\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 \label{eq1} \begin{gathered} -\Delta_p u_{\Omega} = 0\quad \mbox{in } K\backslash \bar{\Omega}, \; 10: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 $$\label{ict} c_K \leq \frac{6.252}{R_K}.$$ 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$is a solution to the Dirichlet problem \label{pe1} \begin{gathered} -\Delta_p u_w = 0 \quad\mbox{in } K\backslash \bar{w}, \;10: B(o,R) \subset K \}$. 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 $00$ such that $d( x_{n_i} ,\partial \Omega_{n_i}) \geq \eta$. This implies that $B(x_{n_i},\eta) \subset \bar{D}\backslash {\Omega_{n_i}}$ and by the continuity of the inclusion for the Hausdorff convergence, we have $\bar {B}(x,\eta)\subset \bar{D}\backslash \Omega$. This is impossible, because $x\in \partial \Omega$. Since by assumption $\Omega_n$ satisfies the $\epsilon$-cone property we have $C(\hat {x}_n,\zeta(\hat{x}_n), \epsilon,\epsilon)\subset \bar{D}\backslash \Omega_n$. There also exists a subsequence of $\zeta(\hat{x}_n)$ which converges on $\zeta = \zeta(x)$. By passing to the limit, we have $C(x,\zeta(x), \epsilon,\epsilon) \subset \bar {D}\backslash \Omega$, and then $C(x,\zeta(x), \frac{\epsilon}{2},\frac{\epsilon}{2}) \subset \bar {D}\backslash \Omega$. Let us take $y\in B(x,\epsilon)\cap \bar{D}\backslash \Omega$, then there exists $y_n\in \bar {D}\backslash \Omega_n$ such that $y_n$ converges on $y$ and we have $\|y_n -x_n \|$ converges on $\|y - x\| < \epsilon$ and $\|x_n - \hat{x}_n\|$ converges on $0$. Then, for $n$ big enough, we have $\| y_n - \hat{x}_n\|<\epsilon$. The $\epsilon$-cone property implies that $C(y_n,\zeta(\hat{x}_n), \epsilon,\epsilon) \subset D\backslash \bar{\Omega}_n$ and by the continuity of the inclusion for the Hausdorff convergence, we obtain $\bar{C(y,\zeta(x), \epsilon,\epsilon)} \subset \bar{D}\backslash \Omega$ then $C(y,\zeta(x), \frac{\epsilon}{2},\frac{\epsilon}{2}) \subset \bar{D}\backslash \Omega$. This means that the $\frac{\epsilon}{2}-$ cone property is satisfied by $\bar{D}\backslash \Omega$ and then by $\Omega$ too. Let us take $\phi\in L^{1}(D)$, then, \begin{align*} \int_{C(y,\zeta(x), \epsilon,\epsilon)} \phi dx &=\lim_{n\to \infty} \int_{C(y_n,\zeta(\hat{x}_n), \epsilon,\epsilon)} \phi dx\\ &= \lim_{n\to \infty} \int_{C(y_n,\zeta(\hat{x}_n), \epsilon,\epsilon)} \chi_{D\backslash \Omega_n} \phi dx \\ &= \int_{C(y,\zeta(x), \epsilon,\epsilon)} \phi(\chi_D - f ) dx \\ &= \int_{C(y,\zeta(x), \epsilon,\epsilon)} \phi dx - \int_{C(y,\zeta(x), \epsilon,\epsilon)} \phi f dx. \end{align*} We obtain that $\int_{C(y,\zeta(x), \epsilon,\epsilon)} \phi f dx = 0$, for all $\phi\in L^{1}(D)$ and then $f = 0$ on $C(y,\zeta(x), \epsilon,\epsilon)$ a.e. By varying $y\in B(x,\epsilon)\cap \bar{D}\backslash \Omega$ and next $x\in \partial \Omega$, we obtain $f = 0$ on the set $\{x\in D\backslash \Omega; d(x,\partial \Omega)<\epsilon \}$. By the same reasoning for $y\in D\backslash \Omega$ such that $d(y,\partial \Omega) \geq \epsilon$, we show that $f = 0$ on $\{ y\in D\backslash \Omega; d(y,\partial \Omega) \geq \epsilon \}$. We also have just showed that $\chi_{\Omega_{n_k}}$ converges on $\chi_{\Omega}$ a.e. and in $L^{1}(D)$ sense. Now we show that $\bar{\Omega}_{n_k}\stackrel{H}{\to}\bar{ \Omega}$: for a subsequence $\Omega_{n_k}$ such that $\bar{\Omega}_{n_k}\stackrel{H}{\to} G$ and it is sufficient to show that $G = \bar{\Omega}$. Let $\bar {B}(x,\eta) \subset \Omega$ then $\bar{B}(x,\eta) \subset \Omega_{n_k}$ for $n$ large enough, then, $\bar {B}(x,\eta) \subset \bar{\Omega}_{n_k}$. By the continuity of the inclusion for the Hausdorff convergence, we have $\bar {B}(x,\eta) \subset G$ for any ball in $\Omega$. This imply that $\Omega \subset G$ then $\bar{\Omega} \subset G$. Let $F = \bar{D}\backslash \Omega$ and $x\in G\cap F$, we have to show that $x\in \bar{\Omega}$. We remark that, there exists a subsequence $(x_{n_k})_{k\in \mathbb{N}} \subset \bar{\Omega}_n{}_k$ such that $x_{n_k}$ converges on $x$ and $y_{n_k}\in \bar {D}\backslash {\Omega_{n_k}}$ such that $y_{n_k}$ converges on $x$. The sequence $\hat{x}_{n_k}$ belongs to $[x_{n_k},y_{n_k}] \cap \partial \Omega_{n_k}$, then we have $\hat{x}_{n_k}$ which converges on $x$. It is interesting to remark that \begin{gather*} C(\hat{x_{n_k}},\zeta(\hat{x}_{n_k}), \epsilon,\epsilon)\subset\Omega_{n_k} ,\quad \Omega_{n_k} \subset \bar{\Omega}_{n_k},\\ C(\hat{x}_{n_k},-\zeta(\hat{x}_{n_k}), \epsilon,\epsilon)\subset D\backslash\bar{\Omega}_{n_k}, \quad D\backslash\bar{\Omega}_{n_k} \subset \bar{D}\backslash {\Omega}_{n_k}. \end{gather*} We can assume that $\zeta(\hat{x}_{n_k})$ converges on $\zeta(x)$ then \begin{gather*} C(x,\zeta(x), \epsilon,\epsilon) \subset G\\ C(x,-\zeta(x), \epsilon,\epsilon)\subset \bar{D}\backslash \Omega. \end{gather*} Let $\eta>0$ and set $$C_{n_k}(\eta) = \{ z\in C(\hat{x}_{n_k},\zeta(\hat{x}_{n_k}), \epsilon,\epsilon), d(z, \partial C(\hat{x}_{n_k},\zeta(\hat{x}_{n_k}), \epsilon,\epsilon) \geq \eta \}.$$ We remark that $\rho (C_{n_k}(\eta), F_{n_k}) \geq \eta$, and by passing to the limit $\rho (C(\eta), F) \geq \eta$, then $C(\eta) \subset \Omega$. This implies that $\bar{C}(x,\zeta(x),\epsilon,\epsilon) \subset \bar{\Omega}$, then $G\cap F \subset \bar{\Omega}$ and $G\cap F \subset \partial \Omega$. It is easy to see that by an absurdity reasoning, we have $G\backslash \bar{\Omega} = \emptyset$, and then $G \subset \bar{\Omega}$. \end{proof} \section{Shape optimization and monotony results} \begin{theorem}\label{pro1} The problem Find $\Omega \in \mathcal{O}_{\epsilon}$ such that $J(\Omega) = \min \{ J(w), w\in \mathcal{O}_{\epsilon}\}$'' admits a solution. \end{theorem} \begin{proof} Consider the function $\tilde u$ defined by $\tilde u =\begin{cases} u & \mbox{if } x \in K\backslash\bar{\Omega}\\ 1 & \mbox{if } x \in \bar{\Omega} \end{cases}$ $\nabla \tilde u = \begin{cases} \nabla u &\mbox{if } x \in K\backslash\bar{\Omega}\\ 0 &\mbox{if } x \in \bar{\Omega} \end{cases}$ Let $E$ be a functional defined on $W_0^{1,p}(K)$ by $$E(\tilde{u}_w ) = \frac{1}{p}\int_K \|\nabla \tilde{u}_w\|^p dx , \quad 10 this implies that \inf \{J(w), w \in \mathcal{O}_{\epsilon} \}>-\infty. Let \alpha = \inf \{ J(w), w \in \mathcal{O}_{\epsilon}\}. Then, there exists a minimizing sequence (\Omega_n)_{n\in \mathbb{N}} \subset \mathcal{O}_{\epsilon} such that  J(\Omega_n) converges on \alpha. Since the sequence (\Omega_n)_{n\in \mathbb{N}} is bounded, there exists a compact set F such that \bar{\Omega}_n \subset F \subset K. By lemma (\ref{ler4}), there is a subsequence (\Omega_{n_k})_{k\in \mathbb{N}}, and \Omega verifying the \epsilon -cone property such that  \Omega_{n_k} \stackrel{H}{\to} \Omega  and \chi_{\Omega_{n_k}} {\to} \chi_{\Omega} a.e. Let us set  u_{\Omega_n} = u_n and show that the sequence (\tilde u_n)_{n\in \mathbb{N}} is bounded in  W^{1,p}(K). If not, for all s there exists a subsequence denoted \tilde u_n^s\in W_0^{1,p}(K)  such that \int_K\|\nabla \tilde u_n\|^pdx >s and \begin{gather*} \int_K\|\nabla \tilde u^s_n\|^pdx =\int_{K\backslash\bar{\Omega}_n}\|\nabla \tilde u^s_n\|^pdx +\int_{\bar{\Omega}_n}\|\nabla \tilde u^s_n\|^p dx\,, \\ \int_K\|\nabla \tilde u^s_n\|^pdx =\int_{K\backslash\bar{\Omega}_n}\|\nabla \tilde u^s_n\|^p dx . \\ %\int_{\Omega_n}\|\nabla \tilde u^s_n\|^pdx > s\,. \end{gather*} That is, J(\Omega_n) converges on +\infty. Then, \inf\{ J(w), w \in \mathcal{O}_{\epsilon} \} = +\infty  is a contradiction. Since W^{1,p}(K)  is a reflexive space, there exists a subsequence  (u_{n_k})_{k\in \mathbb{N}} and u^* such that  u_{n_k}  converges weakly on u^* in W^{1,p}(K) and$$ \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}, 10: 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 $$\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}$$ The solution u_0 is explicitly determined by $$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}$$ 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 $$\label{eqq1} \begin{gathered} -\Delta_p u = 0 \quad \mbox{in } K\backslash B_r\\ u = 1 \quad \mbox{on } \partial B_r \\ u = 0 \quad \mbox{on } \partial K. \end{gathered}$$ 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 $$\label{eqq2} \begin{gathered} -\Delta_p v = 0 \quad\mbox{in } B_{R_K}\backslash B_r\\ v = 1 \quad\mbox{on } \partial B_r \\ v = u_r \quad\mbox{on } \partial B_{R_K}. \end{gathered}$$ 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 $$\label{i1} \begin{gathered} -\Delta_p u = 0 \quad \mbox{in } K\backslash \bar{\Omega}\\ u = 1 \quad \mbox{on } \partial \Omega \\ u = 0 \quad \mbox{on} \partial K\\ \frac{\partial u}{\partial \nu} = c_{\Omega} \quad \mbox{on }\partial \Omega \end{gathered}$$ 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 $$\label{i2} \begin{gathered} -\Delta_p u_{r_n} = 0 \quad \mbox{in } K\backslash B_{r_n} \\ u_{r_n} = 1 \quad \mbox{on } \partial B_{r_n} \\ u_{r_n} = 0 \quad \mbox{on } \partial K. \end{gathered}$$ Let us consider the problem $$\label{i22} \begin{gathered} -\Delta_p z = 0 \quad \mbox{in } K\backslash \bar{\Omega }\\ z = u_{r_n} \quad \mbox{on } \partial \Omega \\ z = 0 \quad \mbox{on } \partial K. \end{gathered}$$ 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 $$\label{i5} \begin{gathered} -\Delta_p u_1 = 0 \quad \mbox{in } K\backslash \bar{\Omega}_1\\ u_1 = 1 \quad \mbox{on } \partial \Omega_1 \\ u_1 = 0 \quad \mbox{on } \partial K\\ \frac{\partial u_1}{\partial \nu} = c_{\Omega_1} \quad \mbox{on }\partial \Omega_1. \end{gathered}$$ 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\frac{e}{R_K} and r\in ]0,\frac{R_K}{e}[ \cup ]\frac{R_K}{e},R_K[. We have the same reasoning and we show that the problem (\ref{eq1}) admits a solution. \subsection*{Case where  p \neq N } Here the reasoning is identical to the case  p = N. We note that$$ \|\nabla u_0\|_{|{\partial B_{r_n}}} = \big|\frac{p - N}{p-1}\big| \frac{1}{ 1 - (\frac{r}{R_K})^{\frac{N - p}{p-1}}} \frac{1}{r} = g(r)  and $g$ is strictly increasing on $](\frac{ p - 1}{ N - 1})^{\frac{ p- 1}{N - p }} R_K, R_K[$ and a strictly decreasing on $] 0, (\frac{ p- 1}{ N - 1})^{\frac{ p-1}{N - p }} R_K[$. For all $c \geq |\frac{p - N}{p -1}|\frac{1}{| (\frac{ p- 1}{ N - 1})^{\frac{N - 1}{N-p}} - (\frac{ p- 1}{ N - 1})^{\frac{p - 1}{N- p}}|}\frac{1}{R_K} = g((\frac{ p - 1}{ N -1})^{\frac{ p- 1}{N - p }} R_K),$ problem (\ref{eq1}) admits a solution. Let us now prove the assertions (i) and (ii) of theorem (\ref{theo1}). It is easy to have, $0c_{K_1}$ then for all $K$ such that $\mathop{\rm vol}(K) =\mathop{\rm vol}(K^{*})$, we have $c_{K^{*}} \leq c_K$. If there exists $K_1$ such that $\mathop{\rm vol}(K_1) =\mathop{\rm vol}(K^{*})$ and $c_{K^{*}} >c_{K_1}$ then $K_1$ can't be a ball and \$R_{K_1}