\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 171, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/171\hfil Optimal ground state energy]
{Optimal ground state energy of two-phase conductors}

\author[A. Mohammadi, M. Yousefnezhad \hfil EJDE-2014/171\hfilneg]
{Abbasali Mohammadi, Mohsen Yousefnezhad }  % in alphabetical order

\address{Abbasali Mohammadi (corresponding author)\\
 Department of Mathematics, College of Sciences,
 Yasouj University, Yasouj 75918-74934, Iran}
\email{mohammadi@yu.ac.ir}

\address{Mohsen Yousefnezhad \newline
 Department of Mathematical Sciences,
 Sharif University of Technology, Tehran, Iran. \newline
School of Mathematics, Institute for Research in Fundamental Sciences (IPM),
 Tehran, 19395-5746, Iran}
\email{yousefnezhad@mehr.sharif.ir}

\thanks{Submitted May 5, 2014. Published August 11, 2014.}
\subjclass[2000]{49Q10, 35Q93, 35P15, 33C10}
\keywords{Eigenvalue optimization; two-phase conductors;
\hfill\break\indent rearrangements; Bessel function}

\begin{abstract}
 We consider the problem of distributing two conducting materials in a
 ball with fixed proportion in order to minimize the first eigenvalue
 of a Dirichlet operator. It was conjectured that the optimal
 distribution consists of putting the material with the highest conductivity
 in a ball around the center. In this paper, we show that the conjecture
 is false for all dimensions greater than or equal to two.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}\label{intro}

 Let $\Omega$ be a bounded domain in $\mathbb{R}^n$
with a smooth boundary which is to be called the design region and
consider two conducting materials with conductivities $0<\alpha< \beta$. These materials are distributed in $\Omega$ such that the volume of the region $D$
occupied by the material with conductivity $\beta$ is a fixed number $A$ with $0<A<|\Omega|$.
 Consider the two-phase eigenvalue problem
\begin{equation} \label{mainpde}
\begin{gathered}
-\operatorname{div} \big( (\beta\chi_D+\alpha\chi_{D^c}) \nabla u \big)=\lambda u \quad \text{in }\Omega,\\
 u=0 \quad \text{on }\partial \Omega,
\end{gathered}
\end{equation}
where $\beta\chi_D+\alpha\chi_{D^c}$ is the conductivity,
$\lambda$ is the ground state energy or the
smallest positive eigenvalue and $u$ is the corresponding eigenfunction.

 We use the notation $\lambda(D)$ to show the dependence of the eigenvalue on $D$, the
 region with the highest conductivity.
 To determine the profile of this system, which gives the minimum principal
eigenvalue, we should verify the following optimization problem
\begin{equation}\label{mainopt}
\inf_{D\subset \Omega,\,|D|= A} \lambda(D),
\end{equation}
where $\lambda$ has the variational formulation
\begin{equation}\label{lambdavar}
\lambda(D)=\min_{u\in H^1_0(\Omega),\, \|u\|_{L^2(\Omega)}=1}
\int_{\Omega} ( \beta\chi_D+\alpha\chi_{D^c}) |\nabla u|^2 dx.
\end{equation}

 In general, this problem has no solution in any class of usual domains.
Cox and Lipton \cite{coxlipton} gave conditions for an optimal
microstructural design. However, when $\Omega$ is a ball, the symmetry of
the domain implies that there exists a radially symmetric minimizer.
 Alvino et al \cite{Alvino} obtained this result thanks to a comparison
result for Hamilton-Jacobi equations. Conca et al \cite{Conca1}
revived the interest in this problem by giving a new simpler proof
of the existence result only using rearrangement techniques.

 In eigenvalue optimization for elliptic partial differential equations,
one of challenging mathematical problems after the problem of existence
is an exact formula of the optimizer or optimal shape design.
Most papers in this field answered this question just in case $\Omega$
is a ball \cite{chanillo, emamipro, emamielec-1, abbasali2,abbasali}.
This class of problems is difficult to solve due to the lack of the topological
information of the optimal shape.
 For one-dimensional case, Krein \cite{Kerien} showed
 that the unique minimizer of \eqref{mainopt} is obtained by putting the
material with the highest conductivity in an interval in the middle of
the domain. Surprisingly, the exact distribution of the two materials
which solves optimization problem \eqref{mainopt} is still not known
for higher dimensions.

 Let $\Omega=\mathcal{B}(0,\mathcal{R})$ be a ball centered at the origin
with radius $\mathcal{R}$ , the solution of the one-dimensional problem suggests
for higher dimensions that $\mathcal{B}(0,\mathcal{R}^*)$ is a natural
candidate to be the optimal domain.
 This conjecture has been supported by numerical evidence in \cite{Conca2}
using the shape derivative analysis of the first eigenvalue for the two-phase
conduction problem. In addition, it has been shown in \cite{Ketab} employing
the second order shape derivative calculus that $D=\mathcal{B}(0,\mathcal{R}^*)$
is a local strict minimum for the optimization problem \eqref{mainopt} when
$A$ is small enough. In spite of the above evidence, it has been established
in \cite{Conca3} that the conjecture is not
 true in two- or three- dimensional spaces when $\alpha$ and
 $\beta$ are close to each other (low contrast regime) and $A$ is
 sufficiently large. The theoretical base for the result is an asymptotic
 expansion of the eigenvalue with respect to $\beta- \alpha$ as
$\beta\to \alpha$, which allows one to approximate the optimization problem by
 a simple minimization problem.

 In this article, we investigate the conjecture for all dimensions $n\geq 2$.
 We prove that the conjecture is false not only for two- or three- dimensional
spaces, but also for all dimensions $n\geq 2$. We provide a different proof
 of the main result in \cite{Conca3} and we establish it in a vastly simpler way.

\section{Preliminaries}\label{prel}

 To establish the main theorem, we need some preparation. Our proof is based
upon the properties of Bessel functions. In this
section, we state some results from the theory of Bessel functions.
The reader can refer to \cite{Bowman, Watson} for further
information about Bessel functions.

 Consider the standard form of Bessel equation,
 \begin{equation}\label{bessel eq}
 x^2y'' +xy'+(x^2-\nu^2)y=0,
 \end{equation}
 where $\nu$ is a nonnegative real number. The regular solution of \eqref{bessel eq},
 called the Bessel function of the first kind of order $\nu$, is given by
 \begin{equation}\label{besselseri}
J_\nu(x)=\sum_{k=0}^\infty \frac{(-1)^k x^{2k+\nu}}{2^{2k+\nu} \Gamma (\nu+k+1)},
 \end{equation}
where $\Gamma$ is the gamma function. We shall use following recurrence
relations between Bessel functions
 \begin{gather}\label{besselrec1}
J_{\nu-1}(x)+J_{\nu+1}(x)= \frac{2\nu}{x} J_\nu(x), \\
\label{besselrec2}
(x^{-\nu}J_{\nu}(x))'=-x^{-\nu}J_{\nu+1}(x).
 \end{gather}

 Let $j_{\nu,m}$ be the $m$th positive zeros of the function $J_{\nu}(x)$,
 then it is well known that the zeros of $J_{\nu}(x)$ are simple with possible
 exception of $x=0$. In addition, we have the following lemma related to
the roots of $J_{\nu}(x)$, \cite{Bowman, Watson}.

 \begin{lemma}\label{besselroot}
 When $\nu\geq 0$, the positive roots of $J_{\nu}(x)$ and $J_{\nu+1}(x)$
interlace according to the inequalities
\[
j_{\nu,m}< j_{{\nu+1},m}<j_{\nu,{m+1}}.
\]
 \end{lemma}

We will need the following technical assertion later.

\begin{lemma}\label{goodrel}
 If $\nu_1,\nu_2\geq0$, then
 \[
(\nu_2^2-\nu_1^2)\int_0^\tau \frac{J_{\nu_2}(s)J_{\nu_1}(s)}{s}\;ds
= \tau (J'_{\nu_2}(\tau)J_{\nu_1}(\tau)-J_{\nu_2}(\tau)J'_{\nu_1}(\tau)).
 \]
 \end{lemma}

\begin{proof}
Functions $J_{\nu_2}$ and $J_{\nu_1}$ are solutions of Bessel equations
\begin{gather*}
 x^2J_{\nu_2}'' +xJ_{\nu_2}'+(x^2-\nu_2^2)J_{\nu_2}=0, \\
 x^2J_{\nu_1}'' +xJ_{\nu_1}'+(x^2-\nu_1^2)J_{\nu_1}=0.
\end{gather*}
Multiplying the first equation by $J_{\nu_1}$ and the second one by
$J_{\nu_2}$, we have
 \begin{gather*}
\frac{\nu_2^2}{x} J_{\nu_2}J_{\nu_1}=xJ_{\nu_2}''J_{\nu_1}+ J_{\nu_2} '
J_{\nu_1}+x J_{\nu_2}J_{\nu_1}, \\
\frac{\nu_1^2}{x}J_{\nu_2}J_{\nu_1}=xJ_{\nu_1}''J_{\nu_2}+ J_{\nu_1} '
J_{\nu_2}+x J_{\nu_2}J_{\nu_1}.
 \end{gather*}
Subtracting the second equality from the first one,
 \[
 [x(J_{\nu_2} ' J_{\nu_1}-J_{\nu_1} ' J_{\nu_2})]'
= \frac{(\nu_2^2-\nu_1^2)}{x}J_{\nu_2}J_{\nu_1}.
 \]
 Integrating this equation from $0$ to $\tau$, leads to the desired assertion.
\end{proof}

This section closes with some results from the rearrangement theory
 related to our optimization problems. The reader can refer
to \cite{Alvino,bu89} for further information about the theory of rearrangements.

\begin{definition}\label{readef} \rm
Two Lebesgue measurable functions $\rho: \Omega \to \mathbb{R}$,
$\rho_0:\Omega \to \mathbb{R}$, are said to be rearrangements of each other if
 \begin{equation}\label{rea}
|\{x\in \Omega : \rho(x)\geq \tau\}|=|\{x\in \Omega : \rho_0(x)\geq \tau\}|
\quad\forall \tau \in \mathbb{R}.
\end{equation}
\end{definition}

The notation $\rho\sim \rho_0$ means that $\rho$ and $\rho_0$ are rearrangements
of each other. Consider $\rho_0:\Omega \to \mathbb{R}$, the class of rearrangements
generated by $\rho_{0}$, denoted $\mathcal{P}$, is defined as follows
 \[
\mathcal{P}=\{\rho:\rho\sim \rho_{0}\}.
\]

 Let $\rho_{0}= \beta\chi_{D_0}+\alpha\chi_{D_0^c}$ where $ D_0\subset \Omega$ and
$|D_0|=A$. For the sake of completeness, we include following
 technical assertion.

\begin{lemma}\label{chirho}
 A function $\rho$ belongs to the rearrangement class $\mathcal{P}$
if and only if $\rho= \beta\chi_{D}+\alpha\chi_{D^c}$ such that
$ D\subset \Omega$ and $|D|=A$.
 \end{lemma}

 \begin{proof}
 Assume $\rho \in \mathcal{P}$. In view of definition \ref{readef},
 \begin{align*}
|\{x\in \Omega: \rho_0(x)= r\}|
&=|\cap_1 ^\infty \{x\in \Omega: r \leq \rho_0(x)< r +\dfrac{1}{n}\}|\\
&=\lim_{n\to \infty}
|\{x\in \Omega: \rho_0(x)\geq r\}|- |\{x\in \Omega: \rho_0(x) \geq r
+\dfrac{1}{n}\}|\\
&=\lim_{n\to \infty} |\{x\in \Omega: \rho(x)\geq r\}|- |\{x\in \Omega: \rho(x)
\geq r +\dfrac{1}{n}\}|\\
&=|\cap_1 ^\infty \{x\in \Omega: r \leq \rho(x)< r +\dfrac{1}{n}\}|\\
&= |\{x\in \Omega: \rho(x)= r\}|.
\end{align*}
This implies that the level sets of $\rho$ and $\rho_0$ have the same
 measures and this yields the assertion. The other part of the theorem is
concluded from definition \ref{readef}.
 \end{proof}

Let us state here one of the essential tools in studying rearrangement
optimization problems.

\begin{lemma}\label{ber}
Let $\mathcal{P}$ be the set of rearrangements of a fixed function
$\rho_{0}\in L^r(\Omega)$, $r>1$, $\rho_{0}\not\equiv 0$, and let
$q\in L^s(\Omega)$, $s=r/(r-1)$, $q\not\equiv 0$. If there is a decreasing
function $\eta:\mathbb{R}\to \mathbb{R}$ such that $\eta(q)\in \mathcal{P}$, then
\[
\int_{\Omega} \rho q dx \geq \int_{\Omega} \eta(q)q
dx \quad \forall \rho \in\mathcal{P},
\]
and the function $\eta(q)$ is the unique minimizer relative to
$\mathcal{P}$.
\end{lemma}

For a proof of the above lemma see \cite{bu89}.

\section{Disproving the conjecture}\label{main result}

 In this section, we study the conjecture proposed in
\cite{Conca2} when $\Omega$ is a ball in $\mathbb{R}^n$ with $n\geq 2$.
We show that the conjecture is false for $n=2,3$ and for
every $n \geq 4$.
 Indeed, we will establish that a ball could not be a global
minimizer for the optimization problem \eqref{mainopt} when $\alpha$
and $\beta$ are close to each other (low contrast regime) and $A$ is large enough.
 It should be noted that our method is not as complicated as the approach has been
stated in \cite{Conca3} and we deny the conjecture in a simpler way.

 We hereafter regard $\Omega\subset \mathbb{R} ^n$ as the unit ball centered
at the origin. Assume that $\psi$ is the eigenfunction
 corresponding to the principal eigenvalue of the Laplacian
with Dirichlet's boundary condition on $\Omega $. Then, one can
consider $\psi=\psi(r)$ as a radial function which satisfies
 \begin{equation}\label{laplacian}
 \begin{gathered}
 r^2\psi''(r)+(n-1)r\psi'(r)+\lambda r^2\psi(r)=0\quad 0<r<1 ,\\
 \psi'(0)=0 \quad \psi(1)=0,
 \end{gathered}
  \end{equation}
 where the boundary conditions correspond to the continuity of the gradient at 
the origin and Dirichlet's condition on the boundary. In the next lemma,
 we examine the function $|\psi'(r)|$.

 \begin{lemma}\label{maxrho}
Let $\psi$ be the eigenfunction of \eqref{laplacian} associated with the principal 
eigenvalue $\lambda$. Then, function $|\psi'(r)|$ has a unique maximum point 
$\rho_n$ in $(0,1)$.
 \end{lemma}

 \begin{proof}
The solution of \eqref{laplacian} is
 \[
 \psi(r)=r^{1-\frac{n}{2}}J_{\frac{n}{2}-1}(\mu r)\quad 0\leq r \leq 1,
 \]
 where $\mu= j_{\frac{n}{2}-1,1}$. For the reader's convenience, we use 
the change of variable $t=\mu r$ and then
 \[
 \psi(t)=\mu^{\frac{n}{2}-1}\Big(\frac{J_{\frac{n}{2}-1}(t)}{t^{\frac{n}{2}-1}}
\Big)\quad 0\leq t \leq \mu.
 \]
 According to lemma \ref{besselroot}, $j_{\frac{n}{2}-1,1}<j_{\frac{n}{2},1}$ 
and then we see $J_\frac{n}{2}(t)\geq 0$ for $0\leq t \leq \mu$. Therefore,
 \[
 |\psi'(t)|=\mu^{\frac{n}{2}-1}\Big(\frac{J_{\frac{n}{2}}(t)}{t^{\frac{n}{2}-1}}\Big)
\quad 0\leq t \leq \mu,
 \]
 invoking formula \eqref{besselrec2}. To determine the maximum point of this 
function, one should calculate $\frac{d}{dt} (|\psi'(t)|)$. Employing relations
 \eqref{besselrec1} and \eqref{besselrec2},
 \[
 \frac{d}{dt} (|\psi'(t)|)=\frac{\mu^{\frac{n}{2}-1}(t J_{\frac{n}{2}-1}(t)
-(n-1)J_{\frac{n}{2}}(t))}{t^{\frac{n}{2}}}.
 \]
 Then $\frac{d}{dt} (|\psi'(t)|)=0$ yields
 \[
 t J_{\frac{n}{2}-1}(t)-(n-1)J_{\frac{n}{2}}(t)=0.
 \]
 The zeros of the last equation are the fixed points of the function
 \[
 g(t)=(n-1) \frac{J_{\frac{n}{2}}(t)}{J_{\frac{n}{2}-1}(t)} \quad 0< t < \mu.
 \]
 We find that
 \[
 J'_{\frac{n}{2}}(t)J_{\frac{n}{2}-1}(t)-J_{\frac{n}{2}}(t)J'_{\frac{n}{2}-1}(t) 
=\frac{(n-1)}{t} \int_0^t \frac{J_{\frac{n}{2}}(\tau)
J_{\frac{n}{2}-1}(\tau)}{\tau}d\tau,
 \]
 applying lemma \ref{goodrel}. Consequently, $g'(t)>0$ for $0<t<\mu$ and 
$g$ is an increasing function. On the other hand,
 $g(t)$ tends to infinity when $t\to \mu$ and, in view of formula \eqref{besselseri}, 
it tends to zero when $t\to 0$.
 Thus, $g(t)$ has a unique fixed point $\rho_n$ in $(0,\mu)$ which it is the unique 
extremum point of $|\psi'(t)|$. Recall  that 
$t J_{\frac{n}{2}-1}(t)-(n-1)J_{\frac{n}{2}}(t)$ is negative when $t\to \mu$. 
Hence, $\frac{d}{dt} (|\psi'(t)|)$ is negative
 in a neighborhood of $\mu$ and thus, $\rho_n$ is the unique maximum point of
 $\frac{d}{dt} (|\psi'(t)|)$ in $(0,\mu)$.
 \end{proof}

We need the following theorem to deduce the main result.

\begin{theorem}\label{lurianthem}
Assume $D_0$ is a subset of $\Omega$ where $|D_0|=A$ and $u_0$ is the eigenfunction 
of \eqref{mainpde} corresponding to $\lambda(D_0)$. Let $D_1$
be a subset of $\Omega$ where
\begin{equation}\label{tformulap}
|D_1|=A \quad\text{and}\quad D_1=\{x:|\nabla u_0|\leq t\}
\end{equation}
with
\begin{equation}\label{tformula}
t=\inf\{s\in \mathbb{R} : |\{x: |\nabla u_0|\leq s\}|\geq A\}.
\end{equation}
Then, $\lambda(D_1)\leq \lambda(D_0)$.
\end{theorem}
\begin{proof}
 It is well known, from the Krein-Rutman theorem \cite{rutman}, 
that $u_0$ is positive everywhere on $\Omega$. Therefore,
 we infer that all sets $\{x: |\nabla u_0|=s\}$
 have measure zero because of \cite[lemma 7.7]{gilbarg}. 
Then, one can determine set $D_1$ uniquely using the above formula. 
Let us define the decreasing function
\[
 \eta(s)= \begin{cases}
 \beta & 0 \leq s\leq t^2, \\ 
\alpha & s>t^2.
 \end{cases}
 \]
 This yields
 \[
 \eta(|\nabla u_0|^2)=\beta \chi_{D_1}+\alpha \chi_{D_1^c}.
 \]
 From lemma \ref{chirho} and \ref{ber}, we  deduce
 \[
 \int (\beta\chi_{D_1}+\alpha \chi_{D_1^c}) |\nabla u_0|^2dx
\leq\int (\beta\chi_{D_0}+\alpha \chi_{D_0^c}) |\nabla u_0|^2dx,
 \]
 and then we have $\lambda(D_1)\leq \lambda(D_0)$ invoking \eqref{lambdavar}.
\end{proof}


\begin{remark}\label{levelsetofu} \rm
In theorem \ref{lurianthem}, if $D_1\neq D_0$, then
\[
 \int (\beta\chi_{D_1}+\alpha \chi_{D_1^c}) |\nabla u_0|^2dx
<\int (\beta\chi_{D_0}+\alpha \chi_{D_0^c}) |\nabla u_0|^2dx,
 \]
applying the uniqueness of the minimizer in lemma \ref{ber}. 
Thus, we observe that $\lambda(D_1)< \lambda(D_0)$ when $D_1\neq D_0$.
\end{remark}

\begin{remark}\label{approximant} \rm
In \cite{Conca3}, it has been proved that if 
${\rho}_*=\beta\chi_{D_*}+\alpha \chi_{D_*^c}$ is the minimizer of
\begin{equation}\label{appproblem}
\min_{\rho \in\mathcal{P}} \int_{\Omega} \rho |\nabla \psi|^2 dx,
\end{equation}
then the set $D_*$ is an approximate solution for \eqref{mainopt}, 
under the assumption of low contrast regime. By arguments similar to
those in the proof of theorem \ref{lurianthem}, one can determine the unique
 minimizer of problem \eqref{appproblem}, 
${\rho}_*=\beta\chi_{D_*}+\alpha \chi_{D_*^c}$,
using formulas \eqref{tformulap} and \eqref{tformula}. 
Recall from lemma \ref{maxrho} that $|\psi'(r)|$ has a unique maximum point 
$\rho_n$ in $(0,1)$ and it is a continuous function on $[0,1]$ with $|\psi'(0)|=0$. 
Then the unique symmetrical domain $D_*$ which 
${\rho}_*=\beta\chi_{D_*}+\alpha \chi_{D_*^c}$ is the solution of \eqref{appproblem} 
is of two possible types. The set $D_*$ is a ball centered at the origin if 
$A\leq |\mathcal{B}(0, \rho_n)|$
and it is the union of a ball and an annulus touching the outer boundary of 
$\Omega$ if $A> |\mathcal{B}(0, \rho_n)|$.
This result has been established in \cite{Conca3} for $n=2,3$.
\end{remark}

 Now we are ready to state our main result. We establish that locating the material 
with the highest  conductivity in a ball centered at the origin is not the 
minimal distribution since we can find another radially symmetric
distribution of the materials which has a smaller basic frequency.

\begin{theorem}\label{bestresult}
Let $D_0=\mathcal{B}(0,\rho)\subset \Omega$ be a ball centered at the origin with
$|D_0|=A$. If $\beta$ is sufficiently close to $\alpha$ and $\rho> \rho_n$, 
then there is a set $D_1\subset \Omega$ with $|D_1|=A$ containing a radially 
symmetric subset of $D_0^c$ where $\lambda(D_1)<\lambda(D_0)$.
\end{theorem}

\begin{proof}
Suppose $u_0$ is the eigenfunction of \eqref{mainpde} associated with 
$\lambda=\lambda(D_0)$ such that $\|u_0\|_{L^2(\Omega)}=1$. Utilizing theorem \ref{lurianthem} 
and remark \ref{levelsetofu}, we conclude $\lambda(D_1)<\lambda(D_0)$ provided
\[
D_1=\{x: |\nabla u_0|\leq  t\}, \quad 
t=\inf\{s\in \mathbb{R} : |\{x: |\nabla u_0|\leq s\}|\geq  A\},
\]
and $D_0\neq D_1$. One can  observe that $u_0$ satisfies the 
transmission problem
\begin{equation} \label{traneq}
 \begin{gathered}
 -\beta \Delta v_1 = \lambda v_1\quad \text{in } D_0, \\
-\alpha \Delta v_2 = \lambda v_2\quad \text{in } D_0^c, \\
v_1(x)=v_2(x) \quad\text{on }\partial D_0,  \\
\beta\frac{\partial}{\partial \mathfrak{n}} v_1
=\alpha\frac{\partial}{\partial \mathfrak{n}}  v_2\quad \text{on } \partial D_0m \\
v_2(x)=0\quad\text{on } \partial \Omega,
 \end{gathered}
\end{equation}
where $\mathfrak{n}$ is the unit outward normal. According to the
above representation, $u_0$ is an analytic function in the closure
of sets $D_0$ and $D_0^c$ employing the analyticity theorem
\cite{john}.

 We should assert that $D_0\neq D_1$. To this end, let us note that $u_0$ is a radial
function and so $u_0(x)=y(r)$, $r=\|x\|$, where the function $y$ solves
\begin{equation} \label{radialeq}
  \begin{gathered}
 y''(r)+\frac{n-1}{r}y'(r)+\frac{\lambda}{\beta} y(r)
 =0 \quad \text{in }(0,\rho) \\
y''(r)+\frac{n-1}{r}y'(r)+\frac{\lambda}{\alpha} y(r)  =0 \quad \text{in }(\rho,1)  \\
y(\rho^-)=y(\rho^+) \\ 
\beta y'(\rho^-)=\alpha  y'(\rho^+) \\
y'(0)=0,\quad y(1)=0.
 \end{gathered}
\end{equation}

We introduce $y_1(r)$ and $y_2(r)$ as the solutions of
\eqref{radialeq} in $[0, \rho]$ and $[\rho, 1]$ respectively. We claim
that if
\begin{equation}\label{maininequality}
|y_2'(1)|< z=\underset{r\in[0,\rho]}{\max} |y_1'(r)|,
\end{equation}
then $D_1$ contains a radially symmetric subset of
$D_0^c$ and so $D_1$ is not equal to $D_0$.

 Recall that level sets of $|\nabla u_0|$ have measure zero.
 Hence, if $|y_2'(r)|>z$ for all $r$ in $[\rho,1]$
then $D_1=\{x:\; |\nabla u_0|\leq t\}=D_0$ with $t=z$. On the
other hand, if $|y_2'(1)|<z$ then we have $t<z$ to satisfy
the condition $|D_1|=A$, in view of the continuity of the function $|y_2'(r)|$.
 In other words, $D_1$  should include a radially symmetric subset of
$D_0^c$. This discussion proves our claim.

 It remains to verify inequality \eqref{maininequality}. 
This is a standard result of the perturbation theory
 of eigenvalues that $u_0$ tends to $\psi$ with $\|\psi\|_{L^2(\Omega)}=1$ and 
$\lambda$ converges to $\alpha \mu$ when $\beta$
 decreases to $\alpha$ \cite{rellich}. The convergence of the eigenfunctions holds 
in the space $H^1_0(\Omega)$. Hence it yields that
 $y(r)$ and $y'(r)$ converge to $\psi(r)$ and $\psi'(r)$ almost everywhere in $\Omega$, 
respectively. Since
 $y'(r)$ and $\psi'(r)$ are continuous functions on the sets $[0,\rho]$ and 
$[\rho,1]$, the convergence
 is pointwise\cite{lieb}. In summary, $|y_1'(r)|$ converges to 
$|\psi'(r)|$ pointwise for all $r$ in $[0, \rho]$
 and $|y_2'(r)|$ converges to $|\psi'(r)|$ pointwise in $[\rho, 1]$. 
Additionally, $\|y_2'(\rho)|-|y_1'(\rho)\|$
 converges to zero when $\beta$ approaches $\alpha$. Invoking lemma \ref{maxrho}, 
we see that $|\psi'(\rho)|-|\psi'(1)|=d_n>0$
 when $\rho>\rho_n$. Thus, if $\beta$ is close to $\alpha$ enough, we have
 \begin{equation}\label{lastrel1}
 \|y_2'(\rho)|-|y_2'(1)\|> {d_n}/2,
 \end{equation}
 and 
 \begin{equation}\label{lastrel2}
 |y_2'(\rho)|\to |\psi'(\rho)|,\quad 
|y_2'(1)|\to |\psi'(1)|,\quad 
|y_2'(\rho)|\to |y_1'(\rho)|,
 \end{equation}
 as $\beta$ converges to $\alpha$. 
Applying \eqref{lastrel1} and \eqref{lastrel2}, leads us to inequalities
 \[
 |y_2'(1)|<|y_1'(\rho)| \leq z.
 \]
\end{proof}

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