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\markboth{\hfil Uniqueness results for Bernoulli problems\hfil EJDE--2002/102}
{EJDE--2002/102\hfil P. Cardaliaguet \& R. Tahraoui \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 102, pp. 1--16. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
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Some uniqueness results for Bernoulli interior free-boundary
problems in convex domains
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\thanks{ {\em Mathematics Subject Classifications:} 35R35.
\hfil\break\indent
{\em Key words:} Bernoulli free-boundary problem, convex solutions,
Borell's inequality.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted January 3, 2002. Published December 10, 2002.} }
\date{}
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\author{Pierre Cardaliaguet \& Rabah Tahraoui }
\maketitle
\begin{abstract}
We establish the existence of a elliptic family of convex solutions
for Bernoulli interior free-boundary problems in bounded convex
domains. We also proved that there is a unique solution to the
problem associated with the so-called Bernoulli constant, and give
an estimate from above for this constant.
\end{abstract}
\newtheorem{Theorem}{Theorem}[section]
\newtheorem{Lemma}[Theorem]{Lemma}
\newtheorem{Corollary}[Theorem]{Corollary}
\newtheorem{Remark}[Theorem]{Remark}
\numberwithin{equation}{section}
\section{Introduction}
Let $\Omega$ be an open bounded subset of $\mathbb{R}^N$ and let $\lambda>0$ be
fixed. For a sub-domain $D\subset \Omega$, the capacity
potential $u_D$ of $D$ in $\Omega$ is defined as the solution to
\begin{equation}\label{MainEq}
\begin{gathered}
-\Delta u= 0 \quad\mbox{in } \Omega\backslash \overline{D}\\
u=0 \quad\mbox{on }\partial \Omega\\
u=1 \quad \mbox{on } \partial D.
\end{gathered}
\end{equation}
The interior Bernoulli free-boundary problem is stated as follows:
Find a sub-domain $D$ such that capacity potential $u_D$ satisfies
$$
\forall x\in \partial D, \quad \frac{\partial u_D(x)}{\partial
n_x}=-\lambda\,,
$$
where $n_x$ is the outward normal to $D$ at $x$.
In the sequel, such a domain is called
a solution of Bernoulli problem of level $\lambda$. Bernoulli problem has been
extensively studied and we refer the reader to the survey paper \cite{FR} for
several motivations and references.
It is known that this problem does not have
a solution for any positive level $\lambda$. For instance, when $\Omega$ is
convex, it is proved in \cite{HS2} that there is some positive constant
$\lambda_\Omega$ such that
Bernoulli problem has a solution of level $\lambda$ if and only if
$\lambda\geq\lambda_\Omega$. This constant $\lambda_\Omega$ is called
Bernoulli constant.
It is also known that even if there are solutions to the problem for some
$\lambda$, there is no uniqueness
in general. For instance, if $\Omega$ is a ball, there are exactly two
solutions to the
problem for any $\lambda >\lambda_\Omega$, while for $\lambda_\Omega$
the solution is unique.
Let us now briefly describe the structure of the solutions when $\Omega$ is
a ball.
All the solutions are balls, with the same center as
$\Omega$ (c.f. \cite{Rei}). For any $\lambda>\lambda_\Omega$, let us denote
by $\widetilde{D_\lambda}$ the largest solution of level $\lambda$ and by
$\widehat{D_\lambda}$ the smallest one.
Then the family of largest solutions
$(\widetilde{D_\lambda})_{\lambda> \lambda_\Omega}$ forms a continuous
increasing family of balls, while the family of smallest solutions
$(\widehat{D_\lambda})_{\lambda> \lambda_\Omega}$ forms a continuous
decreasing family of balls. In Beurling terminology \cite{Be}, the
increasing family is
called an {\it elliptic family of solutions} while the decreasing family is
called a {\it hyperbolic family of solutions}. The unique solution
corresponding to the Bernoulli constant $\lambda_\Omega$ is called {\it
parabolic}. It is the limit of the $(\widetilde{D_\lambda})$ and of the
$(\widehat{D_\lambda})$ when $\lambda\to\lambda_\Omega$. Finally, the limit of
the elliptic family when $\lambda\to+\infty$ is equal to $\Omega$, while the
limit of the hyperbolic family when $\lambda\to+\infty$ is reduced to the
center of the ball. In particular, the boundary of the solutions of Bernoulli
problem completely cover $\Omega$ but the center.
A very interesting - and open - question is whether for general convex
bounded sets $\Omega$ the structure of the solutions of Bernoulli problem
enjoys similar features. In this paper we try to provide some positive
evidence towards this conjecture.
Let us first recall that, for any bounded and convex domain $\Omega$, and for any
fixed
volume $\sigma>0$, there is at least one convex solution to Bernoulli free
boundary problem of volume $\sigma$. This has already been proved
in \cite{Ack81}, Theorem 3 and in \cite{CS}, Theorem 5.1.
In the first part of this paper, we give a new - and we hope enlighting - proof of
this result.
In \cite{HS2}, Henrot and Shahgholian
proved the existence, for any $\lambda\geq \lambda_\Omega$, of a maximal convex
solution $\widetilde{D_\lambda}$ of level $\lambda$. Moreover the family
$(\widetilde{D_\lambda})_{\lambda\geq \lambda_\Omega}$ turns out to be increasing
with respect to the inclusion. In order to prove that this family is an elliptic family of solutions,
it remains to show that it is continuous. This has partially been established by
Acker in \cite{Ack}, Theorem 6.5., where it is proved the existence of some
constant $\lambda_0$ (with $\lambda_0\geq \lambda_\Omega$)
sufficently large such that
the subfamily $(\widetilde{D_\lambda})_{\lambda\geq \lambda_0}$
is continuous (\cite{Ack}, Theorem 6.5 p. 1418). This implies the uniqueness
of the solutions among the sets with a boundary close of the boundary of $\Omega$.
One of the main contribution of this paper is the fact that the full
family $(\widetilde{D_\lambda})_{\lambda\geq \lambda_\Omega}$ is continuous.
More precisely we prove the following result:
\begin{Theorem}\label{uniqueness1}
The family $(\widetilde{D_\lambda})_{\lambda>
\lambda_\Omega}$ is elliptic (i.e., increasing
and continuous) and we have the following inclusion: For any
$\lambda_0$ and $\lambda_1$ larger than $\lambda_\Omega$, for any $s\in[0,1]$,
\begin{equation}\label{InclConv}
(1-s)\widetilde{D_{\lambda_0}}+s\widetilde{D_{\lambda_1}} \subset
\widetilde{D_{\gamma_s}}\,,
\end{equation}
where $\gamma_s=1/[(1-s)/\lambda_0+s/\lambda_1]$.
\end{Theorem}
\paragraph{Remark:}
Since $\gamma_s\leq (1-s)\lambda_0+s\lambda_1$ and the family
$(\widetilde{D_\lambda})_{\lambda\geq
\lambda_\Omega}$ is increasing, Theorem \ref{uniqueness1} implies that
$$
(1-s)\widetilde{D_{\lambda_0}}+s\widetilde{D_{\lambda_1}} \subset
\widetilde{D}_{(1-s)\lambda_0+s\lambda_1}\,.
$$
This property can be viewed as a ``concavity property" of the familly
$(\widetilde{D_\lambda})_{\lambda\geq
\lambda_\Omega}$.
The continuity of the family is a simple consequence of the ``inequality of
concavity" (\ref{InclConv}).
This continuity is important to get uniqueness
results.
Indeed, by standard comparison principle, the ellipticity of the family implies
that any classical solution of Bernoulli problem $D$
containing $\widetilde{D_{\lambda_\Omega}}$ belongs to the family
$(\widetilde{D_\lambda})_{\lambda\geq \lambda_\Omega}$. Namely
there is some $\lambda\geq\lambda_\Omega$ with $D=D_\lambda$. This remark
can be found in \cite[Theorem 6.4]{Ack}.
Let us also point out that Theorem \ref{uniqueness1} (or, more precisely its proof)
implies that $(\widetilde{D_\lambda})_{\lambda\geq \lambda_\Omega}$ is the unique
elliptic family of convex solution (cf. Corollary \ref{Corolle}).
Concerning the parabolic solution, our main contribution is the following
result.
\begin{Theorem}\label{uniqueness2}
There is exactly one solution $\widetilde{D_{\lambda_\Omega}}$ of level
$\lambda_\Omega$.
\end{Theorem}
More precisely, we show in Theorem \ref{parabolic} below that $\widetilde{D_{\lambda_\Omega}}$
is the unique {\it subsolution} of level
$\lambda_\Omega$ (the definition of a subsolution is given later).
This result is much deeper than Theorem \ref{uniqueness1}.
Indeed, it cannot simply rely upon an ``inequality of concavity" of the form (\ref{InclConv}).
In fact the key point for the proof of Theorem \ref{uniqueness1} is an inequality due
to Borell in \cite{Bo}, whereas for Theorem \ref{uniqueness2} it is necessary to investigate
the cases of equality in Borell inequality. This later result has been obtained by the authors
in \cite{CT}.
Since the Bernoulli constant plays a crucial role in this study, we complete the paper by giving
a new estimate from above for the Bernoulli constant. This inequality is optimal in the sense
that it is exact for balls.
In conclusion, this paper gives a fairly complete picture for
the elliptic solutions and for a parabolic solution of the interior Bernoulli free boundary
problem. However the
question of the (conjectured) uniqueness of the hyperbolic family of solutions
remains open.
Let us just give a possible starting point in this
direction: Following \cite{FR}, it is known that, when the volume
$\sigma\to0^+$,
the solutions of Bernoulli problem of volume $\sigma$ become closer and
closer to balls and
concentrate at some points of $\Omega$ called the harmonic centers of
$\Omega$. It is known
that the harmonic center of a convex bounded domain is unique
(this is proved in \cite{CF} for $N=2$, and in \cite{CT} for $N\geq3$).
Therefore the solutions of volume $\sigma$ concentrate to
this unique harmonic center when $\sigma\to0^+$.
Let us finally explain how this paper is organized. In section
\ref{preuvetheo1}, we give a new proof for the existence, for any
volume $\sigma$, of a solution of Bernoulli problem of volume
$\sigma$. Sections 3 and 4 are respectively devoted to
the proof of Theorems \ref{uniqueness1} and \ref{uniqueness2}.
In the last part of the paper we give some estimate for the Bernoulli
constant.
\section{Existence of convex solutions}\label{preuvetheo1}
The aim of this section is to give a
new proof of the following result.
\begin{Theorem}[\cite{CF,Ack81}]\label{existence}
If $\Omega$ is an open bounded convex domain of $\mathbb{R}^N$, then, for any volume
$\sigma\in(0,|\Omega|)$ there is at least one convex solution of Bernoulli
problem of volume $\sigma$.
More precisely, there is a solution which is a minimizer of the problem:
$$
\inf\{ \mathop{\rm cap}(D)\;|\; D\Subset\Omega\;\mbox{\rm
is convex and $|D|=\sigma$}\}\,.
$$
\end{Theorem}
\paragraph{Remark:} It would be interesting to know
if, for any $\sigma \geq |\widetilde{D_{\lambda_\Omega}}|$,
the set $\widetilde{D_{\lambda}}$ of volume $\sigma$ is a minimizer
of the problem.
For proving Theorem \ref{existence}, let us
introduce the following function: $\forall \sigma>0$,
\begin{equation}\label{defF}
F(\sigma)=\inf\{ \mathop{\rm cap}(D)\;|\; D\Subset\Omega\;\mbox{\rm
is convex and $|D|=\sigma$}\}\,,
\end{equation}
where $|D|$ stands for the volume of $D$ and $\mathop{\rm cap}(D)$ is the capacity of
$D$ in
$\Omega$, i.e.,\begin{equation}\label{capa}
\mathop{\rm cap}(D)=\inf \{\int_\Omega |\nabla u|^2\;|\; u\in H_0^1(\Omega),\;
u\geq 1\;{\rm in}\; D\}=\int_{\partial D}|\nabla u_D|\,.
\end{equation}
Standard arguments show that the infimum in the problem defining $F$ is
attained (see for instance \cite{AAC,BDM}).
\begin{Lemma}
The function $F$ is monotonically increasing.
\end{Lemma}
The proof follows from the fact that the capacity $D\to \mathop{\rm cap}(D)$
is monotonically increasing with respect to the inclusion.
\begin{Lemma}\label{derivable}
Let $\sigma$ be a point of derivability of $F$. Then any convex domain
realizing the minimum
in (\ref{defF}) is a solution to Bernoulli problem of level
$\lambda=\sqrt{F'(\sigma)}$.
\end{Lemma}
\paragraph{Remark:} The behaviour of the function $F$ seems to be
extremely relevant for describing the general
behaviour of the solutions of the Bernoulli free boundary problem.
In particular the question of the derivability of $F$, of its convexity
(or its concavity) properties are of crucial importance. From
Lemma \ref{derivable} one could
expect $F$ to be convex on $[\bar{\sigma}, |\Omega|)$
and concave on $(0, \bar{\sigma}]$, where $\bar{\sigma}$
is the volume of the solution of level $\lambda_\Omega$.
\paragraph{Proof of Lemma \ref{derivable}:} We
follow several arguments of Acker \cite{Ack}. Let $D$ realize the minimum
in (\ref{defF}).
From Poincar\'e's variational formula for the capacity (which
can be applied since the boundary of $D$ is Lipschitz, see for instance
\cite{FR}), we have
\begin{equation}\label{Popo}
\frac{d}{dh}\mathop{\rm cap}(D-hB)_{|_{h=0}}= -\int_{\partial D} |\nabla u_D|^2
\end{equation}
where we have set
$$
D-hB=\{x\in D\;|\;d_{\partial D}(x)> h\}\,,
$$
where $d_{\partial D}(x)$ denotes the distance of the point $x$ to the set
$\partial D$.
Since, for $h$ sufficiently small, we have
$$
|D-hB|=|D|-h|\partial D|+o(h)=\sigma -h|\partial D|+o(h)\,,
$$
we can deduce that
$$
F(\sigma-h|\partial D|+o(h))\leq \mathop{\rm cap}(D-hB)\,.
$$
Then, using (\ref{Popo}) and equality $F(\sigma)=\mathop{\rm cap}(D)$, we get:
\begin{equation}\label{un}
F'(\sigma)|\partial D| \geq \int_{\partial D} |\nabla u_D|^2\,.
\end{equation}
Let us now consider $D_h=\{u_{D}>1-h\}$. We know that
$\mathop{\rm cap}(D_h)= \mathop{\rm cap}(D)/(1-h)$.
Moreover,
$$
|D_h|=|D|+h\int_{\partial D}\frac{1}{|\nabla u_D|}+o(h)\,.
$$
Let us recall that $\nabla u\neq 0$ in $\Omega\backslash D$ (see \cite{Le}).
Hence, since $|D|=\sigma$, this gives
$$
F(\sigma+h\int_{\partial D}\frac{1}{|\nabla u_D|}+o(h))\leq
\mathop{\rm cap}(D_h)=\frac{\mathop{\rm cap}(D)}{1-h}\,.
$$
Therefore,
\begin{equation}\label{deux}
F'(\sigma)\Big(\int_{\partial D}\frac{1}{|\nabla u_D|}\Big)\leq
\mathop{\rm cap}(D)=\int_{\partial D}|\nabla u_D|\,.
\end{equation}
Putting (\ref{un}) and (\ref{deux}) together gives:
$$
\Big(\int_{\partial D}\frac{1}{|\nabla u_D|}\Big)
\Big(\int_{\partial D} |\nabla u_D|^2\Big)\;\leq |\partial D|
\Big(\int_{\partial D}|\nabla u_D|\Big)=
\Big(\int_{\partial D}1\Big)\Big(\int_{\partial D}|\nabla u_D|\Big),
$$
where we have used equality (\ref{capa}). Let us set for simplicity
$S=\partial D$ and $a(x)=|\nabla u_D(x)|$. Then the previous inequality can be
rewritten as
$$
\int_{S\times S} \Big(\frac{a(y)^2}{a(x)}-a(x)\Big)\leq 0\,.
$$
Since this expression is symmetric with respect to $x$ and $y$, it implies
$$
\int_{S\times S}\Big(
\frac{a(y)^2}{a(x)}+\frac{a(x)^2}{a(y)}-a(x)-a(y)\Big)\leq 0\,,
$$
i.e.,
$$
\int_{S\times S}
\frac{a(y)^2}{a(x)}\Big[1+\frac{a(x)^3}{a(y)^3}-\frac{a(x)^2}{a(y)^2}-
\frac{a(x)}{a(y)}\Big]\leq 0\,.
$$
Since the polynomial $t\to 1+t^3-t^2-t$ is positive for $t\geq0$ unless $t=1$,
the previous inequality implies that
$$
a(x)=a(y)\;\;\mbox{\rm for almost every } (x,y)\in S\times S\,.
$$
Therefore $a=|\nabla u_D|$ is constant on $S=\partial D$. This means that
$D$ is
a solution of Bernoulli problem. Using (\ref{un}) and (\ref{deux}) shows easily
that it is a solution of level $\lambda=\sqrt{F'(\sigma)}$.
\hfill$\square$
\paragraph{Proof of Theorem \ref{existence}:}
Let $\sigma>0$ be fixed. Since $F$ is
almost everywhere derivable, there is a sequence $(\sigma_n)$ converging to
$\sigma$ such that $F'(\sigma_n)$ exists. Let $D_n$ be a minimizer for
$F(\sigma_n)$. Then, since the $D_n$ are convex and bounded, they converge,
up to
a subsequence again denoted $(D_n)$ to some convex set $D$ with $|D|=\sigma$.
Moreover, from standard arguments in convex analysis, we also have that
$|\partial D_n|$ converges to $|\partial D|>0$. Let us now prove that the
sequence $F'(\sigma_n)$ is bounded. Indeed, we have
$$
\mathop{\rm cap}(D_n)= \int_{\partial D_n}|\nabla u_{D_n}|=|\partial
D_n|\sqrt{F'(\sigma_n)}\,,
$$
because, from Lemma \ref{derivable}, $D_n$ is a solution of Bernoulli
problem of
level $\sqrt{F'(\sigma_n)}$. Since $\mathop{\rm cap}(D_n)=F(\sigma_n)$ and $1/|\partial
D_n|$ are
bounded, we have proved that $F'(\sigma_n)$ is bounded. Thus $(F'(\sigma_n))$
converges (up to a subsequence again denoted $(F'(\sigma_n))$) to some
$\lambda\geq0$. Then standard arguments show that $D$, as a limit of convex
solutions of Bernoulli problem of level $F'(\sigma_n)$, is also a convex
solution of Bernoulli problem of level $\lambda$. Since $|D|=\sigma$, this
completes the proof of Theorem \ref{existence}. \hfill$\square$
\section{The elliptic family of solutions}
The aim of this section is to prove Theorem
\ref{uniqueness1}. Let us first recall the main results of \cite{HS2}
concerning the construction of the
maximal solution $\widetilde{D_\lambda}$ of Bernoulli problem of level
$\lambda$.
A {\it subsolution of the Bernoulli problem of level $\lambda$} is a
set $D\Subset \Omega$ such that
$$
u_D\; \mbox{\rm is Lipschitz continuous and }
\frac{\partial u_D}{\partial \nu_x}\geq-\lambda\;{\rm on}\; \partial
D\,.
$$
Let us introduce for any $\lambda$ the family of subsolutions:
$$
{\cal F}_\lambda=\{D\Subset \Omega\;|\;
u_D\; \mbox{\rm is Lipschitz continuous and }
\frac{\partial u_D}{\partial \nu_x}\geq-\lambda\;{\rm on}\; \partial
D\}\,, $$
where $\nu_x$ denotes the outward normal to $D$ at $x$, and
$u_D$ is the capacity potential of $D$, i.e., the solution of
(\ref{MainEq}). Let us point out that, if a domain $D$ is a solution of
Bernoulli problem
of level $\lambda$, then $D$ belongs to ${\cal F}_\lambda$. Let us set
$$
\lambda_\Omega=\inf\{\lambda\;|\; {\cal F}_\lambda\neq\emptyset\}\,.
$$
Then it is proved in \cite{HS2} that $\lambda_\Omega>0$ and that
$\forall\lambda\geq\lambda_\Omega$,
the set ${\cal F}_\lambda$ is not empty.
The main result of
\cite{HS2} states that, for any $\lambda\geq\lambda_\Omega$, the set
$$
\widetilde{D_\lambda}=\overline{\mathop{\rm Co}}\Big(\bigcup_{D\in{\cal
F}_\lambda}D\Big) $$
is the maximal solution of Bernoulli problem of level $\lambda$, where
$\overline{\mathop{\rm Co}}(A)$ denotes the closed convex hull of a set $A$.
To prove Theorem \ref{uniqueness1}, we need some preliminary results about an inequality
due to Borell \cite{Bo} that we describe now. Let $D_0$ and $D_1$ be two
convex, open subdomains of
$\Omega$. For $s\in [0,1]$,
we denote by $D_s$ the following set:
\begin{align*}
D_s=&(1-s)D_0+s D_1\\
=&\{x\in\mathbb{R}^N,\; \exists x_0\in D_0,\;\exists
x_1\in D_1\;{\rm with}\; x=(1-s)x_0+s x_1\}\,.
\end{align*}
Let us recall that $D_s$ is a convex, open subdomain of $\Omega$.
Following Borell, we denote by $\widetilde{u}_s$ the function
\begin{equation}\label{deftu}
\forall x\in \Omega\backslash D_s,\;\;
\widetilde{u}_s(x)=\sup_{x_0,x_1}\min\{u_{D_0}(x_0),u_{D_1}(x_1)\}
\end{equation}
where the supremum is taken over the $x_0\in\Omega\backslash D_0$ and
$x_1\in\Omega\backslash D_1$ such that
$x=(1-s)x_0+s x_1$.
Borell's inequality states that
\begin{equation}\label{borell}
u_{D_s}(\cdot)\geq \widetilde{u}_s(\cdot)\quad\mbox{in } \Omega\backslash \overline{D_s}\,.
\end{equation}
Moreover, $\widetilde{u}_s$ is continuous on
$\overline{\Omega}\backslash D_s$,
$$
\widetilde{u}_s=0 \quad\mbox{on }\partial \Omega, \quad\mbox{and}\quad
\widetilde{u}_s=1 \quad\mbox{on }\partial D\,.
$$
In \cite{CT}, we have refined Borell's inequality in establishing that the map
$\widetilde{u}_s$ is in fact a subsolution of Laplace equation in the viscosity sense
(for the definition of this
notion, see
\cite{CIL}). Namely:
\begin{Lemma}\label{subsol}
In the viscosity sense,
$$
-\Delta \widetilde{u}_s \leq 0\quad\mbox{in }\Omega\backslash \overline{D_s}.
$$
\end{Lemma}
We use this fact in the next section together with the following sharp
estimate of the case
of equality in Borell's inequality, that we have established in \cite{CT}.
\begin{Theorem} \label{strict}
Assume that for some $s\in(0,1)$ the function $\widetilde{u}_s$ is harmonic.
Then $D_0=D_1$.
\end{Theorem}
\paragraph{Remark:}
We shall mainly use this result combined with Lemma \ref{subsol}
and Borell's inequality (\ref{borell}) in the following way:
If $D_0\neq D_1$ and $s\in(0,1)$,
then $u_{D_s}-\widetilde{u}_s$ is a non-negative, non-zero, superharmonic function.
The key point of the proof of Theorem \ref{uniqueness1} is the
following lemma.
\begin{Lemma}\label{IneqCruc}
Let $\lambda_0$ and $\lambda_1$ be not smaller than $\lambda_\Omega$. Then, for
any convex sets $D_0$ and $D_1$, such that $D_0\in{\cal F}_{\lambda_0}$ and
$D_1\in{\cal F}_{\lambda_1}$, for any $s\in[0,1]$, the set $D_s=(1-s)D_0+sD_1$
belongs to ${\cal F}_{\gamma_s}$, where
$$
\gamma_s=\frac{1}{\frac{1-s}{\lambda_0}+\frac{s}{\lambda_1}}\,.
$$
\end{Lemma}
\paragraph{Remark:} Note that $\gamma_s\leq(1-s)\lambda_0+s\lambda_1$
and thus
$$
D_s=(1-s)D_0+sD_1 \;\in\;{\cal F}_{(1-s)\lambda_0+s\lambda_1}\,.
$$
\paragraph{Proof of Lemma \ref{IneqCruc}:}
For simplicity, we set $u_0=u_{D_0}$ and $u_1=u_{D_1}$.
Following Gabriel \cite{Ga1, Ga2, Ga3} and Lewis \cite{Le}, the level sets
of the functions
$u_i$ are smooth and strictly convex, and $\nabla u_i\neq 0$ in
$\Omega\backslash \overline{D_i}$. From Lemma 2.2 in \cite{HS2}, $D_s$ belongs
to ${\cal F}_{\gamma_s}$ if and only if $$
\frac{\partial u_{D_s}(x)}{\partial
\nu_x}\geq -\gamma_s\;\mbox{\rm for almost all } x\in\partial D_s \; ,
$$
where $\nu_x$ denotes the outward normal to $D_s$ at $x$. The partial
derivative has to be understood in the sense
$$
\frac{\partial u_{D_s}(x)}{\partial
\nu_x}=\lim_{h\to0^+}\frac{u_{D_s}(x+h\nu_x)-1}{h}
$$
and exists almost everywhere on $\partial D_s$ (see \cite{Da}). Let us now
fix some
$x\in\partial D_s$ point where the previous limit exists and where $D_s$ has a
unique unit normal $\nu_x$. From Borell's inequality (\ref{borell}), we have
$$
\lim_{h\to0^+}\frac{u_{D_s}(x+h\nu_x)-1}{h}\geq
\limsup_{h\to0^+}\frac{\widetilde{u}_s(x+h\nu_x)-1}{h}\,.
$$
Let us now recall some results of \cite{Bo} (see also \cite{CT}):
First it is proved that $\widetilde{u}_s$ is ${\cal C}^1$ in
$\Omega\backslash\overline{D_s}$. Second, it is also proved that for any $x\in
\Omega\backslash\overline{D_s}$, there exists a unique pair $(x_0,x_1)$
belonging to
$(\Omega\backslash\overline{D_0})\times(\Omega\backslash\overline{D_1})$,
such that
$$
\widetilde{u}_s(x)=u_0(x_0)=u_1(x_1) \;{\rm and }\; x=(1-s)x_0+sx_1\,.
$$
Moreover,
$\nabla \widetilde{u}_s(x)$ is given by
$$
\frac{\nabla \widetilde{u}_s(x)}{|\nabla \widetilde{u}_s(x)|}=
\frac{\nabla u_0(x_0)}{|\nabla u_0(x_0)|}=
\frac{\nabla u_1(x_1)}{|\nabla u_1(x_1)|}
$$
and
$$
\frac{1}{|\nabla \widetilde{u}_s(x)|}=\frac{1-s}{ |\nabla u_0(x_0)|}+
\frac{s}{|\nabla u_1(x_1)|}\,.
$$
Let us now consider $\xi_h\in [x, x+h\nu_x]$ such that
\begin{equation}\label{frac}
\frac{\widetilde{u}_s(x+h\nu_x)-1}{h}= \langle\nabla \widetilde{u}_s(\xi_h),\nu_x\rangle\,.
\end{equation}
Note that $\xi_h\to x$ when $h\to0^+$.
We now apply the results of \cite{Bo} recalled above to the point $\xi_h$:
There are $\xi_h^i\in\Omega\backslash
\overline{D_i}$ such that
$$
\widetilde{u}_s(\xi_h)=u_0(\xi_h^0)=u_1(\xi_h^1) \;{\rm and }\; \xi_h=(1-s)\xi_h^0+s\xi_h^1
$$
and
$$
\frac{\nabla \widetilde{u}_s(\xi_h)}{|\nabla \widetilde{u}_s(\xi_h)|}=
\frac{\nabla u_0(\xi_h^0)}{|\nabla u_0(\xi_h^0)|}=
\frac{\nabla u_1(\xi_h^1)}{|\nabla u_1(\xi_h^1)|}
$$
and, finally,
\begin{equation}\label{frac2}
\frac{1}{|\nabla \widetilde{u}_s(\xi_h)|}=\frac{1-s}{ |\nabla u_0(\xi_h^0)|}+
\frac{s}{|\nabla u_1(\xi_h^1)|}\,.
\end{equation}
Since $D_i$ belongs to ${\cal F}_{\lambda_i}$, we have
$$
|\nabla u_0(\xi_h^0)|\leq \lambda_0\;{\rm and }\;
|\nabla u_1(\xi_h^1)|\leq \lambda_1\,.
$$
Hence, from (\ref{frac2}) and the previous inequalities, we obtain
\begin{equation}\label{tutu}
|\nabla \widetilde{u}_s(\xi_h)|\leq 1/[(1-s)/ \lambda_0+s/ \lambda_1]=\gamma_s\,.
\end{equation}
Let us now consider a sequence $h_n\to0^+$ such that
\begin{equation}\label{frac3}
\limsup_{h\to0^+}\frac{\widetilde{u}_s(x+h\nu_x)-1}{h}=\lim_n
\frac{\widetilde{u}_s(x+h_n\nu_x)-1}{h_n}\,.
\end{equation}
Let us set $\xi_n=\displaystyle{\xi_{h_n}}$. Since
$a_n=-\nabla \widetilde{u}_s(\xi_n)/|\nabla \widetilde{u}_s(\xi_n)|$ is an outward normal to the convex
set $\{\widetilde{u}_s\geq \widetilde{u}_s(\xi_n)\}$ at $\xi_n$, a standard passage to the limit shows
that $a=\lim_n a_n$ is an outward normal to the set $D_s$ at $x$ because
the $\xi_n$
converge to $x$,
$x$ belongs to $\partial D_s$ and the
convex set $\{\widetilde{u}_s\geq \widetilde{u}_s(\xi_n)\}$ converges to the convex set
$D_s$. Since, from our assumption, $D_s$ has a unique outward
normal at $x$, namely $\nu_x$, we have $a=\nu_x$. Hence, from (\ref{frac}) and
(\ref{frac3}), we get $$
\limsup_{h\to0^+}\frac{\widetilde{u}_s(x+h\nu_x)-1}{h}= \lim_n
\langle \nabla \widetilde{u}_s(\xi_n),\nu_x\rangle=
-\lim_n |\nabla \widetilde{u}_s(\xi_n)|\,.
$$
Using (\ref{tutu}), we prove the desired result:
$$
\lim_{h\to0^+}\frac{u_{D_s}(x+h\nu_x)-1}{h}\geq -\gamma_s\,.
$$
\hfill$\square$
\paragraph{Proof of Theorem \ref{uniqueness1}:} We first prove
(\ref{InclConv}). Let $\lambda_0$ and $\lambda_1$ be fixed.
From \cite{HS2}, the convex sets
$\widetilde{D_{\lambda_0}}$ and $\widetilde{D_{\lambda_1}}$ belong respectively
to ${\cal F}_{\lambda_0}$ and to ${\cal F}_{\lambda_1}$. Lemma \ref{IneqCruc}
then states that, for any $s\in[0,1]$, the convex set
$$
D_s= (1-s) \widetilde{D_{\lambda_0}}+s\widetilde{D_{\lambda_1}}
$$
belongs to ${\cal F}_{\gamma_s}$, where $\gamma_s$ is defined as in Lemma
\ref{IneqCruc}.
Accordingly, from the construction of the solution $\widetilde{D_{\lambda}}$,
we have
$$
D_s\subset \widetilde{D_{\gamma_s}}
$$
This proves (\ref{InclConv}).
We now prove the continuity of the family
$(\widetilde{D_{\lambda}})_{\lambda>\lambda_\Omega}$. Let
$\lambda>\lambda_\Omega$ be fixed. From the construction of the solution
$\widetilde{D_{\lambda}}$ and standard stability results, we have easily that
$$
\widetilde{D_{\lambda}}=\mathop{\rm Int}\Big(\bigcap_{\lambda'>\lambda}
\widetilde{D_{\lambda'}}\Big)\,.
$$
This proves the continuity on the right. For proving the continuity on the
left,
let us set
$$
D=\bigcup_{\lambda'<\lambda}
\widetilde{D_{\lambda'}}\,.
$$
We already know that $D\subset \widetilde{D_{\lambda}}$ and we want to prove
the equality. We argue by contradiction, by assuming that $D\neq
\widetilde{D_{\lambda}}$.
Let us first notice that $D$ is a convex solution of Bernoulli problem of level
$\lambda$, as limit of convex solutions of Bernoulli problem of level
$\lambda'$, with $\lambda'\to\lambda$. Thus its boundary is smooth (see for
instance \cite{HS2}). Since $\widetilde{D_{\lambda}}$ is also a solution of the
Bernoulli problem of level $\lambda$, the strong maximum principle implies that
the boundary of the sets $D$ and $\widetilde{D_{\lambda}}$ are disjoint. Hence
$D\Subset \widetilde{D_{\lambda}}$.
We now choose some $\lambda_0\in (\lambda_\Omega,\lambda)$. Let $s$ be
the largest real in $(0,1)$ such that
$$
(1-s)\widetilde{D_{\lambda_0}}+s\widetilde{D_{\lambda}}\subset D\,.
$$
Let us notice that $s$ belongs to $(0,1)$ and that, if we set
$D_s=(1-s)\widetilde{D_{\lambda_0}}+s\widetilde{D_{\lambda}}$, the
boundaries of
$D_s$ and $D$ have a non empty intersection. Let $x$ belong to this
intersection. From Lemma \ref{IneqCruc}, we know that $D_s$ belongs to ${\cal
F}_{\gamma_s}$, where $\gamma_s=1/[(1-s)/\lambda_0+s/\lambda]$. Let us notice
that $\gamma_s$ is smaller than $\lambda$. Since $D_s\subset D$, we have
$u_{D_s}\leq u_D$. Let $\nu_x$ be the normal to $D_s$ and $D$ at $x$. We
have $$
\lambda=\lim_{h\to0^+}\frac{1-u_D(x+h\nu_x)}{h}\leq
\lim_{h\to0^+}\frac{1-u_{D_s}(x+h\nu_x)}{h}\leq \gamma_s\,.
$$
Hence there is a contradiction, since we have in fact $\gamma_s<\lambda$. This
completes the proof.
\hfill$\square$
The same proof shows that the family $(\widetilde{D_\lambda})_{\lambda>
\lambda_\Omega}$ is the unique elliptic family of convex solutions. Namely:
\begin{Corollary}\label{Corolle}
Let $D_0$ and $D_1$ be two convex solutions of Bernoulli problem respectively of level
$\lambda_0$ and $\lambda_1$. Assume that
$D_0\Subset D_1$ and $\lambda_0<\lambda_1$.
Then $D_1=\widetilde{D_{\lambda_1}}$ .
\end{Corollary}
\paragraph{Proof:} Replace in the proof of Theorem \ref{uniqueness1} the set
$D_{\lambda_\Omega}$ by $D_0$, $D$ by $D_1$ and $\widetilde{D_\lambda}$ by
$\widetilde{D_{\lambda_1}}$.
\hfill$\square$
\section{Uniqueness of the parabolic solution}
We finally investigate the special case of the parabolic solution, i.e., the
solution of level $\lambda_\Omega$. Our aim is to establish the uniqueness of
the solution. We are in fact going to prove a stronger result. Namely:
\begin{Theorem}\label{parabolic}
With the notations of the previous section, we have
$$
{\cal F}_{\lambda_\Omega}=\{\widetilde{D_{\lambda_\Omega}}\}\,.
$$
\end{Theorem}
Note that Theorem \ref{parabolic} implies Theorem \ref{uniqueness2}, since a
solution $D$ of the Bernoulli problem of level $\lambda_\Omega$ always belongs
to ${\cal F}_{\lambda_\Omega}$.
\paragraph{Proof of Theorem \ref{parabolic}:}
We argue by contradiction. Let us assume that there is an open set $D$
belonging to ${\cal F}_{\lambda_\Omega}$, with
$D\neq \widetilde{D_{\lambda_\Omega}}$.
The definition of $ \widetilde{D_{\lambda_\Omega}}$
implies that $D\subset \widetilde{D_{\lambda_\Omega}}$.
From Lemma 2.4 of \cite{HS2}, we know that the convex hull of $D$, denoted
by $D_0$
also belongs to ${\cal F}_{\lambda_\Omega}$. We claim that
$D_0\Subset \widetilde{D_{\lambda_\Omega}}$. Indeed, otherwise,
there should exist some $x$ belonging
to the intersection of the boundary of $D$
and the boundary of $\widetilde{D_{\lambda_\Omega}}$.
Since $D\subset\widetilde{D_{\lambda_\Omega}}$ and $D\neq
\widetilde{D_{\lambda_\Omega}}$,
Hopf maximum principle then would imply that
$|\nabla u_D(x)|>|\nabla u_{\widetilde{D_{\lambda_\Omega}}}(x)|$
at this point $x$. But this is impossible since $|\nabla u_D(x)|\leq \lambda_\Omega$
because $D$ is a sub-solution and
$|\nabla u_{\widetilde{D_{\lambda_\Omega}}}(x)|=\lambda_\Omega$
because $\widetilde{D_{\lambda_\Omega}}$ is a solution of Bernoulli problem
of level $\lambda_\Omega$. Hence we have proved that
$D_0\Subset \widetilde{D_{\lambda_\Omega}}$.
We now consider the convex combination
$D_s=(1-s)D_0+s\widetilde{D_{\lambda_\Omega}}$ for some $s\in(0,1)$.
To achieve the proof of our Theorem, it suffices to prove
that $D_s$ belongs in fact to ${\cal F}_{\lambda_\Omega-\epsilon}$, for some
$\epsilon>0$. Indeed this leads to a contradiction because ${\cal
F}_{\lambda_\Omega-\epsilon}$ is empty from the definition of $\lambda_\Omega$.
We now prove
that $D_s$ belongs to ${\cal F}_{\lambda_\Omega-\epsilon}$ for some
$\epsilon>0$. At this step, we have to underline that
Borell's inequality is no longer enough for proving that $D_s$ belongs to
some
${\cal F}_{\lambda_\Omega-\epsilon}$. Indeed, Borell's inequality
only gives that $D_s$ belongs to ${\cal F}_{\lambda_\Omega}$ (see Lemma
\ref{IneqCruc}). Therefore we have to use a stronger argument:
This argument is
Theorem \ref{strict}, which states that, since
$D_0\neq\widetilde{D_{\lambda_\Omega}}$, the map $\widetilde{u}_s$
defined by (\ref{deftu}) {\it cannot} be a solution of Laplace equation. Since
$\widetilde{u}_s$ is a subsolution of this equation (cf. Lemma \ref{subsol}), this shows
that the map $u_{D_s}-\widetilde{u}_s$ is a {\it non-negative} viscosity supersolution of
Laplace equation,
vanishing at the boundary $\partial\Omega\cup\partial D_s$. Using the fact that
$D_s$ is convex and bounded, Hopf maximum principle states that there is a
neighborhood $U$ of $\partial D_s$ and some positive constant $\epsilon$ such
that
$$
\forall x\in U\backslash D_s, \quad u_{D_s}(x)-\widetilde{u}_s(x)\geq \epsilon
d_{D_s}(x)\,, $$
where $d_{D_s}(x)$ denotes the distance from the point $x$ to the set $D_s$.
We now
argue as in the proof of Lemma \ref{IneqCruc}: We have, for almost every
$x\in\partial D_s$, where there is a unique outward normal $\nu_x$ to $D_s$ at
$x$,
$$
\lim_{h\to0^+}\frac{u_{D_s}(x+h\nu_x)-1}{h}\geq
\limsup_{h\to0^+}\frac{\widetilde{u}_s(x+h\nu_x)-1}{h}+\epsilon
$$
since $ d_{D_s}(x+h\nu_x)=h$. We can estimate the term in $\limsup$ as in
the proof of Lemma \ref{IneqCruc} (with now
$\lambda_0=\lambda_1=\lambda_\Omega$): This gives
$$
\limsup_{h\to0^+}\frac{\widetilde{u}_s(x+h\nu_x)-1}{h}\geq -\lambda_\Omega\,.
$$
Hence we have
$$
\lim_{h\to0^+}\frac{u_{D_s}(x+h\nu_x)-1}{h}\geq -\lambda_\Omega+\epsilon\,.
$$
Using Lemma 2.2 of \cite{HS2}, this proves that $D_s$ belongs to ${\cal
F}_{\lambda_\Omega-\epsilon}$ and gives the desired contradiction.
\hfill$\square$
\section{Estimate for the Bernoulli constant}
Our aim is to estimate from above the Bernoulli constant $\lambda_\Omega$.
For an estimate from below, let us recall that the following question is
still open (see \cite{FR}): Let
$\Omega$ be an open, bounded, convex subset of $\mathbb{R}^N$ and let $\tilde{\Omega}$
be the ball centered at $0$ with $|\Omega|=|\tilde{\Omega}|$. Do we always
have $\lambda_{ \tilde{\Omega}}\leq \lambda_\Omega$?
To explain our result, we have to introduce some definitions and notation. Let
$\Omega$ be an open, bounded convex subset of $\mathbb{R}^N$. Let us denote
by $F=F(|\cdot|)$ the fundamental solution of Laplace equation in $\mathbb{R}^n$
and, for any $x\in \Omega$, let $H_x(\cdot)$ be the regular part
of the Green function of Laplace equation with Dirichlet boundary condition,
i.e., the solution of
\begin{gather*}
-\Delta H_x(\cdot)=0 \quad\mbox{in }\Omega\\
H_x(\cdot)= F(|\cdot-x|) \quad\mbox{on }\partial \Omega
\end{gather*}
The Robin function $t:\Omega\to\mathbb{R}$ and the harmonic radius $r:\Omega\to\mathbb{R}$
are respectively defined by
$$
\forall x\in\Omega, \quad t(x)=H_x(x)\quad\mbox{and}\quad t(x)=F^{-1}(t(x))\,.
$$
We also denote by $\bar{r}_\Omega$ the maximum of the harmonic radius in $\Omega$
(which exists since $\Omega$ is convex and bounded) and by $\bar{x}_\Omega$
the harmonic center of $\Omega$
(i.e., the point of maximum of the strictly concave function $r(\cdot)$, see \cite{CT}
for instance).
Let us recall that in dimension $N=2$,
the maximum of the harmonic radius is usually called
the conformal radius.
In \cite{HS2}, the following estimate from above of the Bernoulli constant
is given:
If $\Omega$ is an open convex bounded subset of $\mathbb{R}^2$, we have
$$
\lambda_\Omega\leq 6.252/\bar{r}_\Omega\,.
$$
We improve this result as follows:
\begin{Theorem}\label{estimate}
For any dimension $N\geq2$, we have
$$
\lambda_{\Omega}\leq \lambda_{B_{\bar{r}_\Omega}(0)}=
\begin{cases}
\frac{N-2}{|(N-1)^{-\frac{N-1}{N-2}}-(N-1)^{-\frac{1}{N-2}}|}
\frac{1}{\bar{r}_\Omega} & \mbox{if } N\geq3\\
e/\bar{r}_\Omega & \mbox{if } N=2
\end{cases}
$$
\end{Theorem}
\paragraph{Remarks:} \begin{enumerate}
\item Let us recall that the following
inequality holds true for open, bounded and convex sets: If
\begin{equation}\label{inclusion}
\Omega_1\subset\Omega_2, \quad
\lambda_{\Omega_1}\geq\lambda_{\Omega_2}\,.
\end{equation}
This is a straightforward consequence of the construction of \cite{HS2}.
This inequality gives an easy estimate from below of the Bernoulli constant
$$
\lambda_\Omega\geq \lambda_{B_R(0)}
$$
where $R$ is the radius of the smallest ball containing $\Omega$.
\item Let us point out that the estimate from above given in the Theorem
does not derive from inequality
(\ref{inclusion})
because the
ball $B(\bar{x}_\Omega,\bar{r}_\Omega)$ is not contained in $\Omega$,
unless $\Omega$ is a ball.
\item Let us finally
notice that the estimate of the Theorem is optimal for balls, because in
this case
the maximum of the harmonic radius $\bar{r}_\Omega$ is equal to the radius
of the ball.
\end{enumerate}
\paragraph{Proof of Theorem \ref{estimate}:} Let us set $B=B_{\bar{r}_\Omega}(0)$ and
$\lambda=\lambda_{B}$ the Bernoulli constant
of the ball $B$. There is a unique
radius $r\in(0,\bar{r}_\Omega)$ such that $D=B_r(0)$ is the solution of
Bernoulli problem
of level $\lambda$ in $B$ (indeed, in the case of balls, it is known that
any solution
is radial, cf. \cite{Rei}). Let $\bar{G}_x(\cdot)$ and $G_x(\cdot)$ be the
Green functions of
the sets $B$ and $\Omega$ respectively, for the Dirichlet problem,
i.e., the solutions of
$$
\begin{array}{c}
-\Delta \bar{G}_x(\cdot)=\delta_x \quad\mbox{in }B\\
\bar{G}_x(\cdot)=0 \quad \mbox{on }\partial B
\end{array}
\quad\mbox{and}\quad
\begin{array}{c}
-\Delta G_x(\cdot)=\delta_x \quad \mbox{in }\Omega\\
G_x(\cdot)=0 \quad \mbox{on }\partial \Omega
\end{array}
$$
where $\delta_x$ is the Dirac measure at $x$.
Then, for $x=0$, the solution $\bar{G}_0(\cdot)$ is radial and we denote by
$\bar{t}$ its value on $\partial D$.
Let us set
$$
\forall s\in[0,+\infty), \quad
\phi(s)=\begin{cases}
s/\bar{t} & \mbox{if } s\leq \bar{t}\\
1 &\mbox{otherwise}
\end{cases}
$$
Then $\bar{u}(\cdot)= \phi\circ \bar{G}_0(\cdot)$ is nothing but the
capacity potential of $D$ in $B$.
Let us now consider the harmonic transplantation $u_0=\phi\circ G_{\bar{x}}$
(see \cite{He}, \cite{BF}). Then
$u_0$ is clearly the capacity potential of the set
$D_0=\{G_{\bar{x}}>\bar{t}\}$.
Following \cite{BF}, Theorem 18, we have
$$
\mathop{\rm cap}{}_B(D)=\int_{B}|\nabla \bar{u}|^2
=\int_{\Omega}|\nabla u_0|^2=\mathop{\rm cap}{}_\Omega(D_0)
$$
and
$$
|D|\leq |D_0|
$$
(from Theorem 18 of \cite{BF}, part 2, with $f(t)=1$ if $t\geq1$, $f(t)=0$
otherwise).
Accordingly,
\begin{align*}
\mathop{\rm cap}{}_B(D)=&\mathop{\rm cap}{}_\Omega(D_0) \\
\geq& \inf\{ \mathop{\rm cap}{}_\Omega(C)\;|\; C\Subset\Omega, \;
|C|=|D|, \; C\;{\rm convex}\}=F(|D|)\,,
\end{align*}
where $F$ is defined in section \ref{preuvetheo1},
because the capacity is non decreasing with respect to the inclusion.
From Theorem \ref{existence}, we know that there is a open convex set $C$
of volume
$|D|$, which is solution of Bernoulli
problem for some level $\bar{\lambda}>0$.
Moreover, from the proof of Theorem \ref{existence} (see section
\ref{preuvetheo1}),
we can choose $C$ as a minimizer of $F(|D|)$.
Accordingly, we have
$$
\mathop{\rm cap}{}_B(D) =\lambda|\partial D|
\geq \mathop{\rm cap}{}_\Omega(C)=\bar{\lambda}|\partial C|\,.
$$
Since $|D|=|C|$, the isoperimetric inequality states that
$|\partial C|\geq |\partial D|$.
Hence we have proved that
$$
\lambda=\lambda_B\geq \bar{\lambda}\geq \lambda_\Omega\,.
$$
Since $B=B_{\bar{r}_\Omega}(0)$, the proof of Theorem \ref{estimate} is
complete. \hfill$\square$
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\noindent\textsc{Pierre Cardaliaguet} \\
Universit\'e de Bretagne Occidentale,
D\'epartement de Math\'ematiques,\\
6, avenue Victor-le-Gorgeu, B.P. 809, 29285 Brest Cedex, France\\
e-mail: Pierre.Cardaliaguet@univ-brest.fr \smallskip
\noindent\textsc{Rabah Tahraoui }\\
C.E.R.E.M.A.D.E., Universit\'e Paris IX Dauphine, \\
Place du Mar\'echal de Lattre de Tassigny,
75775 Paris Cedex 16 France\\
and \\
I.U.F.M. de Rouen, 2, rue du Tronquet, B.P. 18, \\
Mont Saint-Aignan 76131 France\\
e-mail: tahraoui@ceremade.dauphine.fr
\end{document}