\documentclass[reqno]{amsart} \pdfoutput=1\relax\pdfpagewidth=8.26in\pdfpageheight=11.69in\pdfcompresslevel=9 \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 66, pp. 1--5.\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 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/66\hfil Liouville's theorem and the restricted mean property] {Liouville's theorem and the restricted mean property for Biharmonic Functions} \author[Mohamed El Kadiri \hfil EJDE-2004/66\hfilneg] {Mohamed El Kadiri} \address{Mohamed El Kadiri \hfill\break B. P. 726, Sal\'e-Tabriquet, Sal\'e, Morocco} \email{elkadiri@fsr.ac.ma} \date{} \thanks{Submitted April 10, 2003. Published April 28, 2004.} \subjclass[2000]{31B30} \keywords{Biharmonic function, mean property, Liouville's theorem} \begin{abstract} We prove that under certain conditions, a bounded Lebesgue measurable function satisfying the restricted mean value for biharmonic functions is constant, in $\mathbb{R}^n$ with $n\ge 3$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \section{Introduction} It is well known that a biharmonic function $f$ in $\mathbb{R}^n$, i.e. a solution of the classical biharmonic equation $\Delta^2u=0$, satisfies the biharmonic mean formula on every open ball $B=B(x,r)$ of center $x$ and radius $r>0$ in $\mathbb{R}^n$: $$f(x)=\frac{1}{|B|}\int_B fd\lambda-\frac{r^2}{2(n+2)}\Delta f(x),$$ where $|B|$ denotes the volume of the ball $B$ and $\lambda$ is the Lebesgue measure on $\mathbb{R}^n$. This formula is due to Pizzetti and can be found in \cite{n1}. When the biharmonic function $f$ is bounded, say $$\sup_{x\in \mathbb{R}^n}|f(x)| =M<+\infty,$$ one has $$\sup_{x\in \mathbb{R}^n}|\Delta f(x)|\le \frac{4(n+2)}{r^2}M.$$ Letting $r\to\infty$, this yields $\Delta f\equiv 0$, so that $f$ is a bounded harmonic function on $\mathbb{R}^n$, hence $f$ is constant by the Liouville's classical Theorem. This is the Liouville's property for biharmonic functions. Let us recall that a function $f$ on a domain $\Omega$ of $\mathbb{R}^n$, locally integrable and whose Laplacian in the distribution sense is a function, satisfies the restricted biharmonic mean property if there exists a function $r:\Omega\to \mathbb{R}_+$ such that $00$ and $a>0$, the following conditions are satisfied: \begin{itemize} \item[(i)] $\mu_x(s)\le s$ for every non-negative superharmonic function $s$ on $\mathbb{R}^n$. \item[(ii)] There exist a function $r_0$ on $\mathbb{R}^n$, $00$ such that $\mu_x\ge \gamma \lambda_{B(x,r_0(x))}$ and $(G^{\epsilon_x}-G^{\mu_x})1_{CB(x,\eta r_0(x))}\lambda\le a\rho^2\mu_x$. \end{itemize} \noindent\textbf{Remarks} (See \cite{h3}) 1. The condition (i) is satisfied if and only if $G^{\mu_x}\le G^{\epsilon_x}$. 2. Let $r$ be a Borel measurable function on $\mathbb{R}^n$ such that $00$. Then $f$ is constant. \end{theorem} \section{Liouville's Theorem and the restricted biharmonic mean property} For a ball $B=B(x,r)$ of $\mathbb{R}^n$, put $$w_B(x,z)=G(x,z)-\frac{1}{|B|}\int_B G(y,z)d\lambda(y).$$ It is not difficult to see that the function $w_B$ satisfies the following properties: \begin{enumerate} \item $w_B(x,y)=0$ if $y\notin B$. \item The function $w_B(x,.)$ is invariant under rotations around $x.$ \item One has $$\frac{2(n+2)}{r^2}\int w_B(x,y)d\lambda(y)=1.$$ \end{enumerate} This equality is an immediate consequence of the harmonic mean formula on $B$ applied to the function $\int_{B'}G(.,y)dy$, where $B'$ is any open ball of center $x$ such that ${\overline {B}}\subset B'$. It follows that for $x\in \mathbb{R}$, the measure $\mu_x^r$ on $\mathbb{R}^n$ of density $\frac{2(n+2)}{r^2}w_{B(x)}(x,.)$ with respect to the Lebesgue measure is a probability measure supported by ${\overline {B(x,r)}}$ and invariant under rotations around $x$. In particular, one has, $$\int s(y)d\mu_x^r(y)\le s(x)$$ for any non-negative superharmonic function $s$ on $\mathbb{R}^n$. The map $(x,A)\mapsto \mu_x^r(A)$, where $A$ is a Borel set of $\mathbb{R}^n$, is a Markovian kernel on $\mathbb{R}^n$. Let $r>0$ be a function on $\mathbb{R}^n$ such that there exists a constant $M_0\ge 0$ satisfying $r(x)\le \|x\|+M_0=\rho(x)$ for any $x\in \mathbb{R}^n$. We denote by $\mu_x$ the measure $\mu_x^{r(x)}$. When $r$ is Borel measurable, the Markovian kernel $(x,A)\mapsto \mu_x^{r(x)}(A)$ satisfies the condition (i) of the section above. \begin{lemma} \label{lm3.1} Let $f$ be a bounded function of class $\mathcal{C}^2$ in $\mathbb{R}^n$ such that $\Delta f$ is constant, then $f$ is constant. \end{lemma} \begin{proof} By replacing $f$ by $-f$ if necessary one can assume that $\Delta f=c$ with $c\le 0$, so that $f$ is a superharmonic function which we assume to be $\ge 0$ by adding to it a constant if needed. When $n=2$ the result is then immediate. If $n\ge 3$, $f$ is the Green potential of a measure $c\lambda$, hence $c=0$. We then deduce that $f$ is a bounded harmonic function, hence constant by the Liouville's classical theorem. \end{proof} \begin{theorem} \label{thm3.2} Let $r$ be a real function on $\mathbb{R}^n$, $n\ge 3$, such that $00$ on $U$ such that for any $x\in U$, $B(x,r(x))\subset U$ and \eqref{e1} takes place. If $U=\mathbb{R}^n$ assume that $f$ is bounded and that $r(x)\le \|x\|+M_0$, for some constant $M_0\in \mathbb{R}_+$. Suppose moreover that $\Delta f$ is a continuous or that $r$ is locally bounded from below by a constant $>0$. Then $f$ is harmonic. \end{corollary} We will study to the cases $n=1$ and $n=2$ in a forthcoming work. \smallskip \noindent \textbf{Remarks.} 1. We did not consider if the hypothesis that $\Delta f$ is bounded in the above theorem can be dropped. 2. If the function $r$ is bounded from below by a positive constant, then the condition that $f$ is bounded and the restricted biharmonic mean property in the above theorem implies that that the function $\Delta f$ is bounded. In fact, we have $$\sup_{x\in \mathbb{R}^n}|\Delta f(x)| \le \frac{4(n+2)}{\inf_{x\in \mathbb{R}^n}r(x)} \sup_{x\in \mathbb{R}^n}|f(x)|<+\infty.$$ \begin{thebibliography}{00} \bibitem{e1} M. El Kadiri, \emph{Une r\'eciproque du th\'eor\eme de la moyenne pour les fonctions biharmoniques}, Aequationes Math. 65, no 3 (2003) 280--287. \bibitem{e2} M. El Kadiri, \emph{Sur la propri\'et\'e de la moyenne restreinte pour les fonctions biharmoniques}, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 427-429. \bibitem{h1} W. Hansen, N. Nadirashvili, \emph{A converse to the mean value theorem for harmonic functions}, Acta Math., 171 (1993), 139-163. \bibitem{h2} W. Hansen, N. Nadirashvili, \emph{Mean Values and harmonic functions}, Math. Ann., 297 (1) (1993), 157-170. \bibitem{h3} W. Hansen, N. Nadirashvili, \emph{Liouville's Theorem and the restricted Mean Values Property}, J. Math. Pures App., (9) 74 (1995), no. 2, 185--198. \bibitem{n1} M. Nicolescu, \emph{Les fonctions polyharmoniques}, Paris, Hermann, 1936. \bibitem{v1} V. Volterra, \emph{Alcune osservationi sopra propriet\a atte individuare una funzione}, Atti della Reale Academia dei lincei, 18 (1909), 263-266. \end{thebibliography} \end{document}