\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 07, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/07\hfil $\infty$-eigenvalues]
{Uniform stability of the ball with respect to the first Dirichlet and Neumann
$\infty$-eigenvalues}

\author[J. V. da Silva, J. D. Rossi,  A. M. Salort  \hfil EJDE-2018/07\hfilneg]
{Jo\~{a}o Vitor da Silva, Julio D. Rossi, Ariel M. Salort}

\address{Jo\~{a}o Vitor da Silva \newline
Departamento de Matem\'atica,
FCEyN - Universidad de Buenos Aires.  \newline
IMAS - CONICET Ciudad Universitaria,
 Pabell\'on I (1428) Av. Cantilo s/n.
Buenos Aires, Argentina}
\email{jdasilva@dm.uba.ar}

\address{Julio D. Rossi \newline
Departamento de Matem\'atica,
FCEyN - Universidad de Buenos Aires.  \newline
IMAS - CONICET Ciudad Universitaria,
 Pabell\'on I (1428) Av. Cantilo s/n.
Buenos Aires, Argentina}
\email{jrossi@dm.uba.ar}

\address{Ariel M. Salort \newline
Departamento de Matem\'atica,
FCEyN - Universidad de Buenos Aires.  \newline
IMAS - CONICET Ciudad Universitaria,
 Pabell\'on I (1428) Av. Cantilo s/n.
Buenos Aires, Argentina}
\email{asalort@dm.uba.ar}

\dedicatory{Communicated by Jesus Ildefonso Diaz}

\thanks{Submitted September 9, 2017. Published January 6, 2018.}
\subjclass[2010]{35B27, 35J60, 35J70}
\keywords{$\infty$-eigenvalues estimates; $\infty$-eigenvalue problem;
\hfill\break\indent approximation of domains}

\begin{abstract}
 In this note we analyze how perturbations of a ball
 $ B_r \subset \mathbb{R}^n$  behaves in terms of their first
 (non-trivial) Neumann and Dirichlet   $\infty$-eigenvalues when a volume
 constraint $\mathscr{L}^n(\Omega) = \mathscr{L}^n( B_r)$ is imposed.
 Our main result states that $\Omega$ is uniformly close to a ball when
 it has first Neumann and Dirichlet eigenvalues close to the ones for the
 ball of the same volume $ B_r$. In fact, we show that, if
 $$
 |\lambda_{1,\infty}^D(\Omega) - \lambda_{1,\infty}^D( B_r)|
 = \delta_1 \quad \text{and} \quad
 |\lambda_{1,\infty}^N(\Omega) - \lambda_{1,\infty}^N( B_r)| = \delta_2,
 $$
 then there are two balls such that
 $$
  B_{\frac{r}{\delta_1 r+1}} \subset \Omega \subset
  B_{\frac{r+\delta_2 r}{1-\delta_2 r}}.
 $$
 In addition, we  obtain a result concerning stability of the Dirichlet
 $\infty$-eigenfunctions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks
\numberwithin{equation}{section}

\section{Introduction}\label{Intro}

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain (connected open subset) 
with smooth boundary, $1<p< \infty$ and 
$\Delta_p u := \operatorname{div}(|\nabla u|^{p-2}\nabla u)$ 
(the standard $p$-Laplacian operator).
Historically (cf. \cite{Lindq90}), it well-known that the first eigenvalue 
(referred as \emph{the principal frequency} in physical models) of the $p$-Laplacian
 Dirichlet eigenvalue problem
\begin{equation}\label{eq.p}
\begin{gathered}
  -\Delta_p u  =  \lambda_{1, p}^D(\Omega)|u|^{p-2} u \quad \text{in }  \Omega \\
  u  =  0 \quad \text{on }  \partial \Omega
\end{gathered}
\end{equation}
can be characterized variationally as the minimizer of the (normalized) 
problem
\begin{equation} \label{1er.p}
    \lambda_{1, p}^D(\Omega) := \inf_{u \in W^{1,p}_0 (\Omega) \setminus \{0\}}
   \Big\{  \int_{\Omega} |\nabla u|^p dx  : \int_{\Omega} |u|^pdx = 1\Big\}.
\end{equation}

In the theory of shape optimization and non-linear eigenvalue problems obtaining 
(sharp) estimates for the eigenvalues in terms of geometric quantities of the 
domain (e.g. measure, perimeter, diameter, among others) plays a fundamental 
role due to several applications of these problems in pure and applied sciences. 
We recall that the explicit value to \eqref{1er.p} is known only for some 
specific values of $p$ or for very particular domains $\Omega$. 
Notice that upper bounds for $\lambda_{1, p}^D(\Omega)$ are usually obtained 
by selecting particular test functions in \eqref{1er.p}.
 Nevertheless, lower bounds are a more challenging task. In this direction 
we have the remarkable \emph{Faber-Krahn inequality:
Among all domains of prescribed volume the ball minimizes \eqref{1er.p}}. 
More precisely,
\begin{equation} \label{chi}
   \lambda_{1, p}^D(\Omega) \geq \lambda_{1, p}^D( B),
\end{equation}
where $ B$ is the $n$-dimensional ball such that
$\mathscr{L}^n(\Omega) = \mathscr{L}^n( B)$
(along this paper $\mathscr{L}^n (\Omega)$ will denote the Lebesgue measure
of $\Omega$ that is assumed to be fixed).
Using isoperimetric or isodiametric inequality similar lower bounds 
for \eqref{1er.p} in terms of the perimeter (resp. diameter) of 
$\Omega$ are also available
(cf. \cite{Bhat} and \cite[page 224]{Lindq92}, and the references therein).
Recently, stability estimates for certain geometric inequalities were established 
in \cite{FMP2}, thereby providing an improved version of \eqref{chi} by 
adding a suitable remainder term, i.e.,
$$
\lambda_{1, p}^D(\Omega) \geq \lambda_{1, p}^D( B)\big(1+
\gamma_{p, n} (\mathcal{S}(\Omega))^{2+p}\big),
$$
where $\mathcal{S}(\Omega)$ is the so-called \emph{Fraenkel asymmetry}
of $\Omega$, which is precisely defined as
$$
   \mathcal{S}(\Omega) :=  \inf_{x_0 \in \mathbb{R}^n} \Big\{ \frac{\mathscr{L}^n
(\Omega \Delta  B_{r}(x_0))}{\mathscr{L}^n(\Omega)}
:\mathscr{L}^n( B_{r}(x_0)) = \mathscr{L}^n(\Omega) \Big\},
$$
and $\gamma_{p, n}$ is a constant.
Observe that $\mathcal{S}$ measures the distance of a set $\Omega$
from being a ball. For such quantitative estimates
and further related topics we quote \cite{Bp, Cianci, Fusco}
and references therein.

Our main goal here is to find stability results for the limit case $p=\infty$.

First, we introduce what is known for the limit as $p\to \infty$ in the
 eigenvalue problem for the $p$-Laplacian.
When one takes the limit as $ p\to \infty$ in the minimization problem 
\eqref{1er.p}, one obtains
\begin{equation} \label{lam.infD}
  \lambda_{1,\infty}^D(\Omega) := \lim_{p\to\infty} \sqrt[p]{\lambda_{1, p}^D(\Omega)}
=\inf_{u \in W^{1, \infty}_0(\Omega)\setminus\{0\}} 
\|\nabla u\|_{L^{\infty}(\Omega)}>0,
\end{equation}
see \cite{JLM}.
Concerning the limit equation, also in \cite{JLM} it is proved that any family
of normalized eigenfunctions $\{u_p\}_{p>1}$ to \eqref{1er.p} converges 
(up to a  subsequence) locally uniformly to $u_\infty \in W^{1,\infty}_0 (\Omega)$,
a minimizer for \ref{lam.infD} with $\|u_\infty\|_{L^{\infty}(\Omega)}=1$.
Moreover, the pair $(u_\infty, \lambda_{1, \infty}^D(\Omega))$ is a non-trivial
solution to
\begin{equation}\label{eq.infty.p}
\begin{gathered}
  \min\big\{-\Delta_\infty v_\infty, |\nabla v_\infty|
-\lambda_{1,\infty}^D(\Omega)v_\infty\big\}  =  0 \quad \text{in }  \Omega \\
  v_\infty  =  0 \quad \text{on }  \partial \Omega.
\end{gathered}
\end{equation}
Solutions to \eqref{eq.infty.p} must be understood in the viscosity sense 
(see \cite{CIL} for a survey) and
$ \Delta_\infty u(x) := \nabla u(x)^TD^2u(x) \cdot \nabla u(x) $
is the well-known \emph{$\infty$-Laplace operator}.
In addition,  in \cite{JLM} it is given an interesting and useful geometrical 
characterization for \eqref{lam.infD}:
\begin{equation} \label{lam1}
	\lambda_{1,\infty}^D(\Omega)
= \Big(\max_{x \in \Omega} \operatorname{dist}(x, \partial \Omega)\Big)^{-1}.
\end{equation}
Such an information means that the ``principal frequency'' for the 
$\infty$-eigenvalue problem can be detected from the geometry of the domain: 
it is precisely the reciprocal of radius $\mathfrak{r}_\Omega>0$ of the largest 
ball inscribed in $\Omega$. For more references concerning the first 
eigenvalue \eqref{eq.infty.p} we refer to \cite{KH}, \cite{NRSanAS} and \cite{Yu}.

Now, let us turn our attention to Neumann boundary conditions and consider 
the  eigenvalue problem
\begin{equation}\label{eq.p.n}
\begin{gathered}
  -\Delta_p u  =  \lambda_{1, p}^N(\Omega)|u|^{p-2} u \quad\text{in } \Omega \\
   |\nabla u|^{p-2}\tfrac{\partial u}{\partial \nu}  =  0 \quad\text{on }
 \partial \Omega.
\end{gathered}
\end{equation}
As before, we stress that the first non-zero eigenvalue of \eqref{eq.p.n} 
can also be characterized variationally as the minimizer of the  
normalized problem
\begin{equation} \label{1er.p.n}
    \lambda_{1, p}^N(\Omega) := \inf_{u \in W^{1, p}(\Omega)}
 \Big\{  \int_{\Omega} |\nabla u|^pdx: \int_{\Omega}| u|^p dx 
 = 1\text{ and } \int_\Omega |u|^{p-2}udx =0\Big\}.
\end{equation}
The celebrated \emph{Payne-Weinberger inequality} provides a lower bound  
(on any convex domain $\Omega \subset \mathbb{R}^n$) for the first (non-trivial)
 Neumann $p-$eigenvalue (see \cite{ENT,Valt})
\begin{equation} \label{ggg}
  \lambda_{1, p}^N(\Omega) \geq 
(p-1)\Big(\frac{2\pi}{p\, \operatorname{diam}(\Omega)\, \sin(\frac{\pi}{p})}\Big)^{p}.
\end{equation}
For a  stability estimate for this problem with $p=2$ we refer to \cite{Bp}.

When $p\to \infty$, the minimization problem \eqref{1er.p.n} becomes
\begin{equation} \label{lam.inf}
   \lambda_{1,\infty}^N(\Omega) := \lim_{p\to\infty} \sqrt[p]{\lambda_{1, p}^N(\Omega)}
= \inf_{ \substack{u\in W^{1,\infty}(\Omega)  \\
 \max_{\Omega} u  = -\min_{\Omega} u = 1}}  \|\nabla u\|_{L^\infty(\Omega)},
\end{equation}
see \cite{EKNT,RosSaint}. Concerning the limit equation, also in 
\cite{EKNT,RosSaint}, it is proved that any family of normalized eigenfunctions 
$\{u_p\}_{p>1}$ to \eqref{1er.p.n} converges (up to subsequence) locally 
uniformly to a limit $u_\infty \in W^{1,\infty}_0 (\Omega)$ with 
$\|u_\infty\|_{L^{\infty}(\Omega)}=1$.
Moreover, the pair $(u_\infty, \lambda_{1, \infty}^N(\Omega))$
is a non-trivial solution to
\begin{equation}\label{eq.infty.p.n}
\begin{gathered}
\min\big\{-\Delta_\infty v_\infty, |\nabla v_\infty|
 -\lambda_{1,\infty}^N(\Omega)v_\infty\big\}  =  0 \quad\text{in } \Omega\cap \{v>0\} \\
\max\big\{-\Delta_\infty v_\infty, -|\nabla v_\infty|
 -\lambda_{1,\infty}^N(\Omega)v_\infty\big\}  =  0 \quad\text{in } \Omega\cap \{v<0\} \\
 -\Delta_\infty v_\infty  =  0 \quad\text{in } \Omega \cap \{v=0\}\\
 \frac{\partial v_{\infty}}{\partial \nu} =  0 \quad\text{in } \partial \Omega.
\end{gathered}
\end{equation}
In addition, we have the following geometrical characterization for 
$\lambda_{1,\infty}^N(\Omega)$:
\begin{equation} \label{lam1.n}
	\lambda_{1,\infty}^N(\Omega) = \frac{2}{\operatorname{diam}(\Omega)},
\end{equation}
where the intrinsic diameter of $\Omega$ is defined as
$$
	\operatorname{diam}(\Omega) := \max_{\bar\Omega \times \bar\Omega} d_{\Omega}(x,y)
= \max_{\partial \Omega \times \partial \Omega} d_{\Omega}(x,y),
$$
where $d_{\Omega}(x,y)$ is the geodesic distance given by 
$ d_{\Omega}(x,y)=\inf_{\gamma} \operatorname{Long}(\gamma)$,  
where the infimum is taken over all possible Lipschitz curves in 
$\bar\Omega$ connecting $x$ and $y$.

We remark that in the limit case $p=\infty$, the geometrical characterization 
\eqref{lam1.n} of \eqref{lam.inf} yields several interesting consequences:
\begin{itemize}
  \item[\checkmark] If $\mathscr{L}^n(\Omega) = \mathscr{L}^n( B)$,
$ B$ being a ball, then $\lambda_{1,\infty}^N(\Omega) \leq
 \lambda_{1,\infty}^N( B)$, which establishes a
\emph{Szeg\"{o}-Weinberger type inequality}: among all domains of prescribed 
volume the ball maximizes \eqref{lam.inf}.

  \item[\checkmark] $\lambda_{1,\infty}^N(\Omega) \leq \lambda_{1,\infty}^D(\Omega)$
for any convex $\Omega$ with equality if and only if $\Omega$ is a ball.

  \item[\checkmark] The Payne-Weinberger inequality, \eqref{ggg}, becomes 
an equality when  $p = \infty$.
\end{itemize}

Taking into account the previous historic overview, we arrive to our main result, 
which establishes the stability of the ball with respect to small perturbations 
of their first Dirichlet and Neumann $\infty$-eigenvalues.
 More precisely, if a domain $\Omega\subset \mathbb{R}^n$ has Dirichlet 
and Neumann $\infty$-eigenvalues close enough to those of the ball
$ B_r$ of the same Lebesgue measure, then $\Omega$ is uniformly
``almost'' ball-shaped.

\begin{theorem}\label{Mainthm} 
Let $\Omega$ be an open domain satisfying $\mathscr{L}^n(\Omega)=\mathscr{L}^n( B_r)$. If for some $\delta_i>0$ ($i=1, 2$) small enough it holds that
$$
 |\lambda_{1,\infty}^D(\Omega) - \lambda_{1,\infty}^D( B_r)|
= \delta_1 \quad \text{and} \quad 
|\lambda_{1,\infty}^N(\Omega) - \lambda_{1,\infty}^N( B_r)| = \delta_2,
$$
then there are two balls such that
$$
 B_{\frac{r}{\delta_1 r+1}} \subset \Omega \subset
  B_{\frac{r+\delta_2 r}{1-\delta_2 r}}.
$$ 
\end{theorem}


The previous theorem implies the following convergence result.

\begin{theorem}\label{Mainthm2} 
Let $\{\Omega_k\}_{k \in \mathbb{N}}$ be a family of uniformly bounded 
domains satisfying $\mathscr{L}^n(\Omega_k)=\mathscr{L}^n( B_r)$. If
$$
   |\lambda_{1,\infty}^D(\Omega_k) - \lambda_{1,\infty}^D( B_r)|
=  o(1) \quad \text{and} \quad |\lambda_{1,\infty}^N(\Omega)
- \lambda_{1,\infty}^N( B_r)| =  o(1) \quad \text{as } k \to \infty,
$$
then 
$\Omega_k \to  B_r$ in the sense that the Hausdorff distance
between $\Omega$ and a ball
$ B_r$ approaches zero, i.e.,
$$
d_\mathcal{H} (\Omega_k,   B_r)
:= \max\Big\{\,\sup _{{x\in \Omega_k}}\inf _{{y\in  B_r}}d(x,y),\,
\sup _{{y\in  B_r}}\inf _{{x\in  \Omega_k}}d(x,y)\, \Big\} \to 0.
$$
\end{theorem}

Note that our results imply 
\begin{equation} \label{ecccc}
  \max\Big\{\mathscr{L}^n\Big(\Omega\Delta  B_{\frac{r}{\delta_1 r+1}}
\Big), \mathscr{L}^n\Big(\Omega\Delta  B_{\frac{r+\delta_2 r}{1-\delta_2 r}}\Big)
\Big\}\leq \mathfrak{C}(n, \delta_i, r)r^n.
\end{equation}
where $\mathfrak{C}(n, \delta_i, r)=\omega_n \max\{(\delta_1r+1)^n-1,(n-1)\delta_2\}
\to 0$ as $\delta_i\to 0$.
Hence, we can control the Fraenkel asymmetry of the set, $S(\Omega)$. 
But our results give much more since we have a sort
of uniform control on how far the set is from being a ball 
(for instance, we have convergence in Hausdorff distance
in Theorem \ref{Mainthm2}).


Another important question in this theory consists on how the corresponding
 $\infty$-ground states
(solutions to \eqref{eq.infty.p}) behave in relation to perturbations of the
 $\infty$-eigenvalues of the ball. The next result provides an answer for 
this issue, showing that Dirichlet $\infty$-eigenfunctions are uniformly 
close to a cone when the first Dirichlet and Neumann $\infty$-eigenvalues 
are close to those for the ball.
Note that, in general, the $\infty$-eigenvalue problem \eqref{eq.infty.p} 
may have multiple solutions (the first eigenvalue may not be simple), 
see \cite{HSY} and \cite{Yu}.

\begin{theorem} \label{teo.autofunc.intro}
Let $\Omega$ be an open domain satisfying $\mathscr{L}^n(\Omega)=\mathscr{L}^n( B_r)$.
Given $\varepsilon >0$
there are $\delta_i(\varepsilon)>0$ ($i=1, 2$) small enough such that: if
$$
 |\lambda_{1,\infty}^D(\Omega) - \lambda_{1,\infty}^D( B_r)| < \delta_1
\quad \text{and} \quad 
|\lambda_{1,\infty}^N(\Omega) - \lambda_{1,\infty}^N( B_r)| < \delta_2,
$$
then
$$
  |u(x)-v_{\infty}(x)| < \varepsilon \quad \text{in }  \Omega \cap  B_r,
$$
where $v_{\infty} (x)= 1 - \frac{|x|}{r}$ is the normalized $\infty$-ground 
state to \eqref{eq.infty.p} in $ B_r$.
\end{theorem}

Theorem \ref{teo.autofunc.intro} can be rewritten as follows:

\begin{corollary}\label{CorConv} 
Let $\{u_k\}_{k \in \mathbb{N}}$ be a family of normalized solutions to 
\eqref{eq.infty.p} in $\Omega_k$ such that
$$
 |\lambda_{1,\infty}^D(\Omega_k) - \lambda_{1,\infty}^D( B_r)|
=  o(1) \quad \text{and} \quad
 |\lambda_{1,\infty}^N(\Omega_k) - \lambda_{1,\infty}^N( B_r)|
=  o(1)\quad \text{as } k \to \infty.
$$
Then
$u_k \to v_{\infty}$ locally uniformly in $B_r$,
where 
\[
v_{\infty} (x)= 1 - \frac{|x|}{r}
\]
 is the normalized $\infty$-ground state 
to \eqref{eq.infty.p} in $ B_r$.
\end{corollary}

Our approach can be applied for other classes of operators with $p$-Laplacian 
type structure.
We can deal with $p-$Laplace type problems involving an 
\emph{anisotropic $p$-Laplace operator}
$$
    -\mathcal{Q}_p u := -\operatorname{div}(\mathbb{F}^{p-1}(\nabla u)
\mathbb{F}_{\xi}(\nabla u)),
$$
where $\mathbb{F}$ is an appropriate (smooth) norm of $\mathbb{R}^n$ and 
$1<p< \infty$. The necessary tools for studying the anisotropic Dirichlet 
eigenvalue problem, as well as its limit as $p \to \infty$ can be 
found in \cite{BKJ}.
Here, to obtain results similar to ours, one has to replace Euclidean balls 
with  balls in the norm $\mathbb{F}$.


This article is organized as follows: 
In Section \ref{sect-main} we prove our main stability results
including the behaviour of the corresponding $\infty$-eigenfunctions.
In Section \ref{sect-examples} we collect several
examples that illustrate our results.


\section{Proof of main results} \label{sect-main}

Before proving our main result we introduce some notation which will be used 
throughout this section. Given a bounded domain $\Omega\subset \mathbb{R}^n$ 
and a ball $ B_r\subset\mathbb{R}^n$ of radius $r>0$
we denote $\lambda_{1,\infty}^D(\Omega)$ and $\lambda_{1,\infty}^D( B_r)$
the first Dirichlet eigenvalues \eqref{lam1} in $\Omega$ and  in $ B_r$,
 respectively; analogously, $\lambda_{1,\infty}^N(\Omega)$ and
$\lambda_{1,\infty}^N( B_r)$ stand for the first non-trivial
Neumann eigenvalues \eqref{lam1.n} in $\Omega$ and in  $ B_r$.

We introduce the following class of sets which will play an important role 
in our approach. For non-negative constants $\delta_1$ and $\delta_2$ 
we define the class
\begin{align*}
  \Xi_{\delta_1, \delta_2}( B_r)
&:= \Big\{\Omega \subset \mathbb{R}^n   \text{ bounded domain with } 
   \mathscr{L}^n(\Omega)  =  \mathscr{L}^n( B_r) \\
&\quad :
 |\lambda_{1,\infty}^D(\Omega)-\lambda_{1,\infty}^D( B_r)|  =  \delta_1, \;
 |\lambda_{1,\infty}^N(\Omega)-\lambda_{1,\infty}^N( B_r)|  =  \delta_2
\Big\}.
\end{align*}
 Notice that $\Xi_{0, 0}( B_r)$ consists of the family of all
balls with radius $r>0$. Another important remark is that the elements 
of $\Xi_{\delta_1, \delta_2}( B_r)$ are invariant by rigid transformations
(rotations, translations, etc).

Similarly, we can define the class $\Xi^D_{\delta_1}( B_r)$
(resp. $\Xi^N_{\delta_2}( B_r)$) as being
$\Xi_{\delta_1, \delta_2}( B_r)$ with the restriction on the
Dirichlet (resp. Neumann) eigenvalues only.

In the next lemma we show that a control on the difference of the first
 Dirichlet eigenvalue implies that $\Omega$ contains a large ball.

\begin{lemma}\label{Lemma1}
If $\Omega\in \Xi^D_{\delta_1}( B_r)$ then there exists a ball
such that 
$ B_{\frac{r}{\delta_1 r+1}} \subset \Omega$.
 Moreover,
$$
   \mathscr{L}^n\Big(\Omega\Delta  B_{\frac{r}{\delta_1 r +1}}\Big)
\leq \mathfrak{c}(n, \delta_1, r) r^{n}.
$$
where $\mathfrak{c} =  o(1)$ as $\delta_1 \to 0$.
\end{lemma}

\begin{proof}
According to \eqref{lam1} we have 
$$
   \delta_1=|\lambda_{1,\infty}^D(\Omega) - \lambda_{1,\infty}^D( B_r)|
=\big| \frac{1}{r_\Omega} - \frac{1}{r}\big|.
$$
It follows that
$$
	r_\Omega \geq \frac{r}{\delta_1 r +1} .
$$
and then there is ball such that 
$ B_{\frac{r}{\delta r+1}} \subset \Omega$.
Finally,
\begin{align*}
\mathscr{L}^n(\Omega\triangle  B_{\frac{r}{\delta r+1}} )
& = \mathscr{L}^n(\Omega) - \mathscr{L}^n( B_{\frac{r}{\delta r+1}} )\\
& = \omega_n r^n \Big(1-\frac{1}{(\delta r +1)^n}\Big)\\
& \leq  \omega_n r^n \left((\delta r +1)^n-1\right)\\
&= \mathfrak{c}(n, \delta, r)r^{n}
\end{align*}
and the lemma follows.
\end{proof}

Now, we show that a control on the difference of the first Neumann eigenvalue
implies that $\Omega$ is contained in a small ball.

\begin{lemma}\label{Lemma2}
If $\Omega\in \Xi^N_{\delta_2}( B_r)$ then there is a ball such that
 $\Omega \subset  B_{\frac{r}{1-\delta_2 r}}$.
 Moreover,
$$
  \mathscr{L}^n\Big(\Omega \Delta B_{\frac{r}{1-\delta_2 r}}\Big)
\leq (n-1)\omega_n r^n \delta_2.
$$
\end{lemma}

\begin{proof}
Using \eqref{lam1.n} we have 
$$
   \delta_2=|\lambda_{1,\infty}^N(\Omega) - \lambda_{1,\infty}^N( B_r)|
=\Big| \frac{2}{\operatorname{diam}(\Omega)} - \frac{1}{r}\Big|.
$$
It follows that
$$
	\operatorname{diam}(\Omega)\leq \frac{2r}{1-\delta_2 r }
 = r+\frac{r(1+\delta r)}{1-\delta_2 r}
$$
and then there exists a ball such that
$$
  \Omega \subset  B_{\frac{\operatorname{diam}(\Omega)}{2}} 
=  B_{\frac{r}{1-\delta_2 r}}.
$$
Moreover,
\begin{align*}
\mathscr{L}^n\Big(\Omega\Delta  B_{\frac{\operatorname{diam}(\Omega)}{2}}\Big)
&=\mathscr{L}^n\Big( B_{\frac{\operatorname{diam}(\Omega)}{2}}\Big) - \mathscr{L}^n(\Omega)
\\
& =\omega_n r^n \Big( \Big( 1+\frac{\delta_2}{1-\delta_2 r}\Big)^n -1 \Big)\\
&=\omega_n r^n \delta_2 \sum_{k=2}^n \Big(\frac{\delta_2}{1-\delta_2 r}\Big)^k\\
&\leq (n-1)\omega_n \delta_2 r^n
\end{align*}
and the lemma follows.
\end{proof}

\begin{proof}[Proof of Theorem \ref{Mainthm}]
 The theorem follows as an immediate consequence of Lemmas \ref{Lemma1} 
and \ref{Lemma2}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{Mainthm2}] 
The hypothesis implies that $\Omega_k \in \Xi_{\delta_k, \varepsilon_k}( B_r)$ 
for $\delta_k, \varepsilon_k = o(1)$ as $k \to \infty$. 
For this  reason, by Theorem \ref{Mainthm} there are two balls such that
$$
   B_{\frac{r}{\delta_k r+1}} \subset \Omega_k 
\subset  B_{\frac{r+\varepsilon_k r}{1-\varepsilon_k r}}.
$$
Now, using that all these balls are centered at points that are bounded 
(since we assumed that the family $\Omega_k $
is uniformly bounded), we can extract a subsequence such that the centers 
converge and therefore we conclude that
there is a ball $ B_r$ such that
$\Omega_k \to  B_r$ as $k \to \infty$.
\end{proof}


\begin{proof}[Proof of Theorem \ref{teo.autofunc.intro}]
The proof follows by contradiction. Let us suppose that there exists 
an $\varepsilon_0>0$ such that the thesis of theorem fails to hold. 
This means that for each $k \in \mathbb{N}$ we might find a domain 
$\Omega_k$ and $u_k$, a normalized $\infty$-ground state to \eqref{eq.infty.p} 
in $\Omega_k$, such  that $\Omega_{k} \in \Xi_{\gamma_k, \zeta_k}( B_r)$ 
with $\gamma_k, \zeta_k =  o(1)$ as $k \to \infty$, that is,
$$
 |\lambda_{1,\infty}^D(\Omega_k) - \lambda_{1,\infty}^D( B_r)| < \gamma_k
\quad \text{and} \quad
 |\lambda_{1,\infty}^N(\Omega_k) - \lambda_{1,\infty}^N( B_r)| < \zeta_k,
$$
with $\gamma_k, \zeta_k =  o(1)$ as $k \to \infty$,
together with
\begin{equation}\label{Eqcont}
     |u_k(x) - v_{\infty}(x)|> \varepsilon_0 \quad \text{in } \Omega_k \cap  B_r,
\end{equation}
for every $k \in \mathbb{N}$.

Using our previous results, we can suppose that every $\Omega_k \subset  B_{2r}$. 
Then, by extending $u_k$ to zero outside of $\Omega_k$, we may assume that 
$\{u_k\}_{k\in \mathbb{N}} \subset W_0^{1, \infty}( B_{2r})$. In this context,
standard arguments using viscosity theory show that, up to a subsequence, 
$u_k \to u_{\infty}$ uniformly in
$\overline{ B_{2r}}$, being the limit $u_\infty$ a normalized eigenfunction 
for some domain $\hat{\Omega}$ with $\hat{\Omega} \Subset B_{2r}$.
Moreover, we have that $\lambda^D_{1,\infty} (\Omega_k) 
\to \lambda^D_{1,\infty} (\hat{\Omega})$.

According to Theorem \ref{Mainthm2}, $\Omega_k \to  B_r$ as $k \to \infty$. 
By the previous sentences we conclude that $\hat{\Omega} =  B_r$. 
Now, by uniqueness of solutions to \eqref{eq.infty.p} in $ B_r$ we conclude 
that $u_{\infty} = v_{\infty}$. However, this contradicts \eqref{Eqcont} for 
$k \gg 1$ (large enough). Such a contradiction proves the theorem.
\end{proof}

\section{Examples}\label{sect-examples}

Given a fixed ball $ B$ and a domain $\Omega$ having  both of them the same volume, 
Theorem \ref{Mainthm} says that if the $\infty$-eigenvalues are close each other 
then $\Omega$ is almost ball-shaped uniformly.
The following examples illustrate Theorems \ref{Mainthm} and \ref{Mainthm2}.


\begin{example} \label{examp3.1}\rm
The converse of Theorem \ref{Mainthm} (and Theorem  \ref{Mainthm2}) 
is not true: given a fixed ball $ B$, clearly, there are domains 
$\Omega$ fulfilling \eqref{ecccc} such that the difference between 
the Neumann (and Dirichlet) eigenvalues in $\Omega$ and in $ B$ is not small. 
Let us present some illustrative examples.

\begin{enumerate}
  \item A stadium. Let  $ B$ be the unit ball in $\mathbb{R}^2$ and $\Omega$ 
the stadium domain given in Figure \ref{fig1} (a) with
$\ell=\frac{\pi(1-\varepsilon^2)}{2\varepsilon}$. In this case $\mathscr{L}^n( B)=\mathscr{L}^n(\Omega)=\pi$
for any $0<\varepsilon<1$. However,
$$
\lambda_{1,\infty}^N( B)=1, \quad \lambda_{1,\infty}^N(\Omega)
=\frac{2}{\operatorname{diam}(\Omega)}= \frac{4\varepsilon}{\pi+\varepsilon^2(4-\pi)}<\frac13
\quad \text{if } \varepsilon<\frac14.
$$

\item A ball with holes. If $\Omega=B(0,\sqrt{1+\varepsilon^2}) \setminus B(0,\varepsilon)$
is the domain given in Figure \ref{fig1} (b), then $\mathscr{L}^n( B)=\mathscr{L}^n(\Omega)=\pi$,
 however
$$
\lambda_{1,\infty}^D( B)=1, \quad \lambda_{1,\infty}^D(\Omega)
=\frac{1}{\sqrt{1+\varepsilon^2}}>\frac32 \quad \text{if } \frac34<\varepsilon<1.
$$

\item A ball with thin tubular branches. If  $\Omega$ is the domain given 
in Figure \ref{fig1} (c), the condition $\mathscr{L}^n( B)= \mathscr{L}^n(\Omega)$ gives the relation
$$
	r(r+\varepsilon) + \varepsilon(\tfrac{1}{\pi}+\tfrac{\varepsilon}{2})=1, \quad
 \operatorname{diam}(\Omega)=1+r+\pi(1+r).
$$
For instance, if we take $\varepsilon=10^{-3}$ it follows that $r\sim 0.999465$ and then
$$
\lambda_{1,\infty}^N( B)=\frac{2}{\operatorname{diam}( B)}=1, \quad
\lambda_{1,\infty}^N(\Omega)=\frac{2}{\operatorname{diam}(\Omega)}\sim 0.2415.
$$
\end{enumerate}

\begin{figure}[ht]
\begin{center}
 \includegraphics[width=0.8\textwidth]{fig1} % dibujo1.pdf
\end{center}
\caption{Three examples of domains}
\label{fig1} %\label{dib3} 
\end{figure}

Hence, in view of these examples we conclude that a domain that has 
Dirichlet and Neumann $\infty$-eigenvalues
close to the ones for the ball is close to a ball not only in the sense that
$\mathscr{L}^n\left(\Omega\Delta  B_{r}\right)$ is small but it
can not contain holes deep inside (small holes near the boundary are allowed) 
and can not have thin tubular branches.
\end{example}


\begin{example} \label{examp3.2} \rm
The regular polygon $\mathbb{P}_k$ of $k$-sides ($k\geq 3$) centered at the 
origin such that $\mathscr{L}^n(\mathbb{P}_k) = \mathscr{L}^n( B_r)$ satisfies
$$
   | \lambda_{1,\infty}^D(\mathbb{P}_k) - \lambda_{1,\infty}^D(B_r)| = \delta_1
\quad \text{and} \quad 
|\lambda_{1,\infty}^N(B_r) - \lambda_{1,\infty}^N(\mathbb{P}_k)| = \delta_2,
$$
where
$$
	\delta_1=\frac{1}{r\sqrt{\frac{\pi}{k\tan(\frac{\pi}{k})}}}- \frac{1}{r} 
\quad \text{ and} \quad 
\delta_2 = \frac{1}{r} - \frac{1}{r\sqrt{\frac{2\pi}{k\sin(\frac{2\pi}{k})}}}.
$$
Therefore, we can recover the well known convergence $\mathbb{P}_k \to  B_r$ 
as $k \to \infty$.
\end{example}

\begin{example} \label{examp3.3}\rm
 Given $k \in \mathbb{N}$ and positive constants 
$\mathfrak{a}^k_1,\cdots, \mathfrak{a}^k_n$, the $n-$dimensional ellipsoid given by 
$$ 
\mathcal{E}_k := \Big\{(x_1, \cdots, x_n) : \sum_{i=1}^{n} 
\Big(\frac{x_i}{\mathfrak{a}^k_i}\Big)^2<1\Big\}
$$ 
such that $\mathscr{L}^n(\mathcal{E}_k) = \mathscr{L}^n( B_r)$ satisfies
$$
   | \lambda_{1,\infty}^D(\mathcal{E}_k) - \lambda_{1,\infty}^D(B_r)| =\delta_1
\quad \text{and} \quad 
|\lambda_{1,\infty}^N(B_r) - \lambda_{1,\infty}^N(\mathcal{E}_k)| = \delta_2,
$$
where
$$
	\delta_1  = \frac{1}{\min_i\{\mathfrak{a}^k_i\}} - \frac{1}{r},
 \quad \text{and } \quad
 \delta_2=\frac{1}{r} - \frac{1}{\max_i\{\mathfrak{a}^k_i\}}.
$$
Therefore, we recover the fact that if $\min_i \mathfrak{a}^k_i \to r$ and 
$\max_i \mathfrak{a}^k_i \to r$ as $k \to \infty$, then $\mathcal{E}_k \to  B_r$.
\end{example}


\begin{example} \label{examp3.4} \rm
 Given $r>0$ let $k_0 \in \mathbb{N}$ such that 
$\frac{1}{2\pi} \sqrt{\frac{4}{k^2} + 4 \pi^2r^2}> \frac{1}{k \pi}$ for all 
$k \geq k_0$. For each $k\in \mathbb{N}$ let $\Omega_k$ be the planar stadium 
domain from Figure 1 (a) with $l_k = \frac{1}{k}$ and 
$\varepsilon_k = \frac{1}{2\pi} \sqrt{\frac{4}{k^2} + 4 \pi^2r^2}-\frac{1}{k \pi}$. 
It is easy to check that $\Omega_k \in \Xi_{\frac{1}{\varepsilon_k}-\frac{1}{r}, 
\frac{2}{2\varepsilon_k + \frac{1}{k}}-\frac{1}{r}}( B_r)$. 
 Furthermore, in  this case we have that the eigenfunctions are explicit and 
given by
$$
   u_k(x) = \frac{1}{\varepsilon_k}\operatorname{dist}(x, \partial \Omega_k).
$$
Finally, form Corollary \ref{CorConv}
$$
  u_k(x) \to v_{\infty}(x) = \frac{1}{r}\operatorname{dist}(x, \partial  B_r) 
\quad \text{locally uniformly in $B_r$ as } k \to \infty.
$$
\end{example}




\subsection*{Acknowledgments}
This work was supported by Consejo Nacional de Investigaciones 
Cient\'{i}ficas y T\'{e}cnicas (CONICET-Argentina). 
The authors thank the anonymous referee for the suggestions that improved 
the presentation of the article. 
JVS would like to thank the Dept. of Math. and FCEyN of the Universidad 
de Buenos Aires for providing an excellent working environment 
and scientific atmosphere during his CONICET Postdoctoral Fellowship.

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