\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 109, pp. 1--12.\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-2004/109\hfil Isoperimetric inequality] {Isoperimetric inequality for an interior free boundary problem with p-laplacian operator} \author[I. Ly, D. Seck\hfil EJDE-2004/109\hfilneg] {Idrissa Ly, Diaraf Seck} % in alphabetical order \address{Facult\'e des Sciences Economiques et de Gestion, Universit\'e Cheikh Anta Diop, B.P 5683, Dakar, S\'en\'egal} \email[Idrissa Ly]{ndirkaly@ugb.sn} \email[Diaraf Seck]{dseck@ucad.sn} \date{} \thanks{Submitted April 7, 2004. Published September 14, 2004.} \subjclass[2000]{35R35} \keywords{Bernoulli free boundary problem; starshaped domain; \hfill\break\indent shape optimization; shape derivative; monotony} \begin{abstract} By considering the p-Laplacian operator, we establish an existence and regularity result for a shape optimization problem. From a monotony result, we show the existence of a solution to the interior problem with a free surface for a family of Bernoulli constants. We also give an optimal estimation for the upper bound of the Bernoulli constant. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \section{Introduction} Let us study the following interior Bernoulli problem: Given $K$, a $\mathcal{C}^{2}$-regular and bounded domain in $\mathbb{R}^{N}$, and a constant $c>0$, find a domain $\Omega$ and a function $u_{\Omega}$ such that \begin{equation} \label{eq1} \begin{gathered} -\Delta_p u_{\Omega} = 0\quad \mbox{in } K\backslash \bar{\Omega}, \; 1
0:B(o,R)\subset K \}$. \item For $ N = 2$ and $ p= 2$, the minimal value $ c_K $, depending on $K$ and $ p$ for which problem (\ref{eq1}) has a solution is estimated from above by \begin{equation}\label{ict} c_K \leq \frac{6.252}{R_K}. \end{equation} But this inequality is not optimal, since $K$ is a disk of radius $R$. \end{itemize} In \cite[p. 202]{FR}, by considering the Laplacian, Flucher and Rumpf set the following problem: \begin{quote} Let $K$ be a connected domain and $K^{*}$ a ball such that $\mathop{\rm vol}(K) = \mathop{\rm vol}(K^{*})$. Let $c_K$( respectively $c_{K^{*}}$) be the minimal value of $c$ for which the interior Bernoulli problem (\ref{eq1}) admits a solution. Does $c_K$ satisfy the isoperimetric inequality $ c_K \geq c_{K^{*}}$? \end{quote} In \cite{CT}, Cardaliaguet and Tahraoui gave an estimate from above for the Bernoulli constant, by using the harmonic radius. But they didn't give an answer to the question posed by Flucher and Rumpf. Now, by combining a variational approach and a sequential method, we establish an existence result for non-necessary convex domains. Then we show that $c_K$ satisfies the isoperimetric inequality in the sense that $ \max \{c_K:\mathop{\rm vol}(K)= \mathop{\rm vol}(K^{*})\} \geq c_{K^{*}}$. This comparison answers the question posed by Flucher and Rumpf. The structure of this paper is as follows: In the first part, we present the main result. In the second section, we give auxiliary results. The third part deals with the study of the shape optimization problem and the existence of Lagrange multiplier $\lambda_{\Omega}$. Namely, we study at first the existence result for the shape optimization problem: Find $$ \min\{ J(w), w \in \mathcal{O}_{\epsilon}\} , $$ where \[ \mathcal{O}_{\epsilon} = \{ w \subset K: w \mbox{ is an open set satisfying the $\epsilon$-cone property and $\mathop{\rm vol}(w) = m_0$}\} \] where $\mathop{\rm vol}$ denotes the volume, $m_0$ is a fixed value in $\mathbb{R}_+^{*}$. The functional $J$ is $$ J(w): = \frac{1}{p}\int_{K\backslash \bar{w}} \|\nabla u_w\|^{p} dx , $$ where $ u_w$ is a solution to the Dirichlet problem \begin{equation}\label{pe1} \begin{gathered} -\Delta_p u_w = 0 \quad\mbox{in } K\backslash \bar{w}, \;1
0: B(o,R) \subset K \}$.
Let
$$
\mathcal{E}: = \{ c_K : \mathop{\rm vol}(K) = \mathop{\rm vol}(K^{*}) \},
$$
where $c_K$ is the minimal value for which the interior Bernoulli problem
(\ref{eq1}) admits a solution.
\begin{theorem}\label{theo1}
If the solution $\Omega$ of the shape optimization
problem
$\min\{J(w), w \in \mathcal{O}_{\epsilon}\}$ is
$\mathcal{C}^{2}$-regular, then
for all constant $c>0$ satisfying $c \geq \alpha( R_K,p,N)$,
$\Omega$ is the classical solution of the free-boundary problem (\ref{eq1}).
Moreover:
\begin{itemize}
\item[(i)] The constant $c_K$ satisfies $ 0 0$ this implies that $\inf \{J(w), w \in \mathcal{O}_{\epsilon}
\}>-\infty$. Let $\alpha = \inf \{ J(w), w \in \mathcal{O}_{\epsilon}\}$.
Then, there exists a minimizing sequence
$(\Omega_n)_{n\in \mathbb{N}} \subset \mathcal{O}_{\epsilon}$ such that
$ J(\Omega_n)$ converges on $\alpha$.
Since the sequence $(\Omega_n)_{n\in \mathbb{N}}$ is bounded,
there exists a compact set $F$ such that $\bar{\Omega}_n \subset F \subset K$.
By lemma (\ref{ler4}), there is a subsequence
$(\Omega_{n_k})_{k\in \mathbb{N}}$, and $\Omega$ verifying the
$\epsilon $-cone property such that $ \Omega_{n_k} \stackrel{H}{\to} \Omega $ and
$\chi_{\Omega_{n_k}} {\to} \chi_{\Omega}$ a.e.
Let us set $ u_{\Omega_n} = u_n$ and
show that the sequence $(\tilde u_n)_{n\in \mathbb{N}}$ is bounded in
$ W^{1,p}(K)$.
If not, for all $s$ there exists a subsequence denoted
$\tilde u_n^s\in W_0^{1,p}(K) $ such that
$\int_K\|\nabla \tilde u_n\|^pdx >s$ and
\begin{gather*}
\int_K\|\nabla \tilde u^s_n\|^pdx
=\int_{K\backslash\bar{\Omega}_n}\|\nabla \tilde u^s_n\|^pdx +\int_{\bar{\Omega}_n}\|\nabla
\tilde u^s_n\|^p dx\,, \\
\int_K\|\nabla \tilde u^s_n\|^pdx =\int_{K\backslash\bar{\Omega}_n}\|\nabla
\tilde u^s_n\|^p dx . \\
%\int_{\Omega_n}\|\nabla \tilde u^s_n\|^pdx > s\,.
\end{gather*}
That is, $J(\Omega_n)$ converges on $+\infty$. Then,
$\inf\{ J(w), w \in \mathcal{O}_{\epsilon} \} = +\infty $
is a contradiction.
Since $W^{1,p}(K) $ is a reflexive space, there exists a subsequence
$ (u_{n_k})_{k\in \mathbb{N}}$ and $u^*$ such that
$ u_{n_k} $ converges weakly on $u^*$ in $W^{1,p}(K)$
and
$$
\int_{K\backslash \bar{\Omega}} \|\nabla u^*\|^p dx \leq
\lim\inf\int_{K\backslash \bar{\Omega}_{n_k}} \|\nabla u_{n_k}\|^p dx.
$$
From the above we get $ J(\Omega) \leq J(\Omega_{n_k})$
and $J(\Omega) \leq \inf\{J(w), w \in
\mathcal{O}_{\epsilon} \}$. Finally, we have $J(\Omega) = \min\{
J(w), w \in \mathcal{O}_{\epsilon} \}$.
\end{proof}
\begin{remark} \label{rmk4.1} \rm
On the one hand, it is easy to verify that $u^*$ equals $ u_{\Omega}$
and satisfies
\begin{gather*}
-\Delta_p u^*= 0 \quad \mbox{in } K\backslash \bar{\Omega}\\
u^* = 1 \quad \mbox{on } \partial \Omega \\
u^* = 0 \mbox{on } \partial K
\end{gather*}
On the other hand, we have a regularity of $u_{\Omega}$ solution
to the problem (\ref{pe1}); see \cite{D,Le,T2}.
\end{remark}
For the rest of this article, we assume that $\Omega$ is $\mathcal{C}^{2}$-regular
in order to use the shape derivatives.
This hypothesis is possible because if we work with a class
of domains which are $\mathcal{C}^3$-regular and
verifying the geometric normal property,
we can show that $\Omega$ solution to the shape optimization problem is
$\mathcal{C}^{2}$-regular.
\begin{theorem}\label{t1}
Let $L$ be a compact set of $\mathbb{R}^{N}$. Let
$(f_n)_{(n\in \mathbb{N})}$ be a sequence of functions,
$f_n \in \mathcal{C}^{3}(L)$ with
$$
\big| \frac{\partial f_n}{\partial x_i}\big| \leq M, \quad
\big| \frac{\partial^{2} f_n}{\partial x_i\partial x_j}\big| \leq M ,\quad
\big| \frac{\partial^{3} f_n}{\partial x_i\partial x_j \partial x_k}\big| \leq M,
$$
where $M$ is a positive constant independent of $n$.
We define a sequence $(\Omega_n)_{(n\in \mathbb{N})}$,
by $\Omega_n = \{ x\in L : f_n(x)>0\} $.
We assume that there exists $ \alpha > 0 $ such
that $ |f_n(x)| + | \nabla f_n(x)| \geq \alpha$ for all
$x$ belonging to $L$. We assume in addition that $\Omega_n$
has the geometric normal property.
Then there exists, $\Omega$ a $\mathcal{C}^{2}$-regular domain and a
subsequence of $(\Omega_n)_{(n\in \mathbb{N})}$ denoted
$(\Omega_{n_k})_{(k\in \mathbb{N})}$ such that $\Omega_{n_k} $
converges in the compact sense on $ \Omega $ and
$J(\Omega) = \min \{J(w): w\in \mathcal{O}_{\epsilon} \}$.
\end{theorem}
We remark that$\Omega_n$ and $\Omega$ as above belong to
$\mathcal{O}_{\epsilon}$. For this theorem, we need the
following lemma. Then the proof of Theorem \ref{t1} can be found
in \cite{LS2}.
\begin{lemma}\label{lem5}
Let $(f_n)_{(n\in \mathbb{N})}$ be a sequence functions
defined as in theorem \ref{t1}. One supposes that
$\Omega$ is an open set defined by
\[
\Omega = \{ x\in L : h(x)>0\} \quad \mbox{with}\quad
\partial \Omega = \{ x\in L : h(x) = 0 \}
\]
where $h$ is a continuous function defined on $L$
which is a compact set of $\mathbb{R}^{N}$.
If $f_n$ converges uniformly on $h$, then we have
$\Omega_n$ converges in the compact sense to $\Omega$.
\end{lemma}
\begin{proof}
Let $K_1$ be a compact set included in $\Omega$, and let
$\alpha = \inf_{K_1} h$, we have $\alpha>0$. There exists $n_0$
belonging to $\mathbb{N}$, such that for all $n \geq n_0$ we get
$| f_n - h|_{L^{\infty}(K)} <\alpha$. Then for all $x$
belonging to $K_1$ we have
$ f_n(x)>h(x) - \alpha \geq 0$ for $n \geq n_0$. This implies that
$K_1$ is contained in $\Omega_n$.
Let $L_0$ be a compact subset of ${\bar{\Omega}}^{c}$ by hypothesis
we have
$ \bar{ \Omega } = \Omega\cup \partial \Omega = \{ x\in L : h(x) \geq 0 \}$
then $\beta := \max_{L_0} h<0$. Therefore, there exists $n_1$
belonging to $\mathbb{N}$ such that for all $n \geq n_1$ implies that
$| f_n - h|_{L^{\infty}(L_0)} <-\beta$.
One has $f_n(x) \leq h(x) -\beta $ for all $x$
belonging to $ L_0$. This implies that
$f_n(x) \leq 0$ and then $L_0$ is contained in ${\bar
{\Omega}}_n^c$ because $\{ x\in L :h(x)<0 \} \subset {\bar {\Omega}}_n^c$.
\end{proof}
The next theorem gives necessary conditions of optimality.
\begin{theorem}\label{coro1}
If $\Omega$ is the solution of the shape optimization problem
$\min \{ J(w): w\in \mathcal{O}_{\epsilon} \}$,
then there exists a Lagrange multiplier $\lambda_{\Omega}>0$ such that
$\frac{\partial u}{\partial \nu} = ( \frac{p}{p -
1}\lambda_{\Omega})^{1/p}$ on $\partial \Omega$.
\end{theorem}
\begin{proof}
The main technique used to prove this result is the shape
derivatives as used in \cite{SZ, JS}. For the computations, we
refer to \cite[page 42-52]{L},
\end{proof}
\begin{remark} \label{rmk4.2} \rm
A consequence of the Theorems (\ref{pro1}) and (\ref{coro1})
is that $(\Omega, u_{\Omega})$ satisfies
\begin{gather*}
-\Delta_p u_{\Omega} = 0 \quad \mbox{in } K\backslash \bar{\Omega}, 1 0: B(o, R) \subset K \}$.
Let $ r>0$ such that $B(o, r) \subset B(o,R_K)$. First, we
have to look for a solution $ u_0$ to the problem
\begin{equation}\label{eq2}
\begin{gathered}
-\Delta_p u = 0 \quad \mbox{in } B_{R_K}\backslash B_r\\
u = 0 \quad \mbox{on } \partial B_{R_K} \\
u = 1 \quad \mbox{on } \partial B_r\,.
\end{gathered}
\end{equation}
The solution $u_0$ is explicitly determined by
\begin{equation}
u_0(x) = \begin{cases}
\frac{ \ln\|x\| - \ln R_K}{\ln r - \ln R_K}&\mbox{if } p = N \\[3pt]
\frac{ -\|x\|^{\frac{p -N}{p -1}}
+ R_K^{\frac{p -N}{p -1}}} { R_K^{\frac{p -N}{p -1}}
- r^{\frac{p -N}{p -1}}}&\mbox{if }p\neq N,
\end{cases}
\end{equation}
and
\[
\|\nabla u_0(x)\| = \begin{cases}
\frac{1}{ r( \ln R_K - \ln r)}&\mbox{if } p = N \\[3pt]
\frac{ |\frac{p -N}{p -1}| \|x\|^{\frac{-N+1}{p -1}}}{| r^{\frac{p -N}{p -1}}
- R^{\frac{p -N}{p -1}}|}&\mbox{if }p \neq N.
\end{cases}
\]
In particular $\|\nabla u_0\| >c$ on $\partial B_r$ for $r$ small enough. \\
Now consider the following problem
\begin{equation}\label{eqq1}
\begin{gathered}
-\Delta_p u = 0 \quad \mbox{in } K\backslash B_r\\
u = 1 \quad \mbox{on } \partial B_r \\
u = 0 \quad \mbox{on } \partial K.
\end{gathered}
\end{equation}
The problem (\ref{eqq1}) admits a solution denoted by
$u_r$. This solution is obtained by minimizing the functional
$J$ defined on the Sobolev space
\[
V'= \{ v\in W^{1,p} (K\backslash B_r), v
= 1 \mbox{on $\partial B_r$ and} v = 0 \mbox{ on } \partial K \}
\]
and $J(v) = \frac{1}{p}\int_{K\backslash B_r }\|\nabla v\|^{p}dx$.
Consider the problem
\begin{equation}\label{eqq2}
\begin{gathered}
-\Delta_p v = 0 \quad\mbox{in } B_{R_K}\backslash B_r\\
v = 1 \quad\mbox{on } \partial B_r \\
v = u_r \quad\mbox{on } \partial B_{R_K}.
\end{gathered}
\end{equation}
It is easy to see that $ v = u_r$ is a solution to problem
(\ref{eqq2}). By the comparison principle \cite{T1}, we
obtain $0 \leq u_0 \leq 1$ and $0 \leq u_r \leq 1$. On
$\partial (B_{R_K}\backslash B_r)$, we obtain $u_r \geq u_0$
and then, $u_r \geq u_0$ in $B_{R_K}\backslash B_r$. Finally,
we have $ \|\nabla u_r\| \leq \|\nabla u_0\| \mbox{on}
\partial B_r$.
\subsection*{Case where $p = N$}
\[
\|\nabla u_0\|_{|{\partial B_r}} =\frac{1}{r(\ln R_K - \ln r)} = g(r), \quad
\forall r\in ]0,R_K[.
\]
It is easy to see that $g(r)$ is a strictly decreasing function on
$]0,\frac{R_K}{e}[$ and a strictly increasing function on
$]\frac{R_K}{e}, R_K[$.
Then for all $r\in ]0,R_K[$,
$\|\nabla u_0\|_{|{\partial B_r}} \geq g(\frac{R_K}{e}) = \frac{e}{R_K}$.
\smallskip
\noindent(1) For $c = e/R_K$, let $\delta>0$ be a fixed and
sufficiently small number. To initialize we choose
$r_0\in ]0,\frac{R_K}{e}[ \cup ]\frac{R_K}{e},R_K [$ such that
$ \big| \|\nabla u_0\|_{|{\partial B_{r_0}}} - c \big|>\delta$.
To fix ideas let us consider $r_0\in ]0,\frac{R_K}{e}[$.
The process will be identical if $ r_0\in ]\frac{R_K}{e}, R_K[$.
By varying $r$ in the increasing sense, we will achieve
a step denoted $n$ such that
$$
r_n\in ]0,\frac{R_K}{e}[ \mbox{and}
\left\||\nabla u_0\|_{|{\partial B_{r_n}}} - c \right| <\delta .
$$
Consider $\mathcal{O}_n$ the class of admissible
domains defined as follows
$$
\mathcal{O}_n = \left \{ w\in \mathcal{
O}_{\epsilon}, B_{r_n} \subset w, \partial B_{r_n}\cap
\partial w \neq \emptyset, \mbox{ and } \mathop{\rm vol}(w) = V_0 \right\},
$$
where $ V_0$ denotes a fixed positive constant.
We look for $\Omega\in \mathcal{O}_n$ such that
\begin{equation}\label{i1}
\begin{gathered}
-\Delta_p u = 0 \quad \mbox{in } K\backslash \bar{\Omega}\\
u = 1 \quad \mbox{on } \partial \Omega \\
u = 0 \quad \mbox{on} \partial K\\
\frac{\partial u}{\partial \nu} = c_{\Omega} \quad \mbox{on }\partial \Omega
\end{gathered}
\end{equation}
where $ c_{\Omega}= (\frac{p}{p -1}\lambda_{\Omega})^{1/p}$.
Applying the theorem (\ref{pro1}), the shape optimization problem
$\min\{ J(w) , w\in \mathcal{O}_n \} $ admits
a solution and by theorem (\ref{coro1}),
$\Omega$ satisfies the overdetermined boundary condition
$\frac{\partial u}{\partial \nu} = c_{\Omega}$.
Then problem (\ref{i1}) admits a solution .
Since $\Omega\in \mathcal{O}_n$, we have
$B_{r_n} \subset \Omega$, $\partial B_{r_n}\cap \partial
\Omega \neq \emptyset$ and $u_{r_n}$ satisfies
\begin{equation}\label{i2}
\begin{gathered}
-\Delta_p u_{r_n} = 0 \quad \mbox{in } K\backslash B_{r_n} \\
u_{r_n} = 1 \quad \mbox{on } \partial B_{r_n} \\
u_{r_n} = 0 \quad \mbox{on } \partial K.
\end{gathered}
\end{equation}
Let us consider the problem
\begin{equation}\label{i22}
\begin{gathered}
-\Delta_p z = 0 \quad \mbox{in } K\backslash \bar{\Omega }\\
z = u_{r_n} \quad \mbox{on } \partial \Omega \\
z = 0 \quad \mbox{on } \partial K.
\end{gathered}
\end{equation}
It is easy to see that
$ z = u_{r_n}$ is a solution to the problem (\ref{i22}),
and we get $0 \leq u_{r_n} \leq 1$ and
$0 \leq u \leq 1$. On
$\partial (K\backslash \bar{\Omega})$, we have $u_{r_n } \leq u$.
Since $\partial \Omega \cap \partial B_{r_n} \neq \emptyset$,
let $x_0\in \partial \Omega \cap \partial B_{r_n}$, we have
\[
\lim_{h\to 0}\frac{u_{r_n}(x_0 -\nu h) - u_{r_n}(x_0)}{h}
\leq \lim_{h\to 0}\frac{u(x_0 -\nu h) - u(x_0)}{h},
\]
This is equivalent to
\[
\frac{\partial u_{r_n}}{\partial \nu}(x_0) \geq \frac{\partial u}{\partial \nu}(x_0) =
c_{\Omega}.
\]
Let $ \Omega = \Omega_0$ as the first iteration.
We iterate by looking for
$\Omega_1\in \mathcal{O}_n^{1}$ such that
\begin{equation}\label{i5}
\begin{gathered}
-\Delta_p u_1 = 0 \quad \mbox{in } K\backslash \bar{\Omega}_1\\
u_1 = 1 \quad \mbox{on } \partial \Omega_1 \\
u_1 = 0 \quad \mbox{on } \partial K\\
\frac{\partial u_1}{\partial \nu} = c_{\Omega_1} \quad \mbox{on }\partial \Omega_1.
\end{gathered}
\end{equation}
where $ c_{\Omega_1}=(\frac{p}{p -1}\lambda_{\Omega_1})^{1/p}$, and
$$
\mathcal{O}^{1}_n = \left \{ w, w\in \mathcal{O}_{\epsilon}, \Omega_0
\subset w, \mbox{and} \partial w \cap
\partial B_{r_n} \neq \emptyset \mathop{\rm vol}(w) = V_1 \right \},
where V_1$$ is a strictly positive constant and $V_0