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\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
$0<r(x)\le d(x,C\Omega)$ and
\begin{equation} \label{e1}
f(x)=\frac{1}{|B(x)|}
\int_{B(x)}f(y)dy-\frac{r(x)^2}{2(n+2)}\Delta f(x)
\end{equation}
for every $x\in \Omega$, where $B(x)$ is the open ball
of center $x$ and radius $r(x)$.

In \cite{e1} and \cite{e2}, we proved that, under certain conditions
on the functions  $f$ and $r$,  if  $f$ satisfies
the biharmonic mean property on the balls
$B(x)$ of center $x$ and radius $r(x)$,  then
$f$ is biharmonic. This result extends a result by
Hansen and Nadirashvili \cite{h1,h2} on functions
possessing the restricted (harmonic) mean property
to functions satisfying the restricted biharmonic
property.

Let $r: \mathbb{R}^n\to \mathbb{R}_+$
be such that $0< r(x)\le \|x\|+M_0$,
$n\ge 2$, where $M_0$ is a constant, and let  $f$ be a
Lebesgue measurable bounded function on  $\mathbb{R}^n$
which satisfies
$$f(x)=\frac{1}{|B(x)|}\int_{B(x)}f(y)dy
$$
for every $x\in \mathbb{R}^n$. Hansen and Nadirashvili \cite{h3} proved
that if $f$ is continuous or
if $r$ is  bounded from below on every  compact
set of $\mathbb{R}^n$ by some positive constant,
then $f$ is constant.
Our main goal in this work
is to extend this result to biharmonic functions,
to establish a Liouville's theorem
for functions having the restricted biharmonic mean
property.
For doing this, we  use representation measures
for harmonic functions used in  \cite{e1}, and   derived from
the biharmonic mean formula in \cite{h3}.  We will verify that the
conditions in \cite{h3} are satisfied and then use their results.

By a function we always mean a function with values in
$\overline{\mathbb{R}}$, unless otherwise stated.

The results of this paper extend easily to polyharmonic functions of order
greater than 2. We have treated only the biharmonic
case  for  reasons of simplicity.

\section{Liouville's theorem and the restricted biharmonic
mean property}

Let us recall some results of \cite{h2} and \cite{h3} that will be used
in the sequel.
Let $n\ge 3$ and let $G$ denote the  Green
kernel in $\mathbb{R}^n$ normalized in such a way that
for every $y\in \mathbb{R}^n$,
$\Delta G(.,y)=-\epsilon_y$ in the distribution sense. Recall
that
$$
G(x,y)=\frac{1}{\sigma_n(n-2)}\frac{1}{\|x-y\|^{n-2}}, \quad
\forall (x,y)\in (\mathbb{R}^n)^2,
$$
where $\sigma_n$ is the area of the unit sphere of $\mathbb{R}^n$.

Let  $\mathcal{B}(\mathbb{R}^n)$ be the Borel field.
 A kernel on $\mathbb{R}^n$ is a function
$N:\mathbb{R}^n\times \mathcal{B}(\mathbb{R}^n)
\to {\overline R}_+$ such that
\begin{enumerate}
\item For every $A\in \mathcal{B}(\mathbb{R}^n)$,
$x\mapsto N(x,A)$ is Borel measurable,

\item For every $x\in \mathbb{R}^n$,
$A\mapsto N(x,A)$ is a measure $\mu_x$ on $\mathcal{B}(\Omega)$.
\end{enumerate}
A kernel  $N$ on $\mathbb{R}^n$ is said to be
Markovian if $N(x,\mathbb{R}^n)=1$ for
$x\in \mathbb{R}^n$.


Fix a Markovian kernel
$(x,A)\mapsto \mu_x(A)$ on $\mathbb{R}^n$
such that, for some constants  $M_0\ge 1$, $0<\eta<1$,
$\gamma>0$ 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
$\mathbb{R}^n$, $0<r_0(x)\le \|x\|+M_0=:\rho(x)$ and a constant
$\gamma>0$  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 $0<r\le \|\cdot\|+M_0$ for some constant
$M_0\ge 1$. Then the kernel
$(x,E)\mapsto \lambda_{B(x,r(x))}(E)$
satisfies the conditions (i) and (ii), where
$\lambda_A$ denotes the probability measure
$\frac{1_A}{|A|}\lambda$ if
$|A|\ne 0$ (for Condition
(ii), it is sufficient to take $r_0(x)=r(x)/2$).

\begin{theorem}[{\cite[Cor. 2.4]{h3}}] \label{thm2.1}
Let $f$ be a Lebesgue measurable bounded function
on $\mathbb{R}^n$ such that
$$ f(x)=\int fd\mu_x $$
for every $x\in \mathbb{R}^n$. Assume that $f$ is
continuous or that $r_0$ is locally bounded from below by a constant
$>0$.
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
$0<r(x)\le \|x\|+M_0$, and let  $f$
be a bounded Lebesgue measurable function
whose Laplacian in the distribution sense is a bounded function
and such that
\begin{equation} \label{e2}
f(x)=\frac{1}{|B(x)|}\int_{ B(x)}
fd\lambda-\frac{r(x)^2}{2(n+2)}\Delta f(x),
\end{equation}
for every $x\in \mathbb{R}^n$.
If $f$ is continuous or if $r$ is locally bounded from below by a positive constant,
then $f$ is constant.
\end{theorem}

\begin{proof} As in \cite{e1}, we remark that
if $f$ satisfies \eqref{e2} for every
 $x\in \mathbb{R}^n$, then
$\Delta f(x)=\int \Delta f d\mu_x$ for every
$x\in \mathbb{R}^n$. Following  \cite[Sec.\ 2.1]{v1}, we may
assume that $r$ is Borel measurable. We have already seen that
$r$ satisfies condition (i) of the previous
section, we shall prove that it satisfies also
condition (ii). Then  the theorem follows then from
Theorem \ref{thm2.1} and Lemma \ref{lm3.1}.
Let $r_0(x)=r(x)/2$ for $x\in \Omega$.
Then  it is easy to verify that the condition
$\mu_x\ge \gamma\lambda_{B(x,r_0(x))}$  is satisfied for some
constant  $\gamma$ (which does not depend on  $x$).


For simplicity, let $x=0$ and
assume that  $B(x)=B$, the unit ball of  $\mathbb{R}^n$,
let $w=w_B(0,.)$. Then it is not difficult to see that
$w=\kappa_n(|\cdot|^{-n}+(\frac{n}{2}-1)|\cdot|^2
-\frac{n}{2})$. The function  $w$ is invariant
under rotations around 0 and that
$w\approx (1-|\cdot |)^2$ on
$\{\eta <|\cdot|<1\}$. Let us take
$\psi:]0,1[\to \mathbb{R}_+$
such $\psi(|z|)=w(z)$ and let
$\sigma_t$ be the normalized area  measure
on $\{|\cdot|=1\}$. Then
$$
G_\Omega^{\epsilon_0}-G_\Omega^{\mu_0}
= G_B^{\epsilon_0}-G_B^{\mu_0}
= \mathop{\rm const}\int_0^1(G_B^{\epsilon_0}-G_B^{\sigma_t})\psi(t)
t^{n-1}dt,
$$
where for  $\eta<|z|<1$,
$$
(G_B^{\epsilon_0}-G_B^{\sigma_t})(z)
= \kappa_n(|z|^{2-n}-t^{2-n})^+
\le \mathop{\rm const}(t-|z|)^+,
$$
and then
\begin{align*}
(G_\Omega^{\epsilon_0}-G_\Omega^{\mu_0})(z)
&\le C\int_{|z|}^1(t-|z|)\psi(t)dt\\
&= C\psi(|z|)\int_0^1(t-|z|)dt\\
&=\frac{C}{2}w(z)(1-|z|)^2.
\end{align*}
Then condition (ii) is satisfied for  $r_0(x)=\frac{r(x)}{2}$ and
$\eta=1$. Hence, it follows from
Theorem  2.1 that the function
$\Delta f$ is constant, therefore $f$ is constant
by  Lemma 3.1.
\end{proof}

We say that  a function  $f$  is bounded by another
function $g\ge 0$ on an open set $U$ of $\mathbb{R}^n$
if  $|f|\le g$ on $U$.
By combining the above theorem with \cite[Theorems 9 and 10]{e1}
(\cite[Theorems 2.1 and 2.2]{e2}), we obtain the following result.

\begin{corollary} \label{coro1}
Let $f$ be a locally integrable function
in a  domain $U$ of $\mathbb{R}^n$, $n\ge 3$, whose  Laplacian,
in the distribution sense, is bounded by an harmonic function.
Let $r$ be a real function  $>0$ 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}