\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 128, pp. 1--23.\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 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/128\hfil Existence of solutions]
{Existence of solutions for the one-phase and the multi-layer
free-boundary problems with the p-laplacian operator}

\author[I. Ly,  D. Seck\hfil EJDE-2006/128\hfilneg]
{Idrissa Ly,  Diaraf  Seck}

\address{Facult\'e des Sciences Economiques et de Gestion,
 Universit\'e Cheikh Anta Diop, B.P 5683,   Dakar, S\'en\'egal}
\email{ndirkaly@ugb.sn}

\address {Facult\'e des Sciences Economiques et de Gestion,
 Universit\'e Cheikh Anta Diop, B.P 5683,   Dakar, S\'en\'egal}
\email{dseck@ucad.sn}

\date{}
\thanks{Submitted March 6, 2006. Published October 11, 2006.}
\subjclass[2000]{35R35}
\keywords{Bernoulli free boundary problem; starshaped domain;
 \hfill\break\indent shape optimization; shape derivative; monotonicity}

\begin{abstract}
 By considering the p-laplacian  operator,  we show  the
 existence of a solution to the exterior (resp interior) free
 boundary problem with non constant Bernoulli free boundary
 condition.  In the second part of this article,
 we study the existence of  solutions to the two-layer shape
 optimization problem.  From a monotonicity result, we show
 the existence of classical solutions to the two-layer Bernoulli
 free-boundary  problem  with nonlinear joining conditions.
 Also we extend the existence result to the multi-layer case.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}

\section{Introduction}

In part I, we study the exterior and  interior free-boundary
problem with non-constant Bernoulli  boundary condition.
Given $K$ a $\mathcal{C}^{2}$-regular  bounded domain in  $\mathbb{R}^{N}$ and a
positive continuous function $g$,  such that $g(x)>\alpha>0$
 for all  $ x\in R$,  we find  for  the exterior problem  a
domain $\Omega$  and a  function  $u_{\Omega}$  such that
\begin{equation} \label{eq1}
\begin{gathered}
-\Delta_p  u_{\Omega}  =  0\quad\text{in }
 \Omega\backslash K,\;  1<p<\infty \\
  u_{\Omega}     =  0 \quad\text{on }  \partial \Omega \\
  u_{\Omega}     =  1 \quad\text{on }  \partial K \\
-\frac{\partial u_{\Omega}}{\partial \nu_{e }}  =  g(x) \quad\text{ on }
\partial \Omega
\end{gathered}
\end{equation}
Here   $\Delta_p$ denotes  the p-Laplace operator, i.e.
$\Delta_p u : = \mathop{\rm div} (\|\nabla u\|^{p-2}\nabla u)$   and
$\nu_{e}$ is the normal  exterior  unit  of   $\Omega$. And  for  the
interior problem, we look for a  domain $\Omega$  and a  function
$u_{\Omega}$  such that
\begin{equation} \label{eq2}
\begin{gathered}
-\Delta_p  u_{\Omega}  =  0 \quad\text{in }     K\backslash \bar{\Omega},\;
1<p<\infty \\
  u_{\Omega}           =  1 \quad\text{on }  \partial \Omega \\
  u_{\Omega}           =  0 \quad\text{on }  \partial K \\
\frac{\partial u_{\Omega}}{\partial \nu_{i}}  =  g(x) \quad\text{on } \partial
\Omega
\end{gathered}
\end{equation}
where  $\nu_{i}$  is the  normal interior unit  of $\Omega$. The
problem arises when  a fluid  flows  in porous   medium  around an
obstacle.
 In certain  industrial problems  such   as shape  optimization,
galvanization, we seek to find   level lines of the potential function with
prescribed  pressure.

 Inspired by the pioneering  work of Beurling, where the notion  of
sub and supersolutions  in geometrical case is used,
Henrot  and  Shahgohlian  \cite{HS3}
studied this problem . They  proved  that when $ K
\subset\mathbb{R}^{N}$ is a bounded and convex domain a
generalization  of \cite{HS1,HS2} to the case   of the
non-constant Bernoulli boundary condition.

 By  combining a variational approach and a sequential method, we establish
 an existence  result by generalizing the problems studied in \cite{LS2,LS3}
to the case  of the non-constant  Bernoulli boundary condition.

The structure  of  Part I  is  as follows: In the first part, we
present the main result  which  generalizes results in \cite{LS2,LS3},
 to the  case of non-constant Bernoulli boundary  condition. 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 $\lambda_{\Omega}$  for the
exterior (respectively interior) case. First, we study  the
existence result for the  shape optimization problem   for   the
exterior case: Find
$$
\min\{ J_1(w), w \in \mathcal{O}^{1}_{\epsilon}\} ,
$$
where ${\mathcal{O}}^{1}_{\epsilon}  = \{ w \supset K: w$ is an
open set satisfying the $\epsilon$-cone property,
 $\int_{w}\frac{g^{p}}{c^{p}}(x)dx= V_0 \}$,
where $ V_0$  is a given positive value. The functional  $J_1$ is
$$
J_1(w):  = \frac{1}{p}\int_{w\backslash K} \|\nabla u_w\|^{p} dx ,
$$
where   $ u_w$  is a solution to the Dirichlet  problem
\begin{equation}\label{pe1}
\begin{gathered}
-\Delta_p  u_w  = 0  \quad\mbox{in } w\backslash K, \; 1<p<\infty\\
  u_w           = 0  \quad\mbox{on }  \quad\partial  w \\
   u_w          = 1 \quad \mbox{on} \quad \partial K .
\end{gathered}
\end{equation}
Second, we study  the existence result for the  shape
optimization problem.  For   the interior  case: Find
$$
\min\{ J_2(w), w \in \mathcal{O}^{2}_{\epsilon}\} ,
$$
where $\mathcal{O}^{2}_{\epsilon}  = \{ w \subset K: w$  is an
open set satisfying the $\epsilon$-cone  property,
 $\int_{w}\frac{g^{p}}{c^{p}}(x)dx = W_0 \}$,
where $ W_0$  is a given positive value. The functional  $J_2$ is
$$
J_2(w):  = \frac{1}{p}\int_{K\backslash \bar{w}} \|\nabla
u_w\|^{p} dx ,
$$
where   $ u_w$  is a solution to the Dirichlet  problem
\begin{equation}\label{pe2}
\begin{gathered}
-\Delta_p  u_w   = 0  \quad\mbox{in } K\backslash \bar{w}, \;1<p<\infty\\
  u_w             = 1  \quad\mbox{on }  \quad\partial  w \\
   u_w          = 0 \quad \mbox{on} \quad \partial K .
\end{gathered}
\end{equation}

 Next, we obtain  an optimality condition for the exterior
(respectively interior) case
 $$
-\frac{\partial u}{\partial \nu_{e} }  =  (\frac{p}{1-p}
\lambda_{\Omega})^{1/p} \quad
  (\mbox{respectively }    \frac{\partial u}{\partial \nu_{i} }
=  (\frac{p}{p-1} \lambda_{\Omega})^{1/p})
 \quad\mbox{on } \partial \Omega
$$
Then we conclude this section with a  monotonicity result for  the
exterior (respectively interior)  case.
 The  last part is the proof of the main result for
the exterior (respectively interior)  case.

In  Part II, we  study the multi-layer case.
 Let  $ D_0^{*}$  and   $ D_1^{*}$  be
$\mathcal{C}^{2}$-regular,    compact sets  in  $\mathbb{R}^{N}$ and
star shaped with  respect to the origin   such that $ D_1^{*}$
strictly contains $ D_0^{*}$. We look for  $(D, v,u)$ where $D$ is
$\mathcal{C}^{2}$-regular  domain  such that  $ D_0^{*}
 \subset    D  \subset \bar{D}  \subset D_1^{*}$    and
  $ v$  and $u$   are solutions of the  problems
\begin{equation}\label{q0}
\begin{gathered}
-\Delta_p v   =  0 \quad \mbox{in }   D_1^{*}\backslash D \\
          v  =  1 \quad \mbox{on }  \partial D \\
       v  =  0 \quad \mbox{on } \partial D_1^{*}
\end{gathered} \hspace{15mm}
\begin{gathered}
-\Delta_p u  =  0 \quad \mbox{in }  D\backslash D_0^{*} \\
          u  =  0 \quad \mbox{on }  \partial D \\
       u  =  1 \quad \mbox{on }  \partial D_0^{*}
\end{gathered}
\end{equation}
respectively, and   satisfy  the nonlinear   joining condition
\begin{equation}\label{qq0}
\|\nabla v\|^{p} - \|\nabla u\|^{p}  = \lambda \quad\text{on }  \partial D,\;
   \lambda \in \mathbb{R}.
\end{equation}
This  joining condition  is justified by  the method which
we are  using.

The  described above problem  appears  in several   physical
situations  and  can   be appropriately   interpreted in many
industrial  applications.
In  the case  $p =2$, we  refer   the reader to \cite{A1,A2}
 and the   references  therein.

Inspired  by the  pioneering work of Beurling, where  the notion
of sub and super solutions in the geometrical case  is used,
Acker et al \cite{AHPS} studied
this problem with a more general non linear joining junction.
They assumed that $D_0^{*} $  and $D_1^{*} $ are bounded
convex domains and  $D_1^{*} $  contains   $D_0^{*}$ strictly.
 By considering the p-laplacian  operator,  we show  the
 existence of a solution to the exterior (resp interior) free
 boundary problem with non constant Bernoulli free boundary
 condition.  In the second part of this article,
 we study the existence of  solutions to the two-layer shape
 optimization problem.  From a monotonicity result, we show
 the existence of classical solutions to the two-layer Bernoulli
 free-boundary  problem  with nonlinear joining conditions.
 Also we extend the existence result to the multi-layer case..
They proved there exists  a convex $\mathcal{C}^{1}$  domain $D$,
   $D_0^{*}  \subset  \subset  D  \subset  \subset   D_1^{*}$
which is   a classical solution  of the  two-layer  free-boundary
problem (\ref{q0})-(\ref{qq0}).

Using convex  domains, Acker \cite{A1,A2}
proved the existence  for multi-layer  free boundary   problems
 by using the  operator method in the case where  $p= 2$.
 Laurence  and Stredulinsky \cite{lstre1, lstre2}, use convex domains,
when proving  an existence result  in the case  $ p= 2, N= 2$.

 Now,  by combining   a variational  approach   and
a sequential  method, we establish  an existence result  for
non necessarily convex domains.

The structure   of the  Part II is as follows. In the   first
part, we present  the main result. By considering  the auxiliary
results in the  Part I, we study in the second part
 the shape optimization  problem   and the  existence  of  Lagrange
multiplier  function  $\lambda$.
The  existence  result  for the shape  optimization problem
consists of finding a domain  $D$   such that
$$
J( D)= \min \{ J ( w),  w  \in   \mathcal{O}_{\epsilon}\},
$$
where
$\mathcal{O}_{\epsilon}=\{w \subset  \mathbb{R}^{N}, D_0^{*} \subset
 \subset D  \subset \subset D_1^{*}, w $ verifying the $\epsilon$-cone
property , $\mathop{\rm vol}(w) = m_0\}$, where   $vol$ denotes the volume, $m_0$  is a
fixed  value  in $\mathbb{R}_+^{*}$. The functional   $J$  is defined  on
$\mathcal{O}_{\epsilon}$    by
$$
J( D)  =  \frac{1}{p} \int_{ D_1^{*}\backslash D}\|\nabla v\|^{p}
  +  \frac{1}{p} \int_{ D\backslash D_0^{*}}\|\nabla u\|^{p}, \quad   1<p<\infty,
$$
 where  $v$ and $u$ are  solutions  of
\begin{equation}\label{eq0}
\begin{gathered}
-\Delta_p v   =  0 \quad \mbox{in }   D_1^{*}\backslash D \\
          v  =  1 \quad \mbox{on }  \partial D \\
       v  =  0 \quad \mbox{on }  \partial D_1^{*}
\end{gathered}
\hspace{15mm}
\begin{gathered}
-\Delta_p u  =  0 \quad \mbox{in }   D\backslash D_0^{*} \\
          u  =  0 \quad \mbox{on }  \partial D \\
       u  =  1 \quad \mbox{on }  \partial D_0^{*}
\end{gathered}
\end{equation}
respectively. Next, we obtain   the  joining condition as  an  optimality
condition:
\begin{equation}
\|\nabla v\|^{p} - \|\nabla u\|^{p}  =  \frac{p}{p-1}
\lambda_D\quad\text{on }  \partial D,  \;  \lambda_D\in R\,.
\end{equation}
Then we conclude  this section  with  a monotonicity result. The third
section is devoted   to the proof   of the main  result. And the
last part is devoted  to the extension  to the
multi-layer case.

\section{main results of Part I}

For the  exterior case,
let  $K$ be a $\mathcal{C}^{2}$-regular, star-shaped with respect
to the  origin and bounded domain.

\begin{theorem}\label{theo1}
If  $\Omega$  solution  of the   shape  optimization problem  $\min\{
J_1(w), w \in \mathcal{O}^{1}_{\epsilon}\}$  is
$\mathcal{C}^{2}$-regular domain,  then the free boundary  problem
(\ref{eq1})  admits    a classical unique solution  $\Omega$.
\end{theorem}

For the interior case,
let  $K$ be a $\mathcal{C}^{2}$-regular, star-shaped with respect
to the origin and bounded domain.
Let
$$
\alpha( R_K,p,N) :=
\begin{cases}
e/ R_K &\mbox{if } p = N \\[5pt]
\dfrac{|\frac{p -N}{p -1}|} {\big|({\frac{p
-1}{N-1}})^{\frac{N-1}{N-p}} -
 ({\frac{p -1}{N -1}})^{\frac{p -1}{N-p }}\big|} \frac{1}{R_K}
 &\mbox{if }p\neq N.
\end{cases}
$$
 where  $R_K = \sup \{R>0: B(o,R) \subset K  \}$,
Here  $c_K$ is the minimal  value for which  the interior
Bernoulli problem (\ref{eq2}) admits a solution.

 \begin{theorem}\label{theo2}
If   the solution  $\Omega$  of the shape optimization
problem
 $\min\{J(w), w \in \mathcal{O}^{2}_{\epsilon}\}$  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
\eqref{eq2}.
Moreover
The  constant   $c_K$  satisfies $ 0<c_K  \leq \alpha( R_K,p,N)$.
\end{theorem}

To prove  these theorems  we need  the following results.

\section{Auxiliary results}

For the rest of this article, we consider a fixed, closed domain
$D$ which contains all  the  open subsets used.

 Let  $\zeta$  be  an unitary  vector  of $\mathbb{R}^N$, $\epsilon$ be a
 real  number  strictly positive  and  $y$ be in $\mathbb{R}^N$.
 We call a  cone  with vertex $y$,  of direction  $\zeta$ and
 angle  to the  vertex  and height $\epsilon$,  the set
 defined   by
 $$
 {\mathcal{C}}(y, \zeta,\epsilon,\epsilon)
 = \{ x \in \mathbb{R}^N : |x -y|\leq \epsilon  \mbox{ and }
  |(x-y)\zeta|\geq |x -y|\cos \epsilon\}.
 $$

 Let $\Omega$ be   an open  set   of $\mathbb{R}^N$, $\Omega$ is said to  have   the
 $\epsilon$-cone property  if for  all  $x\in \partial \Omega$ then there
 exists  a direction $\zeta$ and   a strictly positive  real  number
 $\epsilon$ such that
$$
{\mathcal{C}}(y, \zeta,\epsilon,\epsilon)
 \subset \Omega,\quad  \mbox{for all } y \in B(x, \epsilon) \cap \bar{\Omega}.
$$
Let  $K_1$ and $K_2$ be two compact  subsets of  $D$.
Let
\[
d( x , K_1)= \inf_{y\in K_2} d(x,y) ,  \quad
d( x , K_2)= \inf_{y\in K_1} d(x,y)  .
\]
Note that
\[
\rho(K_1,K_2) = \sup_{x\in K_2} d(x, K_1) , \quad \rho(K_2,K_1) =
\sup_{x\in K_1} d(x, K_2) .
\]
Let
$$
d_H (K_1,K_2)  =  \max [ \rho(K_1,K_2),\rho(K_2,K_1) ],
$$
which is called the  Hausdorff  distance of $K_1$  and
 $K_2$.

 Let $(\Omega_n)$    be a sequence  of  open  subsets  of $D$   and
 $\Omega$  be  an open   subset   of $D$.
 We say   that  the sequence   $(\Omega_n)$ converges on
 $\Omega$ in  the Hausdorff sense and we denote  by
$\Omega_n \stackrel{H}{\to} \Omega $   if $\lim_{n \to +\infty}
d_H(\bar{D}\backslash \Omega_n,\bar{D}\backslash \Omega) = 0$.

Let $(\Omega_n)$  be a sequence  of  open  sets  of
$\mathbb{R}^{N}$   and $\Omega$  be  an open  set   of
$\mathbb{R}^{N}$.
 We say that   the sequence $(\Omega_n)$    converges   on $\Omega$ in
 the   sense   of  $L^{p}$,  $1  \leq  p<\infty$
 if   $\chi_{\Omega_n}$   converges  on
 $\chi_{\Omega}$   in   $L_{\rm loc}^{p}(\mathbb{R}^{N}),  \chi_{\Omega}$
being  the characteristic   functions  of $\Omega$.

 Let $(\Omega_n)$    be a sequence  of  open  subsets  of $D$   and
 $\Omega$  be  an open   subset   of $D$.
 We   say   that  the sequence   $(\Omega_n)$  converges  on $\Omega$ in the
 compact  sense  if:
 \begin{enumerate}
    \item  Every  compact $G$  subset of   $\Omega$,    is
    included   in   $\Omega_n$  for  $n$  large   enough,
    \item  every  compact $Q$  subset of   $\bar{\Omega}^{c}$,   is
    included   in   $\bar{\Omega}^{c}_n$  for  $n$  large   enough.
 \end{enumerate}

\begin{lemma}\label{ler0}
Let $\Omega_1$  and  $\Omega_2$ be  two different  domains
star-shaped  with respect to the origin  and bounded. If
$\bar{\Omega}_1  \subset \bar{\Omega}_2$ then  there exists
$0 <t_0 < 1$ such that $t_0{\Omega}_2  \subset \Omega_1$  and
$t_0\partial \Omega_2\cap\partial \Omega_1  \neq \emptyset$.
\end{lemma}

The proof of the above lemma can be found in \cite{L}.

\begin{lemma}\label{ler1}
Let  $(f_n)_{n\in \mathbb{N}}$  be  a sequence  of functions of
$L^{p}(\Omega)$,    $1\leq p <\infty$   and
   $f\in  L^{p}(\Omega)$.  We  suppose $f_n$  converges on
$f$ a.e.   and $\lim_{n\to \infty} \| f_n\|_{p} =  \| f\|_{p}$.
Then we have $\lim_{n\to \infty} \| f_n  - f\|_{p} =0$.
\end{lemma}

For the proof of the above lemma see  for example \cite{kav}.

\begin{lemma}[Brezis-Lieb]\label{ler2}
Let  $(f_n)_{n\in \mathbb{N}}$  be  a  bounded sequence
in $L^{p}(\Omega)$,  $1\leq p <\infty$. We suppose that
$f_n$  converges on $f$   a.e.,  then $f\in  L^{p}(\Omega)$ and
$$
\| f\|_{p} = \lim_{n\to \infty} (\| f_n  - f \|_{p} + \| f_n\|_{p}).
$$
\end{lemma}

For the proof of the above lemma, see for example \cite{kav}.

\begin{lemma}\label{ler4}
Let  $(\Omega_n)_{n\in \mathbb{N}}$  be  a sequence  of open sets
in $\mathbb{R}^N$ having  the  $\epsilon$-cone  property,  with
$\bar{\Omega}_n  \subset  F  \subset D$, $F$ a compact set and $D$
a  ball, then,   there exists an open  set $\Omega$, included in
$F$, which satisfies  the  $\frac{\epsilon}{2}$-cone  property
  and  a subsequence $(\Omega_{n_k})_{k\in \mathbb{N}}$  such that
\begin{gather*}
\chi_{{\Omega}_{n_k}} \stackrel{ L^{1}}{\to}  \chi_{\Omega}, \quad
\Omega_{n_k} \stackrel{H}{\to} \Omega \\
\partial  \Omega_{n_k} \stackrel{H}{\to} \partial \Omega, \quad
\bar{\Omega}_{n_k} \stackrel{H}{\to} \bar{\Omega}.
\end{gather*}
\end{lemma}

The  above lemma  is a well known result in functional analysis
related to shape optimization; its proof can be found for example
in \cite{LS3}.

\section{Shape optimization result and monotonicity result}

For the exterior case, we have the following result, whose
proof can be found in \cite{LS2}.

\begin{proposition}\label{pro1}
The problem: Find    $\Omega \in  \mathcal{O}^{1}_{\epsilon}$
    such that
$ J_1(\Omega) = \min \{   J_1(w),  w\in \mathcal{O}^{1}_{\epsilon}\}$
  admits a solution.
\end{proposition}


For the interior case, we have the following result, whose
proof can be found in \cite{LS3}.

\begin{proposition} \label{pro2}
The problem: Find    $\Omega \in  \mathcal{O}^{2}_{\epsilon}$
 such that
$ J_2(\Omega) = \min \{   J_2(w),  w\in \mathcal{O}^{2}_{\epsilon}\}$
  admits a solution.
\end{proposition}


\begin{remark}\label{re1} \rm
Let us define another class of domains:
$$
{\mathcal{O}}_{0}  = \{ w \supset K: w \mbox{ is an open set of}
   \mathbb{R}^{N}.  \mathcal C^{k}\text{-regular domain, }
  \int_{w}\frac{g^{p}}{c^{p}}(x)dx= V_0 \}
 $$
 where $k \geq 3$.
It is possible to use  the oriented distance, the results in
\cite[theorems 5.3 5.5,5.6]{DZ}, \cite{D1Z} and the Ascoli
theorem to prove the existence of a domain at least of class
$\mathcal C^{k-1}$  which is minimum for the shape optimization
problem.
\end{remark}

For the rest of this article, we assume that $\Omega$  is
$\mathcal{C}^{2}$-regular in order  to use  the shape
derivatives. The next  theorems give  a necessary  condition
optimality condition. We  follow   the    approach  of
Sokolowski-Zolesio \cite{SZ} to define   the shape
derivatives  (see  also \cite{JS}).

For the exterior  case, we have the following result.

\begin{proposition} \label{pro3}
 If  $\Omega$   is  the solution of the shape optimization problem
$$
\min \{  J_1(w):  w\in \mathcal{O}^{1}_{\epsilon}  \},
$$
 then there  exists  a Lagrange multiplier  $\lambda_{\Omega} < 0$ such
that $-\frac{\partial  u}{\partial \nu_{e}} =  ( \frac{p}{1 -
p}\lambda_{\Omega})^{\frac{1}{p}} \frac{g}{c}(x)$   on   $\partial
\Omega$.
\end{proposition}

\begin{proof}
Let  $J_1$  be a functional  defined  on
$\mathcal{O}^{1}_{\epsilon}$  by
$$
J_1(w):  = \frac{1}{p}\int_{w\backslash K} \|\nabla u_w\|^{p} dx ,
$$
where   $ u_w$  is a solution to the Dirichlet  problem
\begin{equation}\label{pe11}
\begin{gathered}
-\Delta_p  u_w   = 0  \quad\mbox{in } \Omega\backslash K, \;1<p<\infty\\
  u_w             = 0  \quad\mbox{on }  \quad\partial  w \\
   u_w          = 1 \quad \mbox{on} \quad \partial K .
\end{gathered}
\end{equation}
We use  classical  Hadamard's  formula  to compute  the  Eulerian
derivative  of  the
 functional  $J_1$  at the point  $\Omega$   in the direction $V$.
A standard   computation,  see \cite{L},  shows
$$
d J_1(\Omega;V) =  \int_{\partial \Omega} \|\nabla u\|^{p-2}\frac{\partial u}{\partial \nu_{e}}
u' ds +  \frac{1}{p}\int_{\partial  \Omega} \|\nabla u\|^{p}V(0).\nu_{e} ds
$$
where
$ u'   =  -\frac{\partial u}{\partial \nu_{e}}V(0).\nu_{e}$ on  $\partial \Omega$.
This implies
$$
dJ(\Omega;V) = \frac{1- p}{p}\int_{\partial \Omega} \|\nabla u\|^{p}V(0).\nu_{e} ds.
$$
Let us take $ J( \Omega) = \int_{ \Omega}  \frac{g^{p}}{c^{p}}(x) dx$.
Then
$$
dJ(\Omega;V)= \int_{ \Omega} \mathop{\rm div}(\frac{g^{p}}{c^{p}}(x)
V(0))dx = \int_{\partial \Omega}
\frac{g^{p}}{c^{p}}(x) V(0).\nu_{e} ds.
$$
$\Omega$  is optimal  then there  exists  a Lagrange  multiplier
$\lambda_{\Omega}\in R$   such that  $dJ_1(\Omega,V) = \lambda_{\Omega} d J(\Omega; V)$.
We  obtain
$$
\int_{\partial \Omega} (\frac{1- p}{p}\|\nabla u\|^{p} -
\lambda_{\Omega} \frac{g^{p}}{c^{p}}(x) V(0).\nu)ds = 0
 \quad\text{for all} \quad  V.
$$
Then
\begin{gather*}
 \|\nabla u\|^{p} =   \frac{ p}{1- p}\lambda_{\Omega}\frac{g^{p}}{c^{p}}(x)
\quad\text{on }  \partial \Omega,\\
\|\nabla u\|  =  ( \frac{ p}{1-p}\lambda_{\Omega})^\frac{1}{p}\frac{g}{c}(x)
\quad\text{on }  \partial \Omega
\end{gather*}
Since $\Omega$  is  $\mathcal{C}^{2}$-regular  and  $u = 0$
on $\partial \Omega$,   we get
$$
-\frac{\partial u}{\partial \nu_{e}}=  ( \frac{ p}{1-
p}\lambda_{\Omega})^\frac{1}{p}\frac{g}{c}(x) \quad\text{on }   \partial
\Omega.
$$
\end{proof}

 For the interior  case, we have the following result.

\begin{proposition}\label{pro4}
 If  $\Omega$   is  the solution of the shape optimization problem
$$
\min \{  J_2(w):  w\in \mathcal{O}^{2}_{\epsilon}  \},
$$
 then there  exists  a Lagrange multiplier  $\lambda_{\Omega}>0$ such
that $\frac{\partial  u}{\partial \nu_{i}} =  ( \frac{p}{p -
1}\lambda_{\Omega})^{\frac{1}{p}}\frac{g}{c}(x)$   on   $\partial
\Omega$.
\end{proposition}

For the proof of the above proposition, we  use  the  same technics
as in  proposition \ref{pro3}.
To conclude  this section, we state  a  monotonicity result.
For the exterior case, we have the following result, whose proof
can be found in \cite{LS2}.

\begin{proposition}\label{pro5}
Suppose that  $K$  is   star-shaped  with respect  to the origin.
Let  $\Omega_1$ and $\Omega_2$  be  two different solutions  to
the shape optimization problem
$\min \{J_1(w),   w\in \mathcal{O}^{1}_{\epsilon}\}$,
 star-shaped  with respect to the origin   such that
${\bar{\Omega}}_1   \subset   {\bar{\Omega}}_2$. The   mapping
which   associates   to every   $\Omega$   the  corresponding Lagrange
multiplier  $\lambda_{\Omega}$  is  strictly increasing  i.e
$\lambda_{\Omega_2}    >   \lambda_{\Omega_1}$.
\end{proposition}


 For the interior case, we have the following result, whose proof
is found in  \cite{LS3}.


\begin{proposition}\label{pro6}
Suppose that  $K$  is   star-shaped  with respect  to the origin.
Let  $\Omega_1$ and $\Omega_2$  be  two different solutions  to
the shape optimization problem $\min \{J_2(w),   w\in
\mathcal{O}^{2}_{\epsilon}\}$,
 star-shaped  with respect to the origin   such that
${\Omega}_1  \subset  {\Omega}_2$ and $\partial
\Omega_1\cap\partial \Omega_2 \neq  \emptyset$. The   mapping
which   associates   to every   $\Omega$   the  corresponding Lagrange
multiplier  $\lambda_{\Omega}$  is decreasing  i.e $\lambda_{\Omega_1}
  \geq   \lambda_{\Omega_2}$.
\end {proposition}

\section{Proof of the main results of Part I}

We  use the preceding properties  to prove the main result.
Exterior  case:

\begin{proof}[Proof of the Theorem \ref{theo1}]
 We choose   a ball $B(O,R)$   centered  at the origin and radius $R$
 and  a ball $B(O,r)$   such  that   $  B(O,r)  \subset  K  \subset  B(O,R)$.
 First, we have to  look   for  a  solution $ u_0$  to the problem
\begin{equation}\label{eq221}
\begin{gathered}
-\Delta_p  u = 0 \quad  \mbox{in }  B_{R}\backslash B_r\\
  u             = 0 \quad  \mbox{on  }  \partial B_{R} \\
  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}
 \dfrac{ \ln\|x\| - \ln R}{\ln r - \ln R}&\mbox{if } p = N \\[5pt]
  \dfrac{ \|x\|^{\frac{p -N}{p -1}}
 - R^{\frac{p -N}{p -1}}} { r^{\frac{p -N}{p -1}}
 - R^{\frac{p -N}{p -1}}}&\mbox{if }p\neq N,
  \end{cases}
  \end{equation}
  and
 \[
 \|\nabla u_0(x)\| = \begin{cases}
 \dfrac{1}{ \|x\|^{2}( \ln R - \ln r)}&\mbox{if } p = N \\[5pt]
 \dfrac{ |\frac{p -N}{p -1}| \|x\|^{\frac{-N-p+2}{p -1}}}{| r^{\frac{p -N}{p -1}}
 - R^{\frac{p -N}{p -1}}|}&\mbox{if }p \neq N.
  \end{cases}
\]
In particular  $\|\nabla u_0\|  <  c$    on    $\partial B_R$   for $R$
big  enough.

Now   consider  the   problem
\begin{equation}\label{eqq11}
\begin{gathered}
-\Delta_p  u = 0 \quad   \mbox{in }  B_R\backslash K\\
  u             =  0 \quad   \mbox{on }  \partial B_R \\
  u             = 1 \quad   \mbox{on }  \partial K.
\end{gathered}
\end{equation}
This problem  admits   a solution  denoted   by
$u_R$. This solution is  obtained  by minimizing  the functional
$J_1$ defined on the Sobolev  space
$$
V'= \{ v\in W_0^{1,p} ( B_R),  v = 1  \mbox{on }  \partial K \}
$$
and  $J_1(v) = \frac{1}{p}\int_{B_R\backslash K }\|\nabla
v\|^{p}dx$.

Consider the problem
\begin{equation}\label{eqq21}
\begin{gathered}
-\Delta_p  v = 0 \quad\mbox{in }  B_{R}\backslash K\\
  v             = 0 \quad\mbox{on }  \partial B_R \\
  v             = u_0 \quad\mbox{on }  \partial K.
\end{gathered}
\end{equation}
It is  easy to see that  $ v = u_r$  is  a   solution  to problem
(\ref{eqq21}).  By   the  comparison  principle \cite{T1}, we
obtain $0  \leq  u_0  \leq  1$  and $0  \leq  u_R  \leq  1$. On
$\partial (B_{R}\backslash K)$, we  obtain $u_R  \geq  u_0$ and
then, $u_R  \geq  u_0$ in $B_{R}\backslash K$. Finally, we have
$\|\nabla u_R\|  \geq  \|\nabla u_0\|$ on
$\partial B_R$.

\subsection*{Case $p = N$}
If  $R_1 < R_0$, we get  $\|\nabla u_0\|_{| \partial B_{R_0}}
 \leq  \|\nabla u_0\|_{| \partial B_{R_1}}$  then the mapping  for
all
$R$  associates   $\|\nabla u_0\|_{| \partial B_{R}}$   is   decreasing.

 Initially, we choose  a radius  $R_0$    big enough  and we compute
$\|\nabla u_0\|_{| \partial B_{R_0}}$  and if
$ \big| \|\nabla u_0\|_{|{\partial B_{r_0}}} - c \big|>\delta$,
 where $\delta>0$ is a fixed and sufficiently   small  number.
We  continue     the process by
 varying  $R$  in   the   increasing  sense, we will  achieve
 a  step  denoted   $N$   such that
 $
 \big|\|\nabla u_0\|_{|{\partial B_{R_N}}} -  c \big| <\delta$.

Consider  $\mathcal{O}_N$  the class  of admissible domains
defined as follows
$$
\mathcal{O}_N = \left \{  w\in \mathcal{ O}_{\epsilon}: w
\subset    B_{R_N}  , \;    \int_{w} \frac {g^{p}}{c^{p}} = V_0
\right\},
$$
where $ V_0$  denotes  a fixed positive constant. We  look   for
$\Omega\in \mathcal{O}_N$  and $\lambda_{\Omega}$  a real such
that
\begin{equation}\label{i1}
\begin{gathered}
-\Delta_p  u  = 0 \quad \mbox{in }  \Omega\backslash K\\
  u             = 0 \quad  \mbox{on }  \partial \Omega \\
u             = 1 \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})^{\frac{1}{p}}\frac{g}{c}(x)$.
Applying  proposition (\ref{pro1}), the shape  optimization
problem $\min\{ J_1(w) ,  w\in \mathcal{O}_N \} $   admits
 a solution  and by  proposition   \ref{pro3},
$\Omega$    satisfies  the overdetermined  boundary condition
$-\frac{\partial u}{\partial \nu}   = c_{\Omega}$.

We have  $ \Omega  \in \mathcal{O}_N  $, then $\Omega \subset B_{R_N}$,
according to the  lemma \ref{ler0}  there  exists $t_0 < 1$ such
that $t_0 B_{R_N} \subset \Omega$, and $t_0\partial B_{R_N}\cap
\partial \Omega  \neq  \emptyset$.
Let us take   $x_0  \in t_0\partial B_{R_N}\cap \partial
\Omega $  and  set
$ u_{t_0}(x) = u_{R_N}(\frac{x}{t_0}),\frac{x}{t_0} \in  B_{R_N}\backslash K$.
$ u_{t_0}$  satisfies
\begin{equation}\label{i1s1}
\begin{gathered}
-\Delta_p  u_{t_0} = 0 \quad \mbox{in } t_0(B_{R_N} \backslash K) \\
  u_{t_0}          =  0 \quad \mbox{on } t_0 \partial B_{R_N} \\
  u_{t_0}          = 1 \quad \mbox{on }  t_0\partial K.
\end{gathered}
\end{equation}
On the  other  hand, we  have   $t_0 B_{R_N} \subset \Omega$, let us take
$w_3=   u_{|t_0 B_{R_N}}$, then $w_3$  satisfies
\begin{equation}\label{i1s2}
\begin{gathered}
-\Delta_p  w_3 = 0 \quad \mbox{in } t_0 B_{R_N} \backslash K \\
  w_3        =   u_{|t_0\partial  B_{R_N}}  \quad \mbox{on } t_0 \partial B_{R_N} \\
  w_3        = 1 \quad \mbox{on }  \partial K.
\end{gathered}
\end{equation}
Let us consider the problem
\begin{equation}\label{ils3}
\begin{gathered}
-\Delta_p  z = 0 \quad \mbox{in }t_0 B_{R_N} \backslash K \\
  z  =   0  \quad \mbox{on } \partial t_0\partial B_{R_N}\\
  z  = {u_{t_0}}_{|\partial K}  \quad \mbox{on }  \partial K.
\end{gathered}
\end{equation}
It is easy to see  that $ z = u_{t_0}  $    is    a solution   to
the problem  (\ref{ils3}). And  we get $0  \leq  u_{t_0}  \leq  1$
and
   $0  \leq  u  \leq  1$. On
 $\partial ( t_0 B_{R_N}   \backslash K)$, we have $u_{t_0 }  \leq  u$,
by  the comparison principle  \cite{T1},  we obtain
$u_{t_0 }  \leq  u\quad\text{in}  ( t_0 B_{R_N}   \backslash K)$.
 We have
\[
\lim_{t\to 0}\frac{u_{t_0}(x_0 -\nu_{e} t) - u_{t_0}(x_0)}{t}
\leq \lim_{t\to 0}\frac{u(x_0 -\nu_{e} t) - u(x_0)}{t},
\]
which is equivalent to
\[
-\frac{\partial u_{t_0}}{\partial \nu_{e}}(x_0) \leq  -
\frac{\partial u}{\partial \nu_{e}}(x_0)\,.
\]
This implies
\[
\|\nabla u_{R_N}(x_0)\|  \leq  - \frac{\partial u}{\partial \nu_{e}(x_0)}
\]
Let us consider $ \Omega = \Omega_0$ as the first iteration and
$$
\mathcal{O}_N ^{1}= \big\{ w\in \mathcal{ O}_{\epsilon}:  w
\subset \Omega_0   \subset B_{R_N}  ,\;
   \int_{w} \frac {g^{p}}{c^{p}} = V_1 \big\}, \quad (V_1 < V_0)
$$
where $ V_1$  denotes  a fixed positive constant.

We iterate  by looking for $\Omega_1\in \mathcal{O}_N^{1}$  and
$\lambda_{\Omega_{1}}$ such that
 such that
\begin{equation}\label{il2}
\begin{gathered}
-\Delta_p  u_1  = 0 \quad \mbox{in }  \Omega_1\backslash K\\
  u_1             = 0 \quad  \mbox{on }  \partial \Omega_1 \\
u_1             = 1 \quad   \mbox{on}  \partial K\\
  -\frac{\partial u}{\partial \nu}
= c_{\Omega_1} \quad \mbox{on }\partial \Omega_1
\end{gathered}
\end{equation}
where
$c_{\Omega_1}=(\frac{-p}{p -1}\lambda_{\Omega_1})^{\frac{1}{p}}\frac{g}{c}(x)$.
Applying  proposition  (\ref{pro1}), the shape  optimization
problem $\min\{ J_2(w) ,  w\in \mathcal{O}_N^{1} \} $   admits
 a solution  and by  proposition \ref{pro3},
$\Omega_1$    satisfies  the overdetermined  boundary condition
$-\frac{\partial u_1}{\partial \nu}   = c_{\Omega_1}$.
We have  $ \Omega  \in \mathcal{O}^{1}_N $, then $\Omega_1 \subset
B_{R_N}$, according the lemma \ref{ler0}   there  exists $t_1 < 1$
such that $t_1 B_{R_N} \subset \Omega$, then
$t_1\partial B_{R_N}\cap \partial \Omega_1 \neq  \emptyset$.

Let us take   $x_1  \in t_1\partial B_{R_N}\cap \partial \Omega_1 $
and  set
$ u_{t_1}(x)= u_{R_N}(\frac{x}{t_1}), \frac{x}{t_1} \in  B_{R_N}\backslash K$.
$ u_{t_1}$ satisfies
\begin{equation}\label{i1ss1}
\begin{gathered}
-\Delta_p  u_{t_1} = 0 \quad \mbox{in } t_1(B_{R_N} \backslash K) \\
  u_{t_1}          =  0 \quad \mbox{on } t_1 \partial B_{R_N} \\
  u_{t_1}          = 1 \quad \mbox{on }  t_1\partial K.
\end{gathered}
\end{equation}
On the  other  hand, we  have   $t_1 B_{R_N} \subset \Omega_1$, let us
take $w_4=   u_{|t_1 B_{R_N}}$, then $w_4$   satisfies
\begin{equation}\label{i1ss2}
\begin{gathered}
-\Delta_p  w_4 = 0 \quad \mbox{in } t_1 B_{R_N} \backslash K \\
  w_4       =   u_{1_{|t_1\partial  B_{R_N}}}  \quad \mbox{on }
t_1 \partial B_{R_N} \\
  w_4       = 1 \quad \mbox{on }  \partial K.
\end{gathered}
\end{equation}
Let us consider the problem
\begin{equation}\label{ilss3}
\begin{gathered}
-\Delta_p  z = 0 \quad \mbox{in }t_1 B_{R_N} \backslash K \\
  z  =   0\quad \mbox{on } t_1 \partial B_{R_N} \\
  z  = {u_{t_1}}_{|\partial K}  \quad \mbox{on }  \partial K.
\end{gathered}
\end{equation}
It is easy to see  that $ z = u_{t_1}  $    is    a solution   to
 (\ref{ilss3}). And  we get $0  \leq  u_{t_1}  \leq 1$ and
   $0  \leq  u_1  \leq  1$. On
 $\partial ( t_1 B_{R_N}   \backslash K)$, we have $u_{t_1 }  \leq  u_1$,
by  the comparison principle  \cite{T1},  we obtain
$u_{t_1 }  \leq  u_1\quad\text{in}  ( t_1 B_{R_N}   \backslash K)$.
 We have
\[
\lim_{t\to 0}\frac{u_{t_1}(x_1 -\nu_{e} t) - u_{t_1}(x_1)}{t}
\leq \lim_{t\to 0}\frac{u_1(x_1 -\nu_{e} t) - u_1(x_1)}{t},
\]
which is equivalent to
\[
-\frac{\partial u_{t_1}}{\partial \nu_{e}}(x_1) \leq  -
\frac{\partial u_1}{\partial
\nu_{e}}(x_1).
\]
This implies
\[
\|\nabla u_{R_N}(x_1)\|   \leq - \frac{\partial u_1}{\partial
\nu_{e}} (x_1).
\]
We  can continue   the   process until  a step  denoted   by  $k$
such that
$$
-\frac{\partial u_k}{\partial \nu_{e}}(x_k) = c_{\Omega_k} $$
For all  $s \in \partial B_{R_N}$, we get
$\|\nabla u_0(s)\|  \leq  \|\nabla u_{R_N}(s)\|$  then there exists
$s_0 \in \partial B_{R_N}$, such that $\|\nabla u_0(s_0)\| > c_{\Omega_k}$.

The sequence  $(c_{\Omega_j})_{(0  \leq  j  \leq  k)}$
 is  strictly  decreasing   and  positive,  then
$(\frac{-p}{p-1}\lambda_{\Omega_j})^{\frac{1}{p}}$ converges on  $c$.
Then there exists $\Omega$ solution  to problem  (\ref{eq1}), the
sequence $ (\Omega_j)_{(0  \leq  j  \leq  k)}$  gives a good
approximation  to  $\Omega$. The uniqueness  of the solution
$\Omega$  is given  by  the monotonicity result.

\subsection*{Case $ p  \neq  N $}
If  $R_1 < R_0$, we get  $\|\nabla u_0\|_{| \partial B_{R_1}}
 \geq \|\nabla u_0\|_{| \partial B_{R_0}}$  then  the mapping  for
all $R$  associates   $\|\nabla u_0\|_{| \partial B_{R}}$   is   decreasing.
 Initially,   we choose  a radius  $R_0$    big enough  and we compute
$\|\nabla u_0\|_{| \partial B_{R_0}}$  and if $ \big| \|\nabla
u_0\|_{|{\partial B_{R_0}}} - c \big|>\delta,   \delta>0$ fixed
and sufficiently   small  number. We  continue     the process by
 varying  $R$  in   the   increasing  sense, we will  achieve
 a  step  denoted   $N$   such that
 $
 \big| \|\nabla u_0\|_{|{\partial B_{R_N}}} -  c \big| <\delta$.
Here  the reasoning  is identical to the case  $ p = N$.
\end{proof}

\subsection*{Interior  case}

\begin{proof}[Proof of the Theorem \ref{theo2}]
 Let  $R_K  = \sup \{  R >0: B(o, R)  \subset  K \}$.
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{eq22}
\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}
 \dfrac{ \ln\|x\| - \ln R_K}{\ln r - \ln R_K} &\mbox{if } p = N \\[5pt]
  \dfrac{ -\|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
 \[
 \|\nabla u_0(x)\| = \begin{cases}
 \dfrac{1}{ r( \ln R_K - \ln r)}&\mbox{if } p = N \\[5pt]
 \dfrac{ |\frac{p -N}{p -1}| \|x\|^{\frac{-N+1}{p -1}}}{| r^{\frac{p -N}{p -1}}
 - R^{\frac{p -N}{p -1}}|}&\mbox{if }p \neq N.
  \end{cases}
\]
In particular  $\|\nabla u_0\| > c$    on    $\partial B_r$   for $r$
small  enough.
Now  let us  consider  the  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}
Then problem  (\ref{eqq1})  admits   a solution  denoted   by
$u_r$. This solution is  obtained  by minimizing  the functional
$J$ defined on the Sobolev  space
\[
V'= \{ v\in W^{1,p} (K\backslash B_r),  v = 1  \mbox{on }
\partial B_r \text{ and }v = 0 \text{ on }  \partial K \}
\]
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
(\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\|$ on
$\partial B_r$.

\subsection*{Case $p = N$}
\[
\|\nabla u_0\|_{|{\partial B_r}} =\frac{1}{r(\ln R_K - \ln r)} =
h(r), \quad   \forall   r\in ]0,R_K[.
\]
It is  easy to see that $h(r)$ 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   $r\in ]0,R_K[$,
$\|\nabla u_0\|_{|{\partial B_r}}  \geq  h(\frac{R_K}{e}) = \frac{e}{R_K}$.

\noindent(1)  For  $ g(x) = 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
 $$
 r_n\in  ]0,\frac{R_K}{e}[   \mbox{and}
 \big|\|\nabla u_0\|_{|{\partial B_{r_n}}} -  c \big| <\delta.
  $$
Consider  $\mathcal{O}_n$  the class  of admissible domains
defined as follows
$$
\mathcal{O}_n = \big\{ w\in \mathcal{ O}_{\epsilon},  B_{r_n}
\subset  w,   \partial B_{r_n}\cap
\partial w \neq  \emptyset,  \mbox{ and }  \int_{w} \frac {g^{p}}{c^{p}}
 = V_0 \big\},
$$
where $ V_0$  denotes  a fixed positive constant. We  look   for
$\Omega\in \mathcal{O}_n$  and $\lambda_{\Omega}$ such that
\begin{equation}\label{i11}
\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})^{\frac{1}{p}}
\frac{g}{c}(x)$.
Applying  the proposition (\ref{pro2}), the shape  optimization
problem $\min\{ J_2(w) ,  w\in \mathcal{O}_n \} $   admits
 a solution  and by proposition   \ref{pro4},
$\Omega$    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
  (\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
\[
\lim_{t\to 0}\frac{u_{r_n}(x_0 -\nu t) - u_{r_n}(x_0)}{t} \leq
\lim_{t\to 0}\frac{u(x_0 -\nu t) - u(x_0)}{t},
\]
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})^{\frac{1}{p}}\frac{g}{c}(x)$,   and
$$
 \mathcal{O}^{1}_n  = \big\{w\in \mathcal{O}_{\epsilon}:  \Omega_0
 \subset  w,  \;  \partial w \cap
\partial B_{r_n} \neq  \emptyset  \int_{w} \frac {g^{p}}{c^{p}} = V_1 \big\},
$$
where  $V_1$ is a strictly positive constant  and  $V_0<V_1$. By
the same reasoning   as  above, we conclude  that
\[
\frac{\partial u_{r_n}}{\partial\nu}(x_1)  \geq  \frac{\partial
u_1}{\partial \nu}(x_1) = c_{\Omega_1}
\]
where $ x_1\in  \partial \Omega_1 \cap \partial B_{r_n}$. We  can
continue   the   process until  a step  denoted   by  $k$ which we
will  determine  and we have
\[
\frac{\partial u_{r_n}}{\partial \nu}(x_k)  \geq  \frac{\partial
u_k}{\partial \nu}(x_k) = c_{\Omega_k}  \quad\mbox{and}\quad
  x_k\in  \partial \Omega_k \cap \partial B_{r_n}.
\]
Finally,  we have  constructed  an increasing sequence of domain
solutions: $\Omega_0  \subset  \Omega_1   \subset  \Omega_2 \dots
\subset  \Omega_k$. By the monotonicity result, we have $c_{\Omega_0}
\geq  c_{\Omega_1}  \geq  c_{\Omega_2} \dots \geq  c_{\Omega_k}$.

Since   $ \|\nabla u_{r_n}\|  \leq  \|\nabla u_0\|$ on $\partial
B_{r_n}$, $k$ is chosen   as follows: At each   point   $s_0\in
\partial B_{r_n}$, we have
$$
c_{\Omega_k}  \leq  \frac{\partial u_0}{\partial \nu}(s_0)  \leq
c_{\Omega_{k-1}}.
$$
Then  we  obtain the inequality
\begin{equation}\label{tp}
   c_{\Omega_k} - \frac{e}{R_K}  \leq  \frac{\partial u_0}{\partial
\nu}(s_0)- \frac{e}{R_K}  \leq  c_{\Omega_{k-1}}- \frac{e}{R_K}.
\end{equation}
The sequence  $(c_{\Omega_j})_{(0  \leq  j  \leq  k)}$
 is decreasing   and  strictly positive,  then   it converges on  $l$.
Passing  to the limit in (\ref{tp}), we obtain that $ l=
\frac{e}{R_K}$ and there exists $\Omega$ solution  to problem
(\ref{eq2}). The sequence $ (\Omega_j)_{(0  \leq  j  \leq  k)}$
gives a good  approximation
 to  $\Omega$. The uniqueness  of the solution  $\Omega$  is given
by  the monotonicity result.

\noindent(2)  For  $g(x)>\frac{e}{R_K}$ 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{eq2}) admits  a
solution.

\subsection*{Case  $ p  \neq  N $}
Here  the reasoning  is identical to the case  $ p = N$. We note
that
$$
\|\nabla u_0\|_{|{\partial B_{r_n}}} = \big|\frac{p - N}{p-1}\big|
\frac{1}{ 1 - (\frac{r}{R_K})^{\frac{N - p}{p-1}}} \frac{1}{r} =
h(r)
$$
and $h$ 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
\[
g(x) \geq  |\frac{p - N}{p -1}|\frac{1}{| (\frac{ p- 1}{ N -
1})^{\frac{N - 1}{N-p}}  - (\frac{ p- 1}{ N - 1})^{\frac{p - 1}{N-
p}}|}\frac{1}{R_K} = h((\frac{ p - 1}{ N -1})^{\frac{ p- 1}{N - p
}} R_K),
\]
problem (\ref{eq1}) admits a solution.

It is easy to  have, $0 < c_K  \leq  \alpha(R_K,p,N)$.
If   $K$  is a ball  of radius  $R$,  an explicit  computation   gives
$c_K   =  \alpha(R,p,N)$  and   for all $0<c< c_K$  problem
(\ref{eq2})  has no solution.
\end{proof}

\section{Main result of Part II}

Let  $ D_0^{*}$  and   $ D_1^{*}$  be  $\mathcal{C}^{2}$-regular,
 compact sets  in  $\mathbb{R}^{N}$    and   starshaped with respect
to the origin   such that $ D_1^{*}$  strictly contains $D_0^{*}$.
We  want to find
$ (v, u)$  solutions of
\begin{equation}\label{q1}
\begin{gathered}
-\Delta_p v   =  0 \quad \mbox{in }   D_1^{*}\backslash D \\
          v  =  1 \quad \mbox{on }  \partial D \\
       v  =  0  \quad \mbox{on }  \partial D_1^{*}
\end{gathered}
\hspace{15mm}
\begin{gathered}
-\Delta_p u  =  0 \quad \mbox{in }  D\backslash D_0^{*} \\
          u  =  0 \quad \mbox{on } \partial D \\
       u  =  1  \quad \mbox{on } \partial D_0^{*} \\
\end{gathered}
\end{equation}
respectively, and satisfy  the non linear joining condition
\begin{equation}\label{qq1}
\|\nabla v\|^{p} - \|\nabla u\|^{p}  = \lambda\quad\text{on }  \partial D,
\end{equation}
where $ \lambda$ is a given real $\in  \mathbb{R}$.

\begin{theorem} \label{theomain}
Let  $ D_0^{*}$  and   $ D_1^{*}$  be  $\mathcal{C}^{2}$-regular,
 compact sets  in  $\mathbb{R}^{N}$    and   starshaped   with  respect to
the origin   such that  $ D_1^{*}$  strictly  contains $ D_0^{*}$.
  One supposes in add that there is $R_0 =\sup \{ R>0: B(O,R) \in
  D_1^{*}\}$and  $ D_0^{*} \in B(O,R_0)$
If  $D$, $\mathcal{C}^{2}$-regular  domain     solution to the
shape  optimization problem
  $ \min \{   J ( w),  w  \in  \mathcal{O}_{\epsilon}\}$  such that
 $D_0^{*} \subset \subset D  \subset \subset D_1^{*}$, then
$D$   is  a solution  of the  two-layer  free  boundary
problem   (\ref{q1})-(\ref{qq1}).
\end{theorem}

To prove  the main result of the  Part II, we need to
establish some results such as   shape optimization and monotonicity
results.

\section{Shape optimization  and monotonicity result}

\begin{theorem}\label{theop}
The problem: Find  $D  \in  \mathcal{O}_{\epsilon}$  such  that
$ J(D) = \min \{ J(w),   w  \in  \mathcal{O}_{\epsilon}\}$
admits  a solution
\end{theorem}

\begin{proof}
Let  $E$ be  a functional  defined  on $W^{1,p}(D_1^{*})\times
W^{1,p}(D_1^{*})$     by
$$
E( \tilde{v},\tilde{u})  =  \frac{1}{p} \int_{ D_1^{*}}\|\nabla \tilde{ v}\|^{p}
+  \frac{1}{p} \int_{ D_1^{*}}\|\nabla \tilde{u}\|^{p},  \quad  1<p<\infty,
$$
where    $ \tilde{v}$ is the extension of $v$ in $D_0$ and
$\tilde{u}$  is the extension  by $0$  in $D_1^{*}\backslash D$ of
$u$. And   $v$   and $ u$   are  solutions  of
\begin{equation}\label{opt1}
\begin{gathered}
-\Delta_p v  =  0 \quad \mbox{in }   D_1^{*}\backslash D \\
          v  =  1 \quad \mbox{on }  \partial D \\
       v  =  0 \quad \mbox{on }  \partial D_1^{*}
\end{gathered}
\hspace{15mm}
\begin{gathered}
-\Delta_p u  =  0 \quad \mbox{in }   D\backslash D_0^{*} \\
          u  =  0 \quad \mbox{on }  \partial D \\
       u     =  1 \quad \mbox{on }  \partial D_0^{*}
\end{gathered}
\end{equation}
Let  $J( D) := E( \tilde{v},\tilde{u}) $. It  is easy to see that
$J(D) \geq 0$, this implies   $\inf \{ J(w) ,
  w  \in  \mathcal{O}_{\epsilon}\}  >-\infty$. Let
$\alpha= \inf  \{ J(w) ,   w  \in  \mathcal{O}_{\epsilon}\}$.
Then there  exists  a  minimizing  sequence
$ (D_n)_{(n \in \mathbb{N})} \subset  \mathcal{O}_{\epsilon}$  such that $J(D_n)$
converges to $\alpha$. Since    the sequence is bounded, there
exists   a compact set  $F$  such that
$ D_0^{*} \subset  \subset  \bar{D_n}  \subset F  \subset  \subset
 D_1^{*}$. By the lemma  \ref{ler4},  there  exists  a
subsequence  $(D_{n_k})_{(n_k \in \mathbb{N})}$  and  $ D$    verifying
the $\epsilon$-cone  property   such that
$$
\chi_{{D}_{n_k}} \stackrel{ L^{1}} {\to}    \chi_{D}  \quad\text{and}
D_{n_k}   \stackrel{H}{\to}   D  .
$$
It is easy  to see the sequence $(v_n,u_n)$  is  bounded  in
$W^{1,p}(D_1^{*})$ see \cite{LS2,LS4}. Since $ W^{1,p}(D_1^{*})$ is
a reflexive space, there  exists
 a subsequence $(v_{n_k},u_{n_k})$  and  $(v^{*},u^{*})$     such that
$ v_{n_k}$    converges weakly  on $v^{*}$  in $ W^{1,p}(D_1^{*})$
and
$ u_{n_k}$    converges weakly  on $u^{*}$  in $ W^{1,p}(D_1^{*})$.
The  norm is lower semi continuous  for the  weak  topology   in
$W^{1,p}(D_1^{*})$, then we have
\begin{align*}
&\frac{1}{p}\int_{D_1^{*}\backslash D}\|\nabla v^{*} \|^{p}
+  \frac{1}{p}\int_{ D\backslash D_0^{*}}\|\nabla u^{*}\|^{p} \\
&\geq \liminf (\frac{1}{p} \int_{ D_1^{*}\backslash D_{n_k}}\|\nabla
v_{n_k} \|^{p} +  \frac{1}{p} \int_{ D_{n_k}\backslash
D_0^{*}}\|\nabla u_{n_k}\|^{p}).
\end{align*}
 From  the above  we get
$J(D) \geq \alpha$,  then $J(D) =\min  \{ J(w) ,
  w  \in  \mathcal{O}_{\epsilon}\}$.
\end{proof}

\begin{remark} \label{tmk7.1} \rm
On the  one  hand, see \cite{LS2} \cite{LS3}, it is  easy    to
verify that $v= v^{*}, u =u^{*}$  and  $ v^{*},u^{*}  $   satisfy
\[
\begin{gathered}
-\Delta_p v^{*}  =  0 \quad \mbox{in } {\mathcal D}'( D_1^{*}\backslash D) \\
          v^{*}  =  1 \quad \mbox{on } \partial D \\
       v^{*}   =  0 \quad \mbox{on } \partial D_1^{*}
\end{gathered}
\hspace{15mm}
\begin{gathered}
-\Delta_p u^{*}  =  0 \quad \mbox{in }   {\mathcal D}'(D\backslash D_0^{*}) \\
          u^{*}  =  0 \quad \mbox{on }  \partial D \\
       u^{*}  =  1 \quad \mbox{on }  \partial D_0^{*}
\end{gathered}
\]
respectively.
On the other hand, we have regularity  for $v, u$   as solutions to
(\ref{opt1}); see \cite{D,Le,T2}.
\end{remark}

\begin{remark} \label{rmk7.2} \rm
The remark 4.1 can be stated for the multilayer case.
The theorem 4.3 and the lemma 4.4 proved in \cite{LS3} are valid
too for the multilayer case.
\end{remark}

For the rest of this article,   we assume  that  $D$  is
$\mathcal{C}^{2}$-regular domain  in order  to use    the shape
derivatives.  We follow   the approach  of Sokolowski-Zolesio to
define  the shape  derivatives \cite{SZ} (see  also  \cite{JS}).

\begin{theorem}\label{theoc}
If  $D$     is a   solution  to the shape optimization problem
$ \min \{   J ( w),  w  \in   \mathcal{O}_{\epsilon}\}$, then
there  exists  a Lagrange  multiplier   function
$\lambda_D \in  \mathbb{R}$  such that
\begin{equation}
\|\nabla v\|^{p} - \|\nabla u\|^{p}  = \frac{p}{p- 1}
\lambda_{D}\quad\text{on}\quad   \partial D.
\end{equation}
\end{theorem}

\begin{proof}[Proof of the theorem  \ref{theoc}]
$$
J( D)  =  \frac{1}{p} \int_{ D_1^{*}\backslash D}\|\nabla v\|^{p}
+  \frac{1}{p} \int_{ D\backslash D_0^{*}}\|\nabla u\|^{p}, \quad
   1<p<\infty,
$$
 where  $v$ and $u$  are solutions  of
\begin{equation}\label{eqc}
\begin{gathered}
-\Delta_p v   =  0 \quad \mbox{in }   D_1^{*}\backslash D \\
          v  =  1 \quad \mbox{on } \partial D \\
       v  =  0 \quad \mbox{on }  \partial D_1^{*}
\end{gathered}
\hspace{15mm}
\begin{gathered}
-\Delta_p u  =  0 \quad \mbox{in }  D\backslash D_0^{*} \\
          u  =  0 \quad \mbox{on }  \partial D \\
       u  =  1 \quad \mbox{on }  \partial D_0^{*}
\end{gathered}
\end{equation}
A  standard   computation , see \cite {L}, shows  the Euleurian
derivative  of the functional  $J$   at the point  $D$  in the
direction   $V$  is
 $d J(D, V) =   A +B $, where
  \begin{gather*}
A =  \int_{D_1^{*}\backslash D} \|\nabla v\|^{p-2}\nabla v'\nabla v dx
    +\frac{1}{p}\int_{D_1^{*}\backslash D} \mathop{\rm div}
    (\|\nabla v\|^{p}) V(0)) dx\\
B =  \int_{D\backslash D_0^{*}} \|\nabla u\|^{p-2}\nabla u'\nabla u dx
    +\frac{1}{p}\int_{D\backslash D_0^{*}} \mathop{\rm div}(\|\nabla
    u\|^{p}) V(0)) dx
\end{gather*}
By the Green formula, we have
\begin{align*}
A &= -\int_{D_1^{*}\backslash D} \mathop{\rm div}(\|\nabla v\|^{p-2}\nabla v) v' dx
    +\frac{1}{p}\int_{\partial(D_1^{*}\backslash D)} \|\nabla v\|^{p-2}
      \frac{\partial v}{\partial \nu_1} v' ds \\
   &\quad +  \frac{1}{p}\int_{\partial(D_1^{*}\backslash D)} \|\nabla v\|^{p}V(0).
   \nu_1 ds.
\end{align*}
In $D_1^{*}\backslash D$, we have $\mathop{\rm div}(\|\nabla v\|^{p-2}\nabla v) = 0$,
then
$$
A = \frac{1}{p}\int_{\partial(D_1^{*}\backslash D)} \|\nabla v\|^{p-2}
  \frac{\partial v}{\partial \nu_1} v' ds
  +  \frac{1}{p}\int_{\partial(D_1^{*}\backslash D)} \|\nabla v\|^{p}V(0).\nu_1 ds.
$$
By the same  reasoning, we obtain
$$
B = \frac{1}{p}\int_{\partial(D\backslash D_0^{*})} \|\nabla u\|^{p-2}
\frac{\partial u}{\partial \nu_2} u' ds
+  \frac{1}{p}\int_{\partial(D\backslash D_0^{*})} \|\nabla u\|^{p}V(0).\nu_2 ds.
 $$
Let us take $ \nu_1 = -\nu_2$  where $\nu_2$ is   the exterior normal
unit  to $D$.
By the  computations,   see \cite{L},    we obtain
\begin{gather*}
u'  =  -\frac{\partial u}{\partial \nu_2}V(0).\nu_2 \quad\text{on }   \partial D,\\
v'  =  -\frac{\partial v}{\partial \nu_1}V(0).\nu_1  \quad\text{on }  \partial D.
\end{gather*}
This implies
 \begin{gather*}
A =  -\int_{\partial  D} \|\nabla v\|^{p}V(0).\nu_1 ds
   +\frac{1}{p}\int_{\partial D} \|\nabla v\|^{p} V(0)\nu_1 ds, \\
B =  -\int_{\partial D} \|\nabla u\|^{p}V(0).\nu_2 ds
+\frac{1}{p}\int_{\partial D} \|\nabla u\|^{p} V(0).\nu_2 dx\,.
\end{gather*}
Then  we have
$$
dJ(D,V) = \frac{1- p}{p}\int_{\partial D} (-\|\nabla
v\|^{p} +\|\nabla u\|^{p})V(0).\nu_2 ds.
$$
Let us take $ J_2( D) = \int_{ D} dx = V_0$, then
$$
dJ_2(D,V)= \int_{ D} \mathop{\rm div}(V(0))dx = \int_{\partial D} V(0).\nu_2\,ds.
$$
There  exists   a Lagrange multiplier $\lambda_D\in R$   such that
$dJ(D,V) = \lambda_D d J_2(D, V)$.
We  obtain
$$
\int_{\partial D}[ \frac{1- p}{p} (-\|\nabla v\|^{p}
+\|\nabla u\|^{p})       - \lambda_D)]V(0).\nu_2 ds = 0
 \quad\text{for all}\quad    V,
$$
then
$\|\nabla v\|^{p} - \|\nabla u\|^{p}  =  \frac{p}{p-1}\lambda_{D}$
 on  $\partial D$.
\end{proof}

\begin{remark} \label{rmk7.3} \rm
The consequence $(D, v, u)$ in theorems  (\ref{theop})  and  (\ref{theoc})
satisfies
\[
\begin{gathered}
-\Delta_p v  =  0 \quad \mbox{in }  D_1^{*}\backslash D \\
          v  =  1 \quad \mbox{on }  \partial D \\
       v     =  0 \quad \mbox{on }  \partial D_1^{*}
\end{gathered}
\hspace{15mm}
\begin{gathered}
-\Delta_p u  =  0 \quad \mbox{in }   D\backslash D_0^{*} \\
          u  =  0 \quad \mbox{on }  \partial D \\
       u  =  1 \quad \mbox{on }  \partial D_0^{*}
\end{gathered}
\]
and  satisfy  the  nonlinear  joining  condition
\[
\|\nabla v\|^{p} - \|\nabla u\|^{p}
=\frac{p}{p-1}\lambda_{D}\quad\text{on }  \partial D.
\]
\end{remark}

To conclude  this section, we   state   a monotonicity  result, in the
following sense.

\begin{theorem}\label {them7.3}
Let  $ D_0^{*}$  and   $ D_1^{*}$  be  $\mathcal{C}^{2}$-regular,
 compact sets  in  $\mathbb{R}^{N}$    and   starshaped with respect
to the origin   such that $ D_1^{*}$  strictly contains
$D_0^{*}$. Let  $D_1$ and $D_2$   be two different solutions   to
the shape optimization  problem
 $ \min \{   J ( w),  w  \in  \mathcal{O}_{\epsilon}\}$  starshaped
with respect to the   origin   such that  $D_1  \subset  D_2$ and
$\partial D_1  \cap  \partial D_2 \neq \ \emptyset$ then
$\lambda_{D_1}   \geq  \lambda_{D_2}$.
\end{theorem}

\begin{proof}
For any  $i \in \{1,2\}$, if   $D_i$  is   the solution   to the
shape optimization problem,    we have  $(v_i,u_i)$  satisfy the problem
\[
\begin{gathered}
-\Delta_p v _i  =  0 \quad \mbox{in }  D_1^{*}\backslash D_i \\
          v_i  =  1 \quad \mbox{on }  \partial D_i \\
       v_i  =  0 \quad\mbox{on }  \partial D_1^{*}
\end{gathered}
\hspace{15mm}
\begin{gathered}
-\Delta_p u_i  =  0 \quad \mbox{in }   D_i\backslash D_0^{*} \\
          u_i  =  0 \quad \mbox{on }  \partial D_i \\
       u_i  =  1 \quad \mbox{on }  \partial D_0^{*}
\end{gathered}
\]
and  the  nonlinear  joining  condition
\begin{eqnarray*}
\|\nabla v_i\|^{p} - \|\nabla u_i\|^{p}
=\frac{p}{p-1}\lambda_{D_i}\quad\text{on }  \partial D_i,
   \lambda_{D_i}\in R.
\end{eqnarray*}
Consider  the problem
\begin{equation}\label{e10}
\begin{gathered}
-\Delta_p v _3  =  0 \quad \mbox{in }   D_1^{*}\backslash D_2 \\
          v_3  =  v_1\quad \mbox{on }  \partial D_2 \\
       v_3  =  0 \quad \mbox{on }  \partial D_1^{*}
\end{gathered}
\end{equation}
It is easy to see  that $v_3 = v_1$  is a solution   to
(\ref{e10}). We get  $ 0 \leq  v_2  \leq  1$  and
  $0  \leq  v_1  \leq1$.
  On   $\partial (D_1^{*}\backslash D_2)$, we have  $v_2  \geq  v_1$.
  By the   comparison  principle  \cite{T1}, we obtain  $v_2  \geq  v_1$
in    $D_1^{*}\backslash D_2$.

Let $ x_0\in \partial D_1\cap \partial D_2 $  and $\nu$ be the exterior
unit normal in $x_0 $, then we get
\[
\frac{ v_2(x_0 +\nu h) - v_2(x_0)}{h}  \geq  \frac{ v_1(x_0
 +\nu h) -  v_1(x_0)}{h},
\]
By passing to the limit,
\[
\lim_{h\to 0}\frac{v_2(x_0 +\nu h) - v_2(x_0)}{h}
\geq  \lim_{h\to 0}\frac{ v_1(x_0 +\nu h) -
v_1(x_0)}{h},
\]
which  implies
\[
 \frac{\partial v_2}{\partial \nu}(x_0)  \geq  \frac{\partial v_1}{\partial  \nu}(x_0).
 \]
It suffices to remark that   $ \frac{\partial v_i}{\partial \nu}(x_0)< 0$
    ($i=1,2$) to conclude that
\begin{equation}\label{eqco1}
 \|\nabla v_1 (x_0)\|^{p}   \geq   \|\nabla v_2 (x_0)\|^{p}.
\end{equation}
Consider  the  problem
\begin{equation}\label{e11}
\begin{gathered}
-\Delta_p u_3  =  0 \quad \mbox{in }   D_1\backslash D_0^{*} \\
          u_3  =  u_2 \quad \mbox{on }  \partial D_1 \\
       u_3  =  1 \quad \mbox{on }  \partial D_0^{*}
\end{gathered}
\end{equation}
It is easy to see  that $  u_3 =  u_2$  is a solution   to
(\ref{e11} ). We get  $ 0 \leq  u_1  \leq  1$  and   $0  \leq  u_2  \leq1$.
  On   $\partial (D_1\backslash D_0^{*})$, we have  $u_2  \geq  u_1$.
  By the   comparison  principle  \cite{T1}, we obtain  $u_2  \geq  u_1$
in    $D_1\backslash D_0^{*}$.

Let $ x_0\in \partial D_1\cap \partial D_2$,  then
\[
\frac{ u_2(x_0 -\nu h) - u_2(x_0)}{h}  \geq  \frac{ u_1(x_0
 -\nu h) -  u_1(x_0)}{h}.
\]
By passing to the limit,
\[
\lim_{h\to 0}\frac{u_2(x_0 -\nu h) - u_2(x_0)}{h}
\geq  \lim_{h\to 0}\frac{ u_1(x_0 -\nu h) -
u_1(x_0)}{h}
\]
which implies
\[
 -\frac{\partial u_2}{\partial \nu}(x_0)  \geq  -\frac{\partial u_1}{\partial
 \nu}(x_0).
\]
 That is, $ \|\nabla  u_2(x_0)\|   \geq   \|\nabla u_1 (x_0)\| $,
Then we have   $ \|\nabla  u_2 (x_0)\|^{p}    \geq   \|\nabla
u_1(x_0)\|^{p}$. This implies
\begin{equation}\label{eqco2}
-\|\nabla  u_1 (x_0)\|^{p}  \geq  -\|\nabla u_2(x_0)\|^{p}.
\end{equation}
By  combining  (\ref{eqco1})  and (\ref{eqco2}),
\[
\|\nabla  v_1(x_0)\|^{p}  - \|\nabla  u_1(x_0)\|^{p}   \geq
\|\nabla v_2\|^{p} - \|\nabla u_2\|^{p}.
\]
Then $\lambda_{D_1}  \geq   \lambda_{D_2}$.
\end{proof}

\section{Proof of the  main result of Part II}

In this section, we  use  the preceding  theorems  to prove
the main result.

\begin{proof}[Proof of the theorem \ref{theomain}]
Let   $R_0 = sup\{   R>0, B(O, R)  \subset  D_1^{*}\}$. Let $ r_0>0,  r>0$
such that  $ B(O, r_0)   \subset  D_0^{*}  \subset  B(O, r)$.
First, we look for  $v_0$  solution of  the problem
\begin{equation}\label{eqmain}
\begin{gathered}
-\Delta_p v_0  =  0 \quad \mbox{in }   B_{R_0}\backslash B_r \\
          v_0  =  1 \quad \mbox{on } \partial B_r \\
       v_0  =  0  \quad \mbox{on }  \partial B_{R_0}
\end{gathered}
\end{equation}
and  second    $u_0$   solution of the problem
\begin{equation}\label{eqmain1}
\begin{gathered}
-\Delta_p u_0  =0 \quad\text{in }  B_r\backslash B_{r_0} \\
          u_0  =0 \quad\text{on } \partial B_r \\
       u_0  =1 \quad\text{on } \partial B_{r_0}
\end{gathered}
\end{equation}
The  problem  (\ref{eqmain})  admits a solution  $v_0$  which  is
explicitly  determined by
\[
 v_0(x) = \begin{cases}
 \dfrac{ \ln\|x\| - \ln R_0}{\ln r - \ln R_0}&\mbox{if } p = N \\[5pt]
  \dfrac{ -\|x\|^{\frac{p -N}{p -1}}
 + R_0^{\frac{p -N}{p -1}}} { R_0^{\frac{p -N}{p -1}}
 - r^{\frac{p -N}{p -1}}}&\mbox{if }p\neq N,
  \end{cases}
\]
and
\[
\|\nabla v_0(x)\| = \begin{cases}
 \dfrac{1}{ \|x\|( \ln R_0 - \ln r)}&\mbox{if } p = N \\[5pt]
 \dfrac{ |\frac{p -N}{p -1}| \|x\|^{\frac{-N+1}{p -1}}}{| r^{\frac{p -N}{p -1}}
 - R_0^{\frac{p -N}{p -1}}|}&\mbox{if }p \neq N.
  \end{cases}
\]
Also the   problem  (\ref{eqmain1})  admits a solution  $u_0$
which  is explicitly  determined by
\[
 u_0(x) = \begin{cases}
 \dfrac{ \ln\|x\| - \ln r}{\ln r_0 - \ln r}&\mbox{if } p = N \\[5pt]
  \dfrac{ -\|x\|^{\frac{p -N}{p -1}}
 - r^{\frac{p -N}{p -1}}} { r_0^{\frac{p -N}{p -1}}
 - r^{\frac{p -N}{p -1}}}&\mbox{if }p\neq N,
\end{cases}
\]
and
\[
\|\nabla u_0(x)\| = \begin{cases}
 \dfrac{-1}{ \|x\|( \ln r_0 - \ln r)}&\mbox{if } p = N \\[5pt]
 \dfrac{ |\frac{p -N}{p -1}| \|x\|^{\frac{-N -p +2}{p -1}}}{| r_0^{\frac{p -N}{p -1}}
 - r^{\frac{p -N}{p -1}}|}&\mbox{if }p \neq N.
  \end{cases}
\]
On $\partial B_r$, let  us take   $h(r) = \| \nabla v_0\|^{p} -\| \nabla
u_0\|^{p}$.

Now  consider  the   problem
\begin{equation}\label{eqm1}
\begin{gathered}
-\Delta_p v  =0 \quad\text{in }  D_1^{*}\backslash B_r \\
          v  =1 \quad\text{on } \partial B_r \\
       v =0 \quad\text{on } \partial  D_1^{*}
\end{gathered}
\end{equation}
Problem (\ref{eqm1})  admits  a solution  denoted by  $v_r$.
This solution  is obtained  by minimizing   the functional $J_1$
on the Sobolev space
$$
\mathcal{V}_1 = \{  v   \in W^{1,p}_0(D_1^{*}\backslash B_r),
v=   1 \text{ on } \partial B_r\} \quad\text{and}\quad
 J_1(v) = \frac{1}{p}\int_{D_1^{*}\backslash B_r}\|\nabla v\|^{p} dx.
$$
\begin{equation}\label{eqm2}
\begin{gathered}
-\Delta_p u  =0 \quad\text{in } B_r \backslash D_0^{*} \\
          u  =1 \quad\text{on } \partial  D_0^{*} \\
       u =0 \quad\text{on } \partial   B_r
\end{gathered}
\end{equation}
Then problem (\ref{eqm2})  admits  a solution  denoted by  $u_r$.
This solution  is obtained  by minimizing   the functional $J_2$
on the Sobolev space
$$
 \mathcal{V}_2 = \{  u   \in W^{1,p}_0(B_r\backslash D_0^{*}),
u=   1   \text{ on }  \partial D_0^{*}\} \quad\text{and}\quad
  J_2(u) = \frac{1}{p}\int_{B_r\backslash D_0^{*}}\|\nabla u\|^{p} dx.
$$
Consider  the problem
\begin{equation}\label{eqm3}
\begin{gathered}
-\Delta_p v  =0 \quad\text{in }  B_{R_0}\backslash B_r \\
          v  =1 \quad\text{on } \partial B_r \\
       v  =  v_r \quad  \mbox{on }  \partial  B_{R_0}
\end{gathered}
\hspace{15mm}
\begin{gathered}
-\Delta_p u  =0 \quad\text{in } B_r \backslash D_0^{*} \\
          u  =  u_0 \quad \mbox{on }  \partial  D_0^{*} \\
       u =0 \quad\text{on } \partial   B_r.
\end{gathered}
\end{equation}
It is easy to see  that $ v = v_r $    and    $u =  u_0$  are
respectively solutions   to the problem (\ref{eqm3}). We  have
$0 \leq  v_0  \leq 1$   and  $0  \leq v_r \leq 1$. We obtain on
$\partial (B_{R_0}\backslash B_r)$,  $v_r \geq v_0$. By the comparison
principle \cite{T1},    we have
  $v_r \geq v_0$   in  $B_{R_0}\backslash B_r$. Finally,
we have $ \|\nabla v_r\|  \leq  \|\nabla v_0\|$  then
\begin{equation}\label{eqm31}
 \|\nabla v_0\|^{p} \geq  \|\nabla v_r\|^{p}\quad\text{on }   \partial B_r.
\end{equation}
Also, we have $ 0 \leq  u_0  \leq 1$   and
 $0  \leq u_r \leq 1$. We obtain  on $ \partial (B_r \backslash
D_0^{*})$, $u_r \geq u_0$. By the  comparison principle \cite{T1},
we have   $u_r \geq u_0$   in   $B_r \backslash D_0^{*}$.
 We get
$ \|\nabla u_r\| \geq \|\nabla u_0\|$  then
\begin{equation}\label{eqm32}
-\|\nabla u_0\|^{p} \geq -\|\nabla u_r\|^{p}\quad\text{on }   \partial
B_r.
\end{equation}
By combining  (\ref{eqm31}) and (\ref{eqm32}), we obtain
\[
\|\nabla v_0\|^{p} -\|\nabla u_0\|^{p} \geq \|\nabla v_r\|^{p} -\|\nabla
u_r\|^{p}\quad\text{on }   \partial B_r.
\]

\subsection*{Case $p = N$} Note that
$$
h(r) =  \frac{1}{r^{p}}\left( \frac{1}{ (\ln R_0 - \ln r)^{p}}
-\frac{1}{ (-\ln r_0 + \ln r)^{p}} \right) ,
\quad\text{for all }   r  \in  ]r_0, R_0[.
$$
Let  $\delta >0$  be a   fixed   and  sufficiently  small number.
To initialize, we choose  $r_1 \in ]r_0, R_0[$,  such that
$|h(r_1) - \lambda|>\delta,   \lambda \in \mathbb{R}$. By varying $r$ in
the increasing    sense,  we will achieve a step denoted $n$ such
that   $r_n \in ]r_0, R_0[$  and  $|h(r_n) - \lambda| < \delta$.

Consider $\mathcal{O}_n$  the class  of admissible    domains
defined   as follows
$$
\mathcal{O}_n = \{  w \in \mathcal{O}_{\epsilon},   B_{r_n} \subset w,
\partial B_{r_n}\cap \partial w \neq \emptyset \quad\text{and}\quad
    \mathop{\rm vol}(w) = V_1\},
$$
where $ V_1$  denotes  a fixed  positive  constant.
We look for   $D  \in \mathcal{O}_n$  such that
\begin{equation}\label{eqm4}
\begin{gathered}
-\Delta_p v   =0 \quad\text{in }  D_1^{*}\backslash D \\
          v =1 \quad\text{on } \partial D \\
       v  =0 \quad\text{on } \partial D_1^{*}
\end{gathered}
\hspace{15mm}
\begin{gathered}
-\Delta_p u  =0 \quad\text{in }  D\backslash D_0^{*} \\
          u =0 \quad\text{on } \partial D \\
       u  =1 \quad\text{on } \partial D_0^{*}
\end{gathered}
\end{equation}
and satisfies  the  nonlinear  joining  condition
\begin{equation}\label{eqj}
\|\nabla v\|^{p} - \|\nabla u\|^{p}
=\frac{p}{p-1}\lambda_{D}\quad\text{on }  \partial D.
\end{equation}

Applying  the theorem \ref{theop},  the  shape optimization
problem $\min\{ J(w),  w  \in  \mathcal{O}_n\}$  admits  a
solution $D$ and by the theorem \ref{theoc}, $D$  satisfies the
joining condition (\ref{eqj}). Since  $ D  \in \mathcal{O}_n$, we
have  $B_{r_n} \subset D$  and $\partial B_{r_n}\cap \partial D \neq
\emptyset$ and  $v_{r_n}$ respectively $u_{r_n}$ satisfy
\begin{equation}\label{eqm5}
\begin{gathered}
-\Delta_p v_{r_n}   =0 \quad\text{in }  D_1^{*}\backslash B_{r_n} \\
          v_{r_n} =1 \quad\text{on } \partial B_{r_n} \\
       v_{r_n}  =0 \quad\text{on } \partial D_1^{*}
\end{gathered}
\hspace{15mm}
\begin{gathered}
-\Delta_p u_{r_n}  =0 \quad\text{in }  B_{r_n}\backslash D_0^{*} \\
          u_{r_n} =0 \quad\text{on } \partial B_{r_n} \\
       u_{r_n}  =1 \quad\text{on } \partial D_0^{*}.
\end{gathered}
\end{equation}
Consider  the  problem
\begin{equation}\label{eqm6}
\begin{gathered}
-\Delta_p z   =0 \quad\text{in }  D_1^{*}\backslash D \\
          z  =  v_{r_n} \quad \mbox{on }  \partial D \\
       z  =0 \quad\text{on } \partial D_1^{*}
\end{gathered}
\hspace{15mm}
\begin{gathered}
-\Delta_p g  =0 \quad\text{in }  D\backslash D_0^{*} \\
          g =0 \quad\text{on } \partial D \\
       g  =  u_{r_n} \quad \mbox{on }  \partial D_0^{*}.
\end{gathered}
\end{equation}
It is  easy to see  that $ z = v_{r_n}$   and   $g=  u_{r_n}$
are respectively solutions to problem (\ref{eqm6}).
We get $0  \leq  v_{r_n} \leq  1$ and  $0  \leq  v \leq  1$.
We  have on $\partial (D_1^{*}\backslash D )$,
$v_{r_n}  \leq  v$. By the  comparison principle
\cite{T1},  we obtain  $v_{r_n}  \leq  v$   in
$(D_1^{*}\backslash D )$. Since
$\partial B_{r_n}\cap \partial D \neq \emptyset$, let's take
 $x_1\in \partial B_{r_n}\cap \partial D $, we have by passing to the limit
$$
\lim_{h \to 0}\frac{v_{r_n}(x_1 +\nu h) - v_{r_n}(x_1)}{h}
\leq  \lim_{h \to 0}\frac{v(x_1 +\nu h) - v(x_1)}{h},
$$
 this is  equivalent to (where $\nu$ is the exterior normal to  $D$)
\begin{equation}\label{eqm61}
\|\nabla v_{r_n}(x_1)\|^{p} \geq  \|\nabla v(x_1)\|^{p}
\end{equation}
We get $0 \leq  u \leq  1\quad\text{and}   0  \leq  u_{r_n} \leq  1$.
We  have on $\partial (D\backslash D_0^{*} )$,
$u_{r_n}  \leq  u$. By the  comparison principle
\cite{T1},  we obtain  $u_{r_n}  \leq  u$   in
$ (D\backslash D_0^{*} )$. Since $\partial B_{r_n}\cap \partial D \neq \emptyset$,
 let us take  $x_1\in \partial  B_{r_n}\cap \partial D $, we have  by passing
to the limit
$$
\lim_{h \to 0}\frac{u_{r_n}(x_1 -\nu h) - u_{r_n}(x_1)}{h}
 \leq  \lim_{h \to 0}\frac{u(x_1 -\nu h) - u(x_1)}{h},
$$
that is
\begin{equation}\label{eqm62}
-\|\nabla u_{r_n}(x_1)\|^{p} \geq  -\|\nabla u(x_1)\|^{p}
\end{equation}
By combining  the relations (\ref{eqm61}) and (\ref{eqm62}), we
obtain
\[
\|\nabla v_{r_n}(x_1)\|^{p}-\|\nabla u_{r_n}(x_1)\|^{p} \geq  \|\nabla
v(x_1)\|^{p}-\|\nabla u(x_1)\|^{p}.
\]
Let us take  $ D = D_1$ as  the  first iteration. We iterate   by
looking $D_2  \in  \mathcal{O}_n^{2}$  such that
\begin{equation}\label{eqm7}
\begin{gathered}
-\Delta_p v_2   =0 \quad\text{in }  D_1^{*}\backslash D_2 \\
          v_2 =1 \quad\text{on } \partial D_2 \\
       v_2  =0 \quad\text{on } \partial D_1^{*}
\end{gathered}
\hspace{15mm}
\begin{gathered}
-\Delta_p u_2  =0 \quad\text{in }  D_2\backslash D_0^{*} \\
          u_2 =0 \quad\text{on } \partial D_2 \\
       u_2  =1 \quad\text{on } \partial D_0^{*},
\end{gathered}
\end{equation}
and satisfies  the  nonlinear  joining  condition
\begin{equation}\label{eqj1}
\|\nabla v_2\|^{p} - \|\nabla u_2\|^{p}
=\frac{p}{p-1}\lambda_{D_2}\quad\text{on }  \partial D_2.
\end{equation}
Also
$$
\mathcal{O}^{2}_n = \{  w \in \mathcal{O}_{\epsilon},   D_1 \subset w,
 \partial B_{r_n}\cap \partial w \neq \emptyset \quad\text{and }
 \mathop{\rm vol}(w) = V_2\},
$$
where $ V_2$  is a strictly   positive  constant  and  $ V_1 < V_2$.
By the same   reasoning as above,  we obtain
\[
\|\nabla v_{r_n}(x_2)\|^{p}-\|\nabla u_{r_n}(x_2)\|^{p} \geq
\|\nabla v(x_2)\|^{p}-\|\nabla u(x_2)\|^{p}   \quad\text{on }   \partial B_{r_n}.
\]
We  can   continue  the process until  a step  denoted  $k$,
which  we will be determined, and   we  have
\[
\|\nabla v_{r_n}(x_k)\|^{p}-\|\nabla u_{r_n}(x_k)\|^{p}  <  \|\nabla
v(x_k)\|^{p}-\|\nabla u(x_k)\|^{p} \quad\text{and}   x_k\in  \partial D_k
\cap \partial B_{r_n}.
\]
Finally, we constructed  an increasing  sequence of domain
solutions
$$
D_1  \subset  D_2  \subset \dots \subset  D_{k-1} \subset D_k.
$$
By the  monotonicity  result, in theorem \ref{them7.3},  we  have
$$
\lambda_{D_1}  \geq  \lambda_{D_2}  \geq \dots \geq  \lambda_{D_{k-1}}
 \geq  \lambda_{D_k}.
$$
Since   $\|\nabla v_{r_n}(x_k)\|^{p}-\|\nabla u_{r_n}(x_k)\|^{p}
 \leq   \|\nabla v_0\|^{p}-\|\nabla u_0\|^{p}$ on $\partial B_{r_n}$,
$k$ is chosen  as follows  in each  point
$ s_0  \in \partial B_{r_n}$,
$$
\lambda_{D_k}  \leq  h(s_0)  \leq  \lambda_{D_{k-1}}.
$$
Then  we  obtain   the  inequality
\begin{equation}\label{eqf}
\lambda_{D_k}
 -\lambda  \leq  h(s_0)-\lambda  \leq  \lambda_{D_{k-1}}-\lambda.
\end{equation}
The  sequence  $(\lambda_{D_j})_{(0 \leq j \leq k)}$  is
decreasing  and  underestimated   because we  cannot  indefinitely
generate   a sequence   domains if not we will leave   $D_1^{*}$.
We  have  $\lambda_{D_k}  \geq \lambda_{D_*'}$  where $D_*'$
is  the greatest  domain  contained in $D_1^{*},    \partial
D_*'\cap \partial B_{r_n} \neq \emptyset.  D_*'$   is solution
to the shape  optimization problem  $\min\{ J(w),
 w  \in  \mathcal{O}_n\}$  and   for all  $k$, we have
$D_k   \subset D_*'$.

The  sequence $(\lambda_{D_j})_{(0 \leq j \leq k)}$ converges to
$l$. By passing to the limit   in (\ref{eqf}), we obtain
$l = \lambda$ and there  exists  $D$   solution  to
 problem  (\ref{q0})-(\ref{qq0}). The   sequence
$(D_j)_{(0 \leq j \leq k)}$   gives  a good approximation  to  $D$.

\subsection*{Case $p  \neq N$}
Here   the reasoning   is  identical  to the case  $ p= N$. We
note that
$$
 h(r) =  (|\frac{ p-N}{N-1}|)^{p}(r ^{-|\frac{ N-1}{p-1}|})^{p}
\Big( \frac{1}{ |  r^{\frac{ p-N}{p-1}} - R_0^{\frac{ p-N}{p-1}}|^{p}} -\frac{1}
{ (r|  r_0^{\frac{ p-N}{p-1}} - r^{\frac{ p-N}{p-1}}|)^{p}}
\Big),
$$
for all   $r  \in  ]r_0, R_0[$.
\end{proof}

\section{The multi-layer case}

Let  $ D_0$  and   $ D_{k+1}$  be  $\mathcal{C}^{2}$-regular ,
compact sets  in  $\mathbb{R}^{N}$    and   starshaped with respect to the
origin
  such that  $ D_{k +1}$  strictly  contains $ D_0$. One supposes that there
is $R_0$ such that   $D_0 \subset B(0, R_0) \subset D_{k+1}$ where
$R_0 = \sup \{ R>0: B(0,R) \subset   D_{k+1}\}$.
We  find  a sequence of  domains, $\mathcal{C}^{2}$-regular, starshaped
    with  respect to the origin  and solution  to the shape
optimization problem $\min\{ J(w),  w  \in  \mathcal{O}_{\epsilon}\}$,
$ D_0  \subset  D_1  \subset D_2 \subset \dots \subset D_k  \subset D_{k +1}$
such that   $(D_i, v_i,u_i)$ is  solution  of
\begin{equation}\label{q1mu}
\begin{gathered}
-\Delta_p v_i   =0 \quad\text{in }  D_{i+1}\backslash D_i \\
          v_i  =1 \quad\text{on } \partial D_i \\
       v_i  =0 \quad\text{on } \partial D_{i+1}
\end{gathered}
\hspace{15mm}
\begin{gathered}
-\Delta_p u_i  =0 \quad\text{in }  D_i\backslash D_{i-1} \\
          u_i  =0 \quad\text{on } \partial D_i \\
       u_i  =1 \quad\text{on } \partial D_{i-1}
\end{gathered}
\end{equation}
and satisfy  the non linear joining condition
\begin{equation}\label{qq1mu}
\|\nabla v_i\|^{p} - \|\nabla u_i\|^{p}  = \frac{p}{p-1}
\lambda_{i}\quad\text{on }  \partial {D_i},     \lambda_i \in \mathbb{R},
    1  \leq  i  \leq   k.
\end{equation}

\begin{theorem}\label{theomul}
Let  $ D_0$  and   $ D_{k+1}$  be  $\mathcal{C}^{2}$-regular ,
compact sets  in  $\mathbb{R}^{N}$    and   starshaped with respect to the
origin   such that  $ D_{k+1}$  strictly contains $ D_0$.
  Then  there  exists  a  sequence domains  $(D_i)_{(1  \leq  i \leq  k)}$,
 $\mathcal{C}^{2}$-regular  domain   solution   to  the shape
 optimization problem
  $ \min \{   J ( w),  w  \in  \mathcal{O}_{\epsilon}\}$  such that
$D_0 \subset D_1 \subset D_2 \dots  \subset D_k \subset D_{k+1}$
  solution  of the    multi-layer  free  boundary
problem   (\ref{q1mu})-(\ref{qq1mu}).
\end{theorem}

To prove this theorem, we use the method presented in the proof of
the  two layer case. In fact we consider at first the domains
$D_0$ and $D_{k+1}$. And according to the two layer case there is
$D_1$ ($D_0 \subset D_1 \subset D_{k+1}$) which is solution to the
problem. And sequentially, we seek $D_i$ ($D_{i-1}\subset D_i
\subset D_{k+1}, i=2,\cdots k$).It is always possible to invoke
the two layer case in order to solve these types of problems.

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