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\controldates{24-SEP-2003,24-SEP-2003,24-SEP-2003,24-SEP-2003}
 
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\issueinfo{9}{11}{}{2003}
\dateposted{September 29, 2003}
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\PII{S 1079-6762(03)00115-X}
\copyrightinfo{2003}{American Mathematical Society}

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\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.
The authors also thank V.\,A.~Il'in, E.\,I.~Moiseev and
A.\,L.~Skubachevskii for their attention and concern.


\bibliographystyle{amsplain}
\begin{thebibliography}{10}

\bibitem{DR}
V. N. Denisov and V. D.~Repnikov,
\textit{The stabilization of a
        solution of a Cauchy problem
        for parabolic equations},
{Differ. Equ.} \textbf{20}, No. 1 (1984), 16--33.
\MR{86c:35013}

\bibitem{DM}
V. N. Denisov and A. B.~Muravnik,
\textit{On stabilization of the solution of the Cauchy problem for quasilinear
parabolic equations},
{Differ. Equ.} \textbf{38}, No. 3 (2002), 369--374.

\bibitem{GT}
D.~Gilbarg and N. S. Trudinger,
\textit{Elliptic partial differential equations of second order},
Springer-Verlag, 1977.
\MR{57:13109}

\bibitem{KPZ}
M. Kardar, G. Parisi, and Y.-C.~Zhang,
\textit{Dynamic scaling of growing
        interfaces},
{Phys. Rev. Lett.} \textbf{56} (1986), 889--892.

\bibitem{MHKZ}
E. Medina, T. Hwa, M. Kardar, and Y.-C.~Zhang,
\textit{Burgers equation with
        correlated noise: Renormalization group analysis
        and applications
        to directed polymers and interface growth},
{Phys. Rev.} \textbf{A39} (1989), 3053--3075.
\MR{89m:82027}

\bibitem{P}
S. I. Poho\v zaev,
\textit{Equations of the type $\Delta u=f(x,u,Du)$},
        {Math. USSR Sb.} \textbf{\bf 41}, No. 2 (1982), 269--280.
\MR{82i:35066}

\bibitem{M}
A. Mielke,
\textit{Essential manifolds for an elliptic problem in an infinite strip},
	{J. Differ. Equations} \textbf{\bf 110}, No. 2 (1994), 322--355.
\MR{95f:35086}

\bibitem{G}
E. De Giorgi,
\textit{Sulla differenziabilita e l'analitica delle estremali degli
        integrali multipli regolaru},
        {Mat. Acad. Sci. Torino, Ser.~3}, No. 1 (1957), 25--43.
\MR{20:172}

\bibitem{KS}
N. V. Krylov and M. V. Safonov,
\textit{A certain property of solutions of parabolic equations with measurable
        coefficients},
{Math. USSR Izv.} \textbf{\bf 16} (1981), 151--164.
\MR{83c:35059}

\bibitem{GM}
A. K. Guscin and V. P. Mihailov,
\textit{Stabilization of solutions of Cauchy's problem for a parabolic
equation},
{Differ. Equ.} \textbf{\bf 7}, No. 2 (1971), 232--242.
\MR{43:7758}

\bibitem{K}
S. Kamin,
\textit{On stabilization of the solutions of the Cauchy problem
        for parabolic equations},
{Proc. Roy. Soc. Edinburgh} \textbf{\bf 76A} (1976), 43--53.
\MR{55:13079}

\bibitem{Z}
V. V. \v Zikov,
\textit{On the stabilization of solutions of parabolic equations},
        {Math. USSR Sb.} \textbf{\bf 33}, No.~4 (1977), 519--537.
\MR{57:13190}

\bibitem{B}
A. V.~Bitsadze,
\textit{Some classes of partial differential equations},
Nauka, 1981 (in Russian).
\MR{83e:35003}

\end{thebibliography}

\end{document}