Uniform stability of the ball with respect to the first Dirichlet and Neumann $\infty-$eigenvalues

TL;DR

This paper proves that domains with first Dirichlet and Neumann infinity-eigenvalues close to those of a ball are themselves close to a ball, establishing a stability result for these eigenvalues and eigenfunctions under volume constraints.

Contribution

It provides a quantitative stability analysis linking eigenvalue proximity to geometric closeness to a ball for the first Dirichlet and Neumann infinity-eigenvalues.

Findings

Domains with eigenvalues close to those of a ball are geometrically close to a ball.

Explicit bounds relate eigenvalue differences to inclusion relations between the domain and a ball.

Stability results for the eigenfunctions are also established.

Abstract

In this note we analyze how perturbations of a ball $\mathfrak{B}_r \subset \mathbb{R}^n$ behaves in terms of their first (non-trivial) Neumann and Dirichlet $\infty-$eigenvalues when a volume constraint $\\mathscr{L}^n(\Omega) = \mathscr{L}^n(\mathfrak{B}_r)$ 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 $\mathfrak{B}_r$. In fact, we show that, if $$ |\lambda_{1,\infty}^D(\Omega) - \lambda_{1,\infty}^D(\mathfrak{B}_r)| = \delta_1 \quad \text{and} \quad |\lambda_{1,\infty}^N(\Omega) - \lambda_{1,\infty}^N(\mathfrak{B}_r)| = \delta_2, $$ then there are two balls such that $$\mathfrak{B}_{\frac{r}{\delta_1 r+1}} \subset \Omega \subset \mathfrak{B}_{\frac{r+\delta_2 r}{1-\delta_2 r}}.$$ In addition, we also obtain a result concerning stability of the Dirichlet $\infty-$eigen-functions.

Equations

$$|{\lambda }_{1,\infty }^D(\Omega )-{\lambda }_{1,\infty }^D({\mathfrak{B}}_r)|={\delta }_1\text{and}|{\lambda }_{1,\infty }^N(\Omega )-{\lambda }_{1,\infty }^N({\mathfrak{B}}_r)|={\delta }_2,$$

$${\mathfrak{B}}_{\frac{r}{{\delta }_{1}r+1}}\subset \Omega \subset {\mathfrak{B}}_{\frac{r+{\delta }_{2}r}{1-{\delta }_{2}r}}.$$

$$\left\{\begin{array}[]{rclcl}-\Delta_{p}u&=&\lambda_{1,p}^{D}(\Omega)|u|^{p-2}u&\text{in}&\Omega\\ u&=&0&\text{on}&\partial\Omega\end{array}\right.$$

$${\lambda }_{1,p}^D(\Omega ):=\underset{u\in W_0^{1,p}(\Omega )\setminus \{0\}}{\inf }\left\{{\int }_{\Omega }|\nabla u{|}^{p}dx:{\int }_{\Omega }|u{|}^{p}dx=1\right\}.$$

$${\lambda }_{1,p}^D(\Omega )\ge {\lambda }_{1,p}^D(\mathfrak{B}),$$

$${\lambda }_{1,p}^D(\Omega )\ge {\lambda }_{1,p}^D(\mathfrak{B})\left(1+{\gamma }_{p,n}(\mathcal{S}(\Omega ){)}^{2+p}\right),$$

$$\mathcal{S}(\Omega ):=\underset{x_0\in {\mathbb{R}}^n}{\inf }\left\{\frac{{\mathcal{L}}^n(\Omega △{\mathfrak{B}}_r(x_0))}{{\mathcal{L}}^n(\Omega )}:{\mathcal{L}}^n({\mathfrak{B}}_r(x_0))={\mathcal{L}}^n(\Omega )\right\},$$

$$\infty$$

$$\left\{\begin{array}[]{rclcl}\min\Big{\{}-\Delta_{\infty}v_{\infty},|\nabla v_{\infty}|-\lambda_{1,\infty}^{D}(\Omega)v_{\infty}\Big{\}}&=&0&\text{in}&\Omega\\ v_{\infty}&=&0&\text{on}&\partial\Omega.\end{array}\right.$$

$$\lambda_{1,\infty}^{D}(\Omega)=\Big{(}\max_{x\in\Omega}\mathrm{dist}(x,\partial\Omega)\Big{)}^{-1}.$$

$$\left\{\begin{array}[]{rclcl}-\Delta_{p}u&=&\lambda_{1,p}^{N}(\Omega)|u|^{p-2}u&\text{in}&\Omega\\ \displaystyle|\nabla u|^{p-2}\tfrac{\partial u}{\partial\nu}&=&0&\text{on}&\partial\Omega.\end{array}\right.$$

$${\lambda }_{1,p}^N(\Omega ):=\underset{u\in W^{1,p}(\Omega )}{\inf }\left\{{\int }_{\Omega }|\nabla u{|}^{p}dx:{\int }_{\Omega }|u{|}^{p}dx=1\text{and}{\int }_{\Omega }|u{|}^{p-2}udx=0\right\}.$$

$${\lambda }_{1,p}^N(\Omega )\ge (p-1){\left(\frac{2\pi }{p\mathrm{diam}(\Omega )\sin (\frac{\pi }{p})}\right)}^p.$$

$$\infty$$

$$\left\{\begin{array}[]{rclcl}\min\Big{\{}-\Delta_{\infty}v_{\infty},|\nabla v_{\infty}|-\lambda_{1,\infty}^{N}(\Omega)v_{\infty}\Big{\}}&=&0&\text{in}&\Omega\cap\{v>0\}\\ \max\Big{\{}-\Delta_{\infty}v_{\infty},-|\nabla v_{\infty}|-\lambda_{1,\infty}^{N}(\Omega)v_{\infty}\Big{\}}&=&0&\text{in}&\Omega\cap\{v<0\}\\ -\Delta_{\infty}v_{\infty}&=&0&\text{in}&\Omega\cap\{v=0\}\\ \displaystyle\frac{\partial v_{\infty}}{\partial\nu}&=&0&\text{in}&\partial\Omega.\end{array}\right.$$

$${\lambda }_{1,\infty }^N(\Omega )=\frac{2}{\mathrm{diam}(\Omega )},$$

$$\mathrm{diam}(\Omega ):=\underset{\bar{\Omega }\times \bar{\Omega }}{\max }{d}_{\Omega }(x,y)=\underset{\partial \Omega \times \partial \Omega }{\max }{d}_{\Omega }(x,y),$$

$$|{\lambda }_{1,\infty }^D(\Omega )-{\lambda }_{1,\infty }^D({\mathfrak{B}}_r)|={\delta }_1\text{and}|{\lambda }_{1,\infty }^N(\Omega )-{\lambda }_{1,\infty }^N({\mathfrak{B}}_r)|={\delta }_2,$$

$${\mathfrak{B}}_{\frac{r}{{\delta }_{1}r+1}}\subset \Omega \subset {\mathfrak{B}}_{\frac{r+{\delta }_{2}r}{1-{\delta }_{2}r}}.$$

$$|{\lambda }_{1,\infty }^D({\Omega }_k)-{\lambda }_{1,\infty }^D({\mathfrak{B}}_r)|=\text{o}(1)\text{and}|{\lambda }_{1,\infty }^N(\Omega )-{\lambda }_{1,\infty }^N({\mathfrak{B}}_r)|=\text{o}(1)\text{as }k\to \infty ,$$

$${\Omega }_k\to {\mathfrak{B}}_r$$

$$d_{\mathcal{H}}(\Omega_{k},\mathfrak{B}_{r}):=\max\Big{\{}\,\sup_{{x\in\Omega_{k}}}\inf_{{y\in\mathfrak{B}_{r}}}d(x,y),\,\sup_{{y\in\mathfrak{B}_{r}}}\inf_{{x\in\Omega_{k}}}d(x,y)\,\Big{\}}\to 0.$$

$$\max \left\{{\mathcal{L}}^n\left(\Omega △{\mathfrak{B}}_{\frac{r}{{\delta }_{1}r+1}}\right),{\mathcal{L}}^n\left(\Omega △{\mathfrak{B}}_{\frac{r+{\delta }_{2}r}{1-{\delta }_{2}r}}\right)\right\}\le \mathfrak{C}(n,{\delta }_i,r)r^n.$$

$$|{\lambda }_{1,\infty }^D(\Omega )-{\lambda }_{1,\infty }^D({\mathfrak{B}}_r)|<{\delta }_1\text{and}|{\lambda }_{1,\infty }^N(\Omega )-{\lambda }_{1,\infty }^N({\mathfrak{B}}_r)|<{\delta }_2,$$

$$|u(x)-v_{\infty }(x)|<\epsilon \text{in}\Omega \cap {\mathfrak{B}}_r,$$

$$v_{\infty }(x)=1-\frac{|x|}{r}$$

$$|{\lambda }_{1,\infty }^D({\Omega }_k)-{\lambda }_{1,\infty }^D({\mathfrak{B}}_r)|=\text{o}(1)\text{and}|{\lambda }_{1,\infty }^N({\Omega }_k)-{\lambda }_{1,\infty }^N({\mathfrak{B}}_r)|=\text{o}(1)\text{as }k\to \infty .$$

$$u_k\to v_{\infty }\text{locally uniformly}\text{in}{\mathfrak{B}}_r,$$

$$v_{\infty }(x)=1-\frac{|x|}{r}$$

$$-{\mathcal{Q}}_{p}u:=-\mathrm{div}({\mathbb{F}}^{p-1}(\nabla u){\mathbb{F}}_{\xi }(\nabla u)),$$

Full text

[TABLE]

then there are two balls such that

[TABLE]

In addition, we also obtain a result concerning stability of the Dirichlet $\infty-$eigen-functions.

1. Introduction

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain (connected open subset) with smooth boundary, $1<p<\infty$ and $\Delta_{p}u\mathrel{\mathop{:}}=\mathrm{div}(|\nabla u|^{p-2}\nabla u)$ (the standard $p$-Laplacian operator). Historically (cf. [13]), it well-known that the first eigenvalue (referred as the principal frequency in physical models) of the $p-$Laplacian Dirichlet eigenvalue problem

[TABLE]

can be characterized variationally as the minimizer of the following (normalized) problem:

[TABLE]

In the theory of shape optimization and nonlinear 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 (p-Dirichlet) 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 (p-Dirichlet). Nevertheless, lower bounds are a more challenging task. In this direction we have the remarkable Faber-Krahn inequality: Among all domains of prescribed volume the ball minimizes (p-Dirichlet). More precisely,

[TABLE]

where $\mathfrak{B}$ is the $n$-dimensional ball such that $\mathscr{L}^{n}(\Omega)=\mathscr{L}^{n}(\mathfrak{B})$ (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 (p-Dirichlet) in terms of the perimeter (resp. diameter) of $\Omega$ are also available (cf. [1] and [14, page 224], and the references therein). Recently, stability estimates for certain geometric inequalities were established in [10], thereby providing an improved version of (1.2) by adding a suitable remainder term, i.e.,

[TABLE]

where $\mathcal{S}(\Omega)$ is the so-called Fraenkel asymmetry of $\Omega$, which is precisely defined as

[TABLE]

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 [2], [4], [9] 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 (p-Dirichlet), one obtains

[TABLE]

see [11]. Concerning the limit equation, also in [11] it is proved that any family of normalized eigenfunctions $\{u_{p}\}_{p>1}$ to (p-Dirichlet) converges (up to a subsequence) locally uniformly to $u_{\infty}\in W^{1,\infty}_{0}(\Omega)$, a minimizer for $\infty$-Dirichlet with $\|u_{\infty}\|_{L^{\infty}(\Omega)}=1$. Moreover, the pair $(u_{\infty},\lambda_{1,\infty}^{D}(\Omega))$ is a nontrivial solution to

[TABLE]

Solutions to (1.3) must be understood in the viscosity sense (cf. [6] for a survey) and $\Delta_{\infty}u(x)\mathrel{\mathop{:}}=\nabla u(x)^{T}D^{2}u(x)\cdot\nabla u(x)$ is the well-known $\infty-$Laplace operator. In addition, also in [11], it is given an interesting and useful geometrical characterization for ($\infty$-Dirichlet):

[TABLE]

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 (1.3) we refer to [12], [15] and [18].

Now, let us turn our attention to Neumann boundary conditions and consider the following eigenvalue problem:

[TABLE]

As before, we stress that the first non-zero eigenvalue of (1.5) can also be characterized variationally as the minimizer of the following normalized problem:

[TABLE]

The celebrated Payne-Weinberger inequality provides a lower bound (on any convex domain $\Omega\subset\mathbb{R}^{n}$) for the first (non-trivial) Neumann $p-$eigenvalue (cf. [8] and [17])

[TABLE]

For a stability estimate for this problem with $p=2$ we refer to [2].

When $p\to\infty$, the minimization problem (p-Neumann) becomes

[TABLE]

see [7] and [16]. Concerning the limit equation, also in [7] and [16], it is proved that any family of normalized eigenfunctions $\{u_{p}\}_{p>1}$ to (p-Neumann) converges (up to subsequence) locally uniformly to $u_{\infty}\in W^{1,\infty}_{0}(\Omega)$ with $\|u_{\infty}\|_{L^{\infty}(\Omega)}=1$. Moreover, the pair $(u_{\infty},\lambda_{1,\infty}^{N}(\Omega))$ is a nontrivial solution to

[TABLE]

In addiction, we have the following geometrical characterization for $\lambda_{1,\infty}^{N}(\Omega)$:

[TABLE]

where the intrinsic diameter of $\Omega$ is defined as

[TABLE]

being $d_{\Omega}(x,y)$ the geodesic distance given by $d_{\Omega}(x,y)=\inf_{\gamma}\text{Long}(\gamma)$, 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 (1.8) of ($\infty$-Neumann) yields several interesting consequences:

• ✓

If $\mathscr{L}^{n}(\Omega)=\mathscr{L}^{n}(\mathfrak{B})$, $\mathfrak{B}$ being a ball, then $\lambda_{1,\infty}^{N}(\Omega)\leq\lambda_{1,\infty}^{N}(\mathfrak{B})$, which establishes a Szegö-Weinberger type inequality: among all domains of prescribed volume the ball maximizes ($\infty$-Neumann).

• ✓

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

• ✓

The Payne-Weinberger inequality, (1.6), becomes an equality when $p=\infty$.

Taking 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 $\mathfrak{B}_{r}$ of the same Lebesgue measure, then $\Omega$ is uniformly “almost” ball-shaped.

Theorem 1.1**.**

Let $\Omega$ be an open domain satisfying $\mathscr{L}^{n}(\Omega)=\mathscr{L}^{n}(\mathfrak{B}_{r})$. If for some $\delta_{i}>0$ ($i=1,2$) small enough it holds that

[TABLE]

then there are two balls such that

[TABLE]

The previous theorem implies the following convergence result.

Theorem 1.2**.**

Let $\{\Omega_{k}\}_{k\in\mathbb{N}}$ be a family of uniformly bounded domains satisfying $\mathscr{L}^{n}(\Omega_{k})=\mathscr{L}^{n}(\mathfrak{B}_{r})$. If

[TABLE]

then

[TABLE]

in the sense that the Hausdorff distance between $\Omega$ and a ball $\mathfrak{B}_{r}$ goes to zero, i.e.,

[TABLE]

Note that our results imply that

[TABLE]

where $\mathfrak{C}(n,\delta_{i},r)=\omega_{n}\max\{(\delta_{1}r+1)^{n}-1,(n-1)\delta_{2}\}\to 0$ 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 1.2).

Another important question in this theory consists on how the corresponding $\infty-$ground states (solutions to (1.3)) 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 (1.3) may have multiple solutions (the first eigenvalue may not be simple), see [5] and [18].

Theorem 1.3**.**

Let $\Omega$ be an open domain satisfying $\mathscr{L}^{n}(\Omega)=\mathscr{L}^{n}(\mathfrak{B}_{r})$. Given $\varepsilon>0$ there are $\delta_{i}(\varepsilon)>0$ ($i=1,2$) small enough such that: if

[TABLE]

then

[TABLE]

where

[TABLE]

is the normalized $\infty-$ground state to (1.3) in $\mathfrak{B}_{r}$.

Theorem 1.3 can be rewritten as follows:

Corollary 1.4**.**

Let $\{u_{k}\}_{k\in\mathbb{N}}$ be a family of normalized solutions to (1.3) in $\Omega_{k}$ such that

[TABLE]

Then,

[TABLE]

where

[TABLE]

is the normalized $\infty-$ground state to (1.3) in $\mathfrak{B}_{r}$.

Our approach can be applied for other classes of operators with $p-$Laplacian type structure. We can deal with $p-$Laplacian type problems involving an anisotropic $p$-Laplacian operator

[TABLE]

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 [3]. Here, to obtain results similar to ours, one has to replace Euclidean balls with balls in the norm $\mathbb{F}$.

The paper is organized as follows: in Section 2 we prove our main stability results including the behavior of the corresponding $\infty-$eigenfunctions and in Section 3 we collect several examples that illustrate our results.

2. Proof of the Main Theorems

Before proving our main result we introduce some notations which will be used throughout this section. Given a bounded domain $\Omega\subset\mathbb{R}^{n}$ and a ball $\mathfrak{B}_{r}\subset\mathbb{R}^{n}$ of radius $r>0$ we denote $\lambda_{1,\infty}^{D}(\Omega)$ and $\lambda_{1,\infty}^{D}(\mathfrak{B}_{r})$ the first Dirichlet eigenvalues (1.4) in $\Omega$ and in $\mathfrak{B}_{r}$, respectively; analogously, $\lambda_{1,\infty}^{N}(\Omega)$ and $\lambda_{1,\infty}^{N}(\mathfrak{B}_{r})$ stand for the first nontrivial Neumann eigenvalues (1.8) in $\Omega$ and in $\mathfrak{B}_{r}$.

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:

[TABLE]

Notice that, $\Xi_{0,0}(\mathfrak{B}_{r})$ consists of the family of all balls with radius $r>0$. Another important remark is that the elements of $\Xi_{\delta_{1},\delta_{2}}(\mathfrak{B}_{r})$ are invariant by rigid movements (rotations, translations, etc).

Similarly, we can define the class $\Xi^{D}_{\delta_{1}}(\mathfrak{B}_{r})$ (resp. $\Xi^{N}_{\delta_{2}}(\mathfrak{B}_{r})$) as being $\Xi_{\delta_{1},\delta_{2}}(\mathfrak{B}_{r})$ 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.

Lemma 2.1**.**

If $\Omega\in\Xi^{D}_{\delta_{1}}(\mathfrak{B}_{r})$ then there exists a ball such that

[TABLE]

Moreover,

[TABLE]

where $\mathfrak{c}=\text{o}(1)$ as $\delta_{1}\to 0$.

Proof.

According to (1.4) we have that

[TABLE]

It follows that

[TABLE]

and then there is ball such that

[TABLE]

Finally,

[TABLE]

and the lemma follows. ∎

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

Lemma 2.2**.**

If $\Omega\in\Xi^{N}_{\delta_{2}}(\mathfrak{B}_{r})$ then there is a ball such that

[TABLE]

Moreover,

[TABLE]

Proof.

Using (1.8) we have that

[TABLE]

It follows that

[TABLE]

and then there exists a ball such that

[TABLE]

Moreover,

[TABLE]

and the lemma follows. ∎

Proof of Theorem 1.1.

The proof of Theorem 1.1 follows as an immediate consequence of Lemmas 2.1 and 2.2. ∎

Next, we will prove Theorem 1.2.

Proof of Theorem 1.2.

The hypothesis implies that $\Omega_{k}\in\Xi_{\delta_{k},\varepsilon_{k}}(\mathfrak{B}_{r})$ for $\delta_{k},\varepsilon_{k}=\text{o}(1)$ as $k\to\infty$. For this reason, by Theorem 1.1 there are two balls such that

[TABLE]

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 $\mathfrak{B}_{r}$ such that $\Omega_{k}\to\mathfrak{B}_{r}$ as $k\to\infty$. ∎

Proof of Theorem 1.3.

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 (1.3) in $\Omega_{k}$, such that $\Omega_{k}\in\Xi_{\gamma_{k},\zeta_{k}}(\mathfrak{B}_{r})$ with $\gamma_{k},\zeta_{k}=\text{o}(1)$ as $k\to\infty$, that is,

[TABLE]

with $\gamma_{k},\zeta_{k}=\text{o}(1)$ as $k\to\infty$, together with

[TABLE]

for every $k\in\mathbb{N}$.

Using our previous results, we can suppose that every $\Omega_{k}\subset\mathfrak{B}_{2r}$. Then, by extending $u_{k}$ to zero outside of $\Omega_{k}$, we may assume that $\{u_{k}\}_{k\in\mathbb{N}}\subset W_{0}^{1,\infty}(\mathfrak{B}_{2r})$. In this context, standard arguments using viscosity theory show that, up to a subsequence, $u_{k}\to u_{\infty}$ uniformly in $\overline{\mathfrak{B}_{2r}}$, being the limit $u_{\infty}$ a normalized eigenfunction for some domain $\hat{\Omega}$ with $\hat{\Omega}\Subset\mathfrak{B}_{2r}$. Moreover, we have that $\lambda^{D}_{1,\infty}(\Omega_{k})\to\lambda^{D}_{1,\infty}(\hat{\Omega})$.

According to Theorem 1.2, $\Omega_{k}\to\mathfrak{B}_{r}$ as $k\to\infty$. By the previous sentences we conclude that $\hat{\Omega}=\mathfrak{B}_{r}$. Now, by uniqueness of solutions to (1.3) in $\mathfrak{B}_{r}$ we conclude that $u_{\infty}=v_{\infty}$. However, this contradicts (2.1) for $k\gg 1$ (large enough). Such a contradiction proves the theorem. ∎

3. Examples

Given a fixed ball $\mathfrak{B}$ and a domain $\Omega$ having both of them the same volume, Theorem 1.1 says that if the $\infty-$eigenvalues are close each other then $\Omega$ is almost ball-shaped uniformly. The following examples illustrate Theorem 1.1 and 1.2.

Example 3.1**.**

The reciprocal in Theorem 1.1 (and Theorem 1.2) is not true: given a fixed ball $\mathfrak{B}$, clearly, there are domains $\Omega$ fulfilling (1.9) such that the difference between the Neumann (and Dirichlet) eigenvalues in $\Omega$ and in $\mathfrak{B}$ is not small. Let us present some illustrative examples.

1. (1)

A stadium. Let $\mathfrak{B}$ be the unit ball in $\mathbb{R}^{2}$ and $\Omega$ the stadium domain given in Figure 1 (a) with $\ell=\frac{\pi(1-\varepsilon^{2})}{2\varepsilon}$. In this case $\mathscr{L}^{n}(\mathfrak{B})=\mathscr{L}^{n}(\Omega)=\pi$ for any $0<\varepsilon<1$. However,

[TABLE] 2. (2)

A ball with holes. If $\Omega=B(0,\sqrt{1+\varepsilon^{2}})\setminus B(0,\varepsilon)$ is the domain given in Figure 1 (b), then $\mathscr{L}^{n}(\mathfrak{B})=\mathscr{L}^{n}(\Omega)=\pi$, however

[TABLE] 3. (3)

A ball with thin tubular branches. If $\Omega$ is the domain given in Figure 1 (c), the condition $\mathscr{L}^{n}(\mathfrak{B})=\mathscr{L}^{n}(\Omega)$ gives the relation

[TABLE]

For instance, if we take $\varepsilon=10^{-3}$ it follows that $r\sim 0.999465$ and then

[TABLE]

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 $\mathscr{L}^{n}\left(\Omega\bigtriangleup\mathfrak{B}_{r}\right)$ is small but it can not contain holes deep inside (small holes near the boundary are allowed) and can not have thin tubular branches.

Example 3.2**.**

The regular polygon $\mathbb{P}_{k}$ of $k-$sides ($k\geq 3$) centered at the origin such that $\mathscr{L}^{n}(\mathbb{P}_{k})=\mathscr{L}^{n}(\mathfrak{B}_{r})$ satisfies

[TABLE]

where

[TABLE]

Therefore, we can recover the well known convergence $\mathbb{P}_{k}\to\mathfrak{B}_{r}$ as $k\to\infty$.

Example 3.3**.**

Given $k\in\mathbb{N}$ and positive constants $\mathfrak{a}^{k}_{1},\cdots,\mathfrak{a}^{k}_{n}$, the $n-$dimensional ellipsoid given by

[TABLE]

such that $\mathscr{L}^{n}(\mathcal{E}_{k})=\mathscr{L}^{n}(\mathfrak{B}_{r})$ satisfies

[TABLE]

where

[TABLE]

Therefore, we recover the fact that if $\min\limits_{i}\mathfrak{a}^{k}_{i}\to r$ and $\max\limits_{i}\mathfrak{a}^{k}_{i}\to r$ as $k\to\infty$, then $\mathcal{E}_{k}\to\mathfrak{B}_{r}$.

Example 3.4**.**

Given $r>0$ let $k_{0}\in\mathbb{N}$ such that $\frac{1}{2\pi}\sqrt{\frac{4}{k^{2}}+4\pi^{2}r^{2}}>\frac{1}{k\pi}$ 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 $\varepsilon_{k}=\frac{1}{2\pi}\sqrt{\frac{4}{k^{2}}+4\pi^{2}r^{2}}-\frac{1}{k\pi}$. It is easy to check that $\Omega_{k}\in\Xi_{\frac{1}{\varepsilon_{k}}-\frac{1}{r},\frac{2}{2\varepsilon_{k}+\frac{1}{k}}-\frac{1}{r}}(\mathfrak{B}_{r})$. Furthermore, in this case we have that the eigenfunctions are explicit and given by

[TABLE]

Finally, form Corollary 1.4

[TABLE]

Acknowledgments

This work was supported by Consejo Nacional de Investigaciones Cien-tíficas y Técnicas (CONICET-Argentina). JVS would like to thank the Dept. of Math. and FCEyN Universidad de Buenos Aires for providing an excellent working environment and scientific atmosphere during his Postdoctoral program.