients\\ at non-linearities}\label{s2} Hereinafter the point $x=(x_1,\dots,x_N,x_{N+1})$ is denoted by $(x',x_{N+1}),$ and the half-space $E^N\times(0,+\infty)$ is denoted by $E_+^{N+1}.$ In this section we consider the following problem: \begin{equation}\label{eq10} \Delta u + g(u)|\nabla u|^2 = 0,~x\in E_+^{N+1}; \end{equation} \begin{equation}\label{eq11} u(x',0)=\varphi(x'),~x'\in E^N; \end{equation} where $g$ is continuous in $(-\infty,+\infty),$ and $\varphi$ is continuous and bounded in~$E^N.$ Following e.g. \cite{B}, we introduce the function \begin{equation}\label{eq12} f(s)\defeq\int_0^se^{\int_0^\tau g(\sigma)d\sigma}d\tau, \end{equation} and prove the following assertions. \begin{theorem}\label{thm4} There exists a unique classical bounded solution of problem~\eqref{eq10}, \eqref{eq11}. \end{theorem} \begin{theorem}\label{thm5} Let $x'\in E^N,\,l\in(-\infty,+\infty),$ and let $u(x)$ be the classical bounded solution of problem~\eqref{eq10}, \eqref{eq11}. Then \[\lim_{x_{N+1}\to+\infty}u(x)=l\,\textit{ if and only if } \,\lim_{R\to+\infty}\frac{N\Gamma(\frac{N}{2})}{2\pi^{\frac{N}{2}}R^N} \int_{|y|\le R}f[\varphi(y)]dy=f(l). \] \end{theorem} \begin{theorem}\label{thm6} Let $l\in(-\infty,+\infty)$, and let $u(x)$ be the classical bounded solution of problem~\eqref{eq10}, \eqref{eq11}. Then \[u(x)\buildrel{x_{N+1}\to+\infty}\over{\longrightarrow}l\, \textit{ uniformly with respect to }\,x'\in E^N\] if and only if \[{N\Gamma({N \over 2})\over 2\pi^{N \over 2}R^N}\int_{|y|\le R} \!\!\!\!f[\varphi(x'+y)]dy \buildrel{R\to+\infty}\over{\longrightarrow}\!f(l)\, \textit{ uniformly with respect to }\,x'\in E^N.\] \end{theorem} \begin{remark} The results of this section are valid also for the equation \[ \frac{\partial^2 u}{\partial y^2} + \!\!\sum^{N}_{i,j=1}\frac{\partial}{\partial x_i} \Bigl(a_{ij}(x,y)\frac{\partial u}{\partial x_j}\Bigr)+ g(u)\Bigg[\sum^{N}_{i,j=1} a_{ij}(x,y)\Bigl(\frac{\partial u}{\partial x_j}\Bigr)^2\!\!+ \Bigl(\frac{\partial u}{\partial y}\Bigr)^2\Bigg]=0, \] where, besides \eqref{eq3} and condition $A,$ we have $a_{ij}\in C^{1,q}(\overline{E^N_+})$ with a positive $q~(i,j=1,\dots,N).$ \end{remark} \section{The case of singular coefficients at non-linearities}\label{s3} In this section, we study the equation \begin{equation}\label{eq13} \Delta u + \alpha u^\beta|\nabla u|^2 = 0, \end{equation} where $\beta\in[-1,0)$ and $\alpha>-1$ for $\beta=-1.$ We will also assume that $\varphi(x')\ge 0$ (apart from its continuity and boundedness), and consider positive solutions of \eqref{eq13}, \eqref{eq11}. For $\beta\in(-1,0)$ we use \eqref{eq12} again, but for $\beta=-1$ the function $f(s)$ is defined as $s^{\alpha+1}.$ Then the following assertions are valid. \begin{theorem}\label{thm7} If $\varphi$ is different from the identical zero, then there exists a uni\-que classical positive bounded solution of problem~\eqref{eq13}, \eqref{eq11}. \end{theorem} \begin{theorem}\label{thm8} Let $\beta=-1$ and let $u(x)$ be the classical positive bounded solution of problem~\eqref{eq13}, \eqref{eq11}, $x'\in E^N,\,l\ge 0.$ Then \[\lim_{x_{N+1}\to+\infty}u(x)=l\,\textit{ if and only if } \,\lim_{R\to+\infty}{N\Gamma({N\over 2})\over 2\pi^{N\over 2}R^N} \int_{|y|\le R}\varphi^{\alpha+1}(y)dy=l^{\alpha+1}.\] \end{theorem} \begin{theorem}\label{thm9} Let $\beta\in(-1,0)$ and let $u(x)$ be the classical positive bounded solution of problem~\eqref{eq13}, \eqref{eq11}, $x'\in E^N,\,l\ge 0.$ Then \[\lim_{x_{N+1}\to+\infty}u(x)=l\,\textit{ if and only if } \,\lim_{R\to+\infty}{N\Gamma({N\over 2})\over 2\pi^{N\over 2}R^N} \int_{|y|\le R}\tilde f[\varphi(y)]dy=f(l), \] where \begin{equation}\label{eq14} \tilde f(s)=\int_0^se^{{\alpha\over 1-\beta}\tau^{1-\beta}}d\tau. \end{equation} \end{theorem} \begin{theorem}\label{thm10} Let $\beta=-1$ and let $u(x)$ be the classical positive bounded solution of problem~\eqref{eq13}, \eqref{eq11}, $l\ge 0.$ Then \[u(x)\buildrel{x_{N+1}\to+\infty}\over{\longrightarrow}l\, \textit{ uniformly with respect to }\,x'\in E^N\] if and only if \[{N\Gamma({N\over 2})\over 2\pi^{N\over 2}R^N}\int_{|y|\le R} \!\!\!\!\varphi^{\alpha+1}(x'+y)dy \buildrel{R\to+\infty}\over{\longrightarrow}\!l^{\alpha+1}\, \textit{ uniformly with respect to }\, x'\in E^N.\] \end{theorem} \begin{theorem}\label{thm11} Let $\beta\in(-1,0)$ and let $u(x)$ be the classical positive bounded solution of problem~\eqref{eq13}, \eqref{eq11}, $l\ge 0.$ Then \[u(x)\buildrel{x_{N+1}\to+\infty}\over{\longrightarrow}l\, \textit{ uniformly with respect to }\,x'\in E^N\] if and only if \[{N\Gamma({N\over 2})\over 2\pi^{N\over 2}R^N}\int_{|y|\le R} \!\!\!\!\tilde f[\varphi(x'+y)]dy \buildrel{R\to+\infty}\over{\longrightarrow}\!\tilde f(l)\, \textit{ uniformly with respect to }\, x'\in E^N.\] \end{theorem} \begin{remark} In the case of Cauchy problem for \textit{parabolic} equations, assertions similar to Theorems~\ref{thm5}--\ref{thm11} were proved in \cite{DM}. \end{remark} \section*{Acknowledgments} The authors are very grateful to L.\,A.~Peletier and S.\,I.~Poho\v zaev for fruitful discussions. 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