On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operators

TL;DR

This paper studies the second eigenvalue of the anisotropic p-Laplacian, providing bounds and inequalities, and explores the behavior as p approaches infinity, extending classical spectral results to anisotropic operators.

Contribution

It establishes a Hong-Krahn-Szego type inequality for the second eigenvalue of the anisotropic p-Laplacian and analyzes its limit as p tends to infinity, which is novel for this class of operators.

Findings

Lower bound for the second eigenvalue among sets of fixed measure

Validity of a Hong-Krahn-Szego type inequality in the anisotropic setting

As p approaches infinity, the eigenvalue problem converges to a limit problem

Abstract

Let $\Omega$ be a bounded open set of $\mathbb R^{n}$, $n\ge 2$. In this paper we mainly study some properties of the second Dirichlet eigenvalue $\lambda_{2}(p,\Omega)$ of the anisotropic $p$-Laplacian \[ -\mathcal Q_{p}u:=-\textrm{div} \left(F^{p-1}(\nabla u)F_\xi (\nabla u)\right), \] where $F$ is a suitable smooth norm of $\mathbb R^{n}$ and $p\in]1,+\infty[$. We provide a lower bound of $\lambda_{2}(p,\Omega)$ among bounded open sets of given measure, showing the validity of a Hong-Krahn-Szego type inequality. Furthermore, we investigate the limit problem as $p\to+\infty$.

Equations

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

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

$$\left\{\begin{array}[]{ll}-\mathcal{Q}_{p}u=\lambda|u|^{p-2}u&\text{in}\ \Omega\\ u=0&\text{on}\ \partial\Omega\end{array}\right.$$

$${\lambda }_1(p,\Omega )=\underset{\phi \in W_0^{1,p}(\Omega )\setminus \{0\}}{\min }\frac{{\int }_{\Omega }{F}^p(\nabla \phi )dx}{{\int }_{\Omega }|\phi {|}^{p}dx}.$$

$${\lambda }_1(p,\Omega )\ge {\lambda }_1(p,{\mathcal{W}}_R)$$

$$\underset{p\to \infty }{\lim }{\lambda }_1(p,\Omega {)}^{\frac{1}{p}}=\frac{1}{{\rho }_F(\Omega )},$$

$${\lambda }_2(p,\Omega ):=\{\begin{array}{ll}\min \{\lambda >{\lambda }_1(p,\Omega ):\lambda \text{is an eigenvalue}\}& \text{if }{\lambda }_1(p,\Omega )\text{ is simple}\\ {\lambda }_1(p,\Omega )& \text{otherwise},\end{array}$$

$${\lambda }_2(p,\Omega )\ge {\lambda }_2(p,\tilde{\mathcal{W}}),$$

$$\underset{p\to \infty }{\lim }{\lambda }_2(p,\Omega {)}^{\frac{1}{p}}=\frac{1}{{\rho }_{2,F}(\Omega )},$$

$$\xi \in {\mathbb{R}}^n\mapsto F(\xi )\in [0,+\infty [,$$

$$F(t\xi )=|t|F(\xi ),t\in \mathbb{R},\xi \in {\mathbb{R}}^n,$$

$$a|\xi |\le F(\xi ),\xi \in {\mathbb{R}}^n,$$

$$F(\xi )\le b|\xi |,\xi \in {\mathbb{R}}^n.$$

$${\nabla }_{\xi }^2[F^p](\xi )\text{ is positive definite in }{\mathbb{R}}^n\setminus \{0\},$$

$${\mathcal{Q}}_{\mathrm{p}}[u]:=\mathrm{div}\left(\frac{1}{p}{\nabla }_{\xi }[F^p](\nabla u)\right)$$

$$\frac{1}{p}\sum _{i,j=1}^n{\nabla }_{{\xi }_i{\xi }_j}^2[F^p](\eta ){\xi }_i{\xi }_j\ge \gamma |\eta {|}^{p-2}|\xi {|}^2,$$

$${\nabla }_{\xi }^2[F^2](\xi )\text{ is positive definite in }{\mathbb{R}}^n\setminus \{0\},$$

$$F^o(v)=\underset{\xi \ne 0}{\sup }\frac{⟨\xi ,v⟩}{F(\xi )}.$$

$$F(v)=\underset{\xi \ne 0}{\sup }\frac{⟨\xi ,v⟩}{F^o(\xi )}.$$

$$|⟨\xi ,\eta ⟩|\le F(\xi )F^o(\eta ),\forall \xi ,\eta \in {\mathbb{R}}^n.$$

$$\mathcal{W}=\{\xi \in {\mathbb{R}}^n:F^o(\xi )<1\}$$

$$⟨{\nabla }_{\xi }F(\xi ),\xi ⟩=F(\xi ),⟨{\nabla }_{\xi }{F}^o(\xi ),\xi ⟩=F^o(\xi ),$$

$$F({\nabla }_{\xi }{F}^o(\xi ))=F^o({\nabla }_{\xi }F(\xi ))=1,\forall \xi \in {\mathbb{R}}^n\setminus \{0\},$$

$$F^o(\xi ){\nabla }_{\xi }F({\nabla }_{\xi }{F}^o(\xi ))=F(\xi ){\nabla }_{\xi }{F}^o({\nabla }_{\xi }F(\xi ))=\xi \forall \xi \in {\mathbb{R}}^n\setminus \{0\}.$$

$$\left\{\begin{array}[]{ll}-\operatorname{\mathcal{Q}_{p}}u=\lambda|u|^{p-2}u&\text{in }\Omega\\ u=0&\text{on }\partial\Omega.\end{array}\right.$$

$${\int }_{\Omega }⟨F^{p-1}(\nabla u){\nabla }_{\xi }F(\nabla u),\nabla \phi ⟩dx=\lambda {\int }_{\Omega }|u{|}^{p-2}u\phi dx$$

$$\lambda =\frac{{\int }_{\Omega }{F}^p(\nabla u)dx}{{\int }_{\Omega }|u{|}^{p}dx}>0.$$

$${\lambda }_1(p,\Omega )=\underset{\phi \in W_0^{1,p}(\Omega )\setminus \{0\}}{\min }\frac{{\int }_{\Omega }{F}^p(\nabla \phi )dx}{{\int }_{\Omega }|\phi {|}^{p}dx}.$$

$$[{\lambda }_1(p,\Omega ){]}^{\frac{1}{p}}\le \frac{s}{p}{\left(\frac{{\int }_{\Omega }|\psi {|}^{s-p}{F}^p(\nabla \psi )dx}{{\int }_{\Omega }|\psi {|}^{s}dx}\right)}^{\frac{1}{p}}\le \frac{s}{p}{\left(\frac{{\int }_{\Omega }{F}^s(\nabla \psi )dx}{{\int }_{\Omega }|\psi {|}^{s}dx}\right)}^{\frac{1}{s}}$$

$$|\Omega {|}^{p/N}{\lambda }_1(p,\Omega )\ge {\kappa }_N^{p/N}{\lambda }_1(p,\mathcal{W}).$$

$$\underset{k\to +\infty }{\lim }{\lambda }_k={\lambda }_1$$

Full text

[TABLE]

where $F$ is a suitable smooth norm of $\mathbb{R}^{n}$ and $p\in]1,+\infty[$. We provide a lower bound of $\lambda_{2}(p,\Omega)$ among bounded open sets of given measure, showing the validity of a Hong-Krahn-Szego type inequality. Furthermore, we investigate the limit problem as $p\to+\infty$.

MSC 2010: 35P15 - 35P30 - 35J60

Keywords: Nonlinear eigenvalue problems - Hong-Krahn-Szego inequality - Finsler metrics

1 Introduction

Let $\Omega$ be a bounded open set of $\mathbb{R}^{n}$, $n\geq 2$. The main aim of this paper is to study some properties of the Dirichlet eigenvalues of the anisotropic $p$-Laplacian operator:

[TABLE]

where $1<p<+\infty$, and $F$ is a sufficiently smooth norm on $\mathbb{R}^{n}$ (see Section 2 for the precise assumptions on $F$), namely the values $\lambda$ such that the problem

[TABLE]

admits a nontrivial solution in $W_{0}^{1,p}(\Omega)$.

The operator in (1) reduces to the $p$-Laplacian when $F$ is the Euclidean norm on $\mathbb{R}^{n}$. For a general norm $F$, $\mathcal{Q}_{p}$ is anisotropic and can be highly nonlinear. In literature, several papers are devoted to the study of the smallest eigenvalue of (2), denoted by $\lambda_{1}(p,\Omega)$, in bounded domains (see Section 2), which has the variational characterization

[TABLE]

Let $\Omega$ be a bounded domain. It is known (see [BFK, DG2]) that $\lambda_{1}(p,\Omega)$ is simple, the eigenfunctions have constant sign and it is isolated; moreover the only positive eigenfunctions are the first eigenfunctions. Furthermore, the Faber-Krahn inequality holds (see [BFK]):

[TABLE]

where $\mathcal{W}_{R}$ is the so-called Wulff shape, that is the ball with respect to the dual norm $F^{o}$ of $F$, having the same measure of $\Omega$ (see Section 2). Many other results are known for $\lambda_{1}(p,\Omega)$. The interested reader may refer, for example, to [BFK, BKJ, BGM, DG3, KN, P, WX]. As matter of fact, also different kind of boundary conditions have been considered as, for example, in the papers [DG, DGP] (Neumann case), [DG2] (Robin case).

Among the results contained in the quoted papers, we recall that if $\Omega$ is a bounded domain, it has been proved in [BKJ] that

[TABLE]

where $\rho_{F}(\Omega)$ is the anisotropic inradius of $\Omega$ with respect to the dual norm (see Section 2 for its definition), generalizing a well-known result in the Euclidean case contained in [JLM].

Actually, very few results are known for higher eigenvalues in the anisotropic case. In [F] the existence of a infinite sequence of eigenvalues is proved, obtained by means of a $\min-\max$ characterization. Actually, as in the Euclidean case, it is not known if this sequence exhausts all the set of the eigenvalues. Here we will show that the spectrum of $-\mathcal{Q}_{p}$ is a closed set, that the eigenfunctions are in $C^{1,\alpha}(\Omega)$ and that admit a finite number of nodal domains. We recall the reference [L2], where many results for the spectrum of the $p$-Laplacian in the Euclidean case have been summarized.

The core of the paper relies in the study of the second eigenvalue $\lambda_{2}(p,\Omega)$, $p\in]1,+\infty[$, in bounded open sets, defined as

[TABLE]

and in analyzing its behavior when $p\to\infty$.

First of all, we show that if $\Omega$ is a domain, then $\lambda_{2}(p,\Omega)$ admits exactly two nodal domains. Moreover, for a bounded open set $\Omega$, we prove a sharp lower bound for $\lambda_{2}$, namely the Hong-Krahn-Szego inequality

[TABLE]

where $\widetilde{\mathcal{W}}$ is the union of two disjoint Wulff shapes, each one of measure $\frac{|\Omega|}{2}$.

In the Euclidean case, such inequality is well-known for $p=2$, and it has been recently studied for any $1<p<+\infty$ in [BF2].

Finally, we address our attention to the behavior of $\lambda_{2}(p,\Omega)$ when $\Omega$ is a bounded open set and $p\to+\infty$. In particular, we show that

[TABLE]

where $\rho_{2,F}(\Omega)$ is the radius of two disjoint Wulff shapes $\mathcal{W}_{1},\mathcal{W}_{2}$ such that $\mathcal{W}_{1}\cup\mathcal{W}_{2}$ is contained in $\Omega$. Furthermore, the normalized eigenfunctions of $\lambda_{p}(2,\Omega)$ converge to a function $u_{\infty}$ that is a viscosity solution to a suitable fully nonlinear elliptic problem (see Section 5). In the Euclidean case, this kind of result has been proved for bounded domains in [JL]. We consider both the nonconnected case and general norm $F$. In a forthcoming paper we will deal with the limit case $p\to 1$.

As enhanced before, the aim of the paper is twofold: first, to consider the case of a general Finsler norm $F$; second, to extend also the results known in the case of domains, to the case of nonconnected sets. We structured the paper as follows. In Section $2$ we recall the main definitions as well as some basic fact of convex geometry, and we fix the precise assumptions on $F$. In Section $3$ we state the general eigenvalue problem for $-\mathcal{Q}_{p}$, recalling some known results, extending them, where it is possible, to the case of nonconnected sets; moreover we provide several properties of the first and of higher eigenvalues and eigenfunctions. In Section $4$ the attention will be focused on the second eigenvalue $\lambda_{2}(p,\Omega)$ of $-\mathcal{Q}_{p}$ proving, among the other properties of $\lambda_{2}$, the Hong-Krahn-Szego inequality. Finally, in Section $5$ we study the limit case $p\to\infty$.

2 Notation and preliminaries

Throughout the paper we will consider a convex even 1-homogeneous function

[TABLE]

that is a convex function such that

[TABLE]

and such that

[TABLE]

for some constant $0<a$. Under this hypothesis it is easy to see that there exists $b\geq a$ such that

[TABLE]

Moreover, throughout the paper we will assume that

[TABLE]

with $1<p<+\infty$.

The hypothesis (6) on $F$ assures that the operator

[TABLE]

is elliptic, hence there exists a positive constant $\gamma$ such that

[TABLE]

for some positive constant $\gamma$, for any $\eta\in\mathbb{R}^{n}\setminus\{0\}$ and for any $\xi\in\mathbb{R}^{n}$.

Remark 2.1**.**

We stress that for $p\geq 2$ the condition

[TABLE]

implies (6).

The polar function $F^{o}\colon\mathbb{R}^{n}\rightarrow[0,+\infty[$ of $F$ is defined as

[TABLE]

It is easy to verify that also $F^{o}$ is a convex function which satisfies properties (4) and (5). Furthermore,

[TABLE]

From the above property it holds that

[TABLE]

The set

[TABLE]

is the so-called Wulff shape centered at the origin. We put $\kappa_{n}=|\mathcal{W}|$, where $|\mathcal{W}|$ denotes the Lebesgue measure of $\mathcal{W}$. More generally, we denote with $\mathcal{W}_{r}(x_{0})$ the set $r\mathcal{W}+x_{0}$, that is the Wulff shape centered at $x_{0}$ with measure $\kappa_{n}r^{n}$, and $\mathcal{W}_{r}(0)=\mathcal{W}_{r}$.

The following properties of $F$ and $F^{o}$ hold true (see for example [BP]):

[TABLE]

Finally, we will recall the following

Definition 1**.**

A domain of $\mathbb{R}^{n}$ is a connected open set.

3 The Dirichlet eigenvalue problem for $-\mathcal{Q}_{p}$

Here we state the eigenvalue problem for $\operatorname{\mathcal{Q}_{p}}$. Let $\Omega$ be a bounded open set in $\mathbb{R}^{n}$, $n\geq 2$, $1<p<+\infty$, and consider the problem

[TABLE]

Definition 2**.**

We say that $u\in W^{1,p}_{0}(\Omega)$, $u\neq 0$, is an eigenfunction of (7), if

[TABLE]

for all $\varphi\in W^{1,p}_{0}(\Omega)$. The corresponding real number $\lambda$ is called an eigenvalue of (7).

Obviously, if $u$ is an eigenfunction associated to $\lambda$, then

[TABLE]

3.1 The first eigenvalue

Among the eigenvalues of (7), the smallest one, denoted here by $\lambda_{1}(p,\Omega)$, has the following well-known variational characterization:

[TABLE]

In the following theorems its main properties are recalled.

Theorem 3.1**.**

If $\Omega$ is a bounded open set in $\mathbb{R}^{n}$, $n\geq 2$, there exists a function $u_{1}\in C^{1,\alpha}(\Omega)\cap C(\overline{\Omega})$ which achieves the minimum in (9), and satisfies the problem (7) with $\lambda=\lambda_{1}(p,\Omega)$. Moreover, if $\Omega$ is connected, then $\lambda_{1}(p,\Omega)$ is simple, that is the corresponding eigenfunctions are unique up to a multiplicative constant, and the first eigenfunctions have constant sign in $\Omega$.

Proof.

The proof can be immediately adapted from the case of $\Omega$ connected and we refer the reader, for example, to [L, BFK]. ∎

Theorem 3.2**.**

Let $\Omega$ be a bounded open set in $\mathbb{R}^{n}$, $n\geq 2$. Let $u\in W_{0}^{1,p}(\Omega)$ be an eigenfunction of (7) associated to an eigenvalue $\lambda$. If $u$ does not change sign in $\Omega$, then there exists a connected component $\Omega_{0}$ of $\Omega$ such that $\lambda=\lambda_{1}(p,\Omega_{0})$ and $u$ is a first eigenfunction in $\Omega_{0}$. In particular, if $\Omega$ is connected then $\lambda=\lambda_{1}(p,\Omega)$ and a constant sign eigenfunction is a first eigenfunction.

Proof.

If $\Omega$ is connected, a proof can be found in [L, DG2]. Otherwise, if $u\geq 0$ in $\Omega$ disconnected, by the maximum principle $u$ must be either positive or identically zero in each connected component of $\Omega$. Hence there exists a connected component $\Omega_{0}$ such that $u$ coincides in $\Omega_{0}$ with a positive eigenfunction relative to $\lambda$. By the previous case, $\lambda=\lambda_{1}(p,\Omega_{0})$ and the proof is completed. ∎

Here we list some other useful and interesting properties that can be proved in a similar way than the Euclidean case.

Proposition 3.3**.**

Let $\Omega$ be a bounded open set in $\mathbb{R}^{n}$, $n\geq 2$, the following properties hold.

For $t>0$ it holds $\lambda_{1}(p,\,t\Omega)=t^{-p}\lambda_{1}(p,\Omega)$. 2. 2.

If $\Omega_{1}\subseteq\Omega_{2}\subseteq\Omega$, then $\lambda_{1}(p,\Omega_{1})\geq\lambda_{1}(p,\Omega_{2})$. 3. 3.

For all $1<p<s<+\infty$ we have $p[\lambda_{1}(p,\Omega)]^{1/p}<s[\lambda_{1}(s,\Omega)]^{1/s}$.

Proof.

The first two properties are immediate from (9). As regards the third property, the inequality derives from the Hölder inequality, similarly as in [L1]. Indeed, taking $\phi=|\psi|^{\frac{s}{p}-1}\psi$, $\psi\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)$, $\psi\geq 0$, we have by (4) that

[TABLE]

By minimizing with respect to $\psi$, we get the thesis. ∎

In addition, the Faber-Krahn inequality for $\lambda_{1}(p,\Omega)$ holds.

Theorem 3.4**.**

Let $\Omega$ be a bounded open set in $\mathbb{R}^{n}$, $n\geq 2$, then

[TABLE]

Moreover, equality sign in (10) holds if $\Omega$ is homothetic to the Wulff shape.

The proof of this inequality, contained in [BFK], is based on a symmetrization technique introduced in [AFLT] (see [ET, FV] for the equality cases).

Using the previous result we can prove the following property of $\lambda_{1}(p,\Omega)$.

Proposition 3.5**.**

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$, $n\geq 2$. The first eigenvalue of (7), $\lambda_{1}(p,\Omega)$, is isolated.

Proof.

We argue similarly as in [L2]. For completeness we give the proof. For convenience we write $\lambda_{1}$ instead of $\lambda_{1}(p,\Omega)$. Let $\lambda_{k}\neq\lambda_{1}$ a sequence of eigenvalues such that

[TABLE]

Let $u_{k}$ be a normalized eigenfunction associated to $\lambda_{k}$ that is,

[TABLE]

By (11), there exists a function $u\in W_{0}^{1,p}(\Omega)$ such that, up to a subsequence

[TABLE]

By the strong convergence of $u_{k}$ in $L^{p}(\Omega)$ and, recalling that $F$ is convex, by weak lower semicontinuity, it follows that

[TABLE]

Hence, $u$ is a first eigenfunction. On the other hand, being $u_{k}$ not a first eigenfunction, by Theorem 3.2 it has to change sign. Hence, the sets $\Omega_{k}^{+}=\{u_{k}>0\}$ and $\Omega_{k}^{-}=\{u_{k}<0\}$ are nonempty and, as a consequence of the Faber-Krahn inequality and of Theorem 3.2, it follows that

[TABLE]

This implies that both $|\Omega^{+}_{k}|$ and $|\Omega_{k}^{-}|$ cannot vanish as $k\to+\infty$ and finally, that $u_{k}$ converges to a function $u$ which changes sign in $\Omega$. This is in contradiction with the characterization of the first eigenfunctions, and the proof is completed. ∎

3.2 Higher eigenvalues

First of all, we recall the following result (see [F, Theorem 1.4.1] and the references therein), which assures the existence of infinite eigenvalues of $-\mathcal{Q}_{p}$. We use the following notation. Let $\mathbb{S}^{n-1}$ be the unit Euclidean sphere in $\mathbb{R}^{n}$, and

[TABLE]

Moreover, let $\mathcal{C}_{n}$ be the class of all odd and continuous mappings from $\mathbb{S}^{n-1}$ to $M$. Then, for any fixed $f\in\mathcal{C}_{n}$, we have $f:\omega\in\mathbb{S}^{n-1}\mapsto f_{\omega}\in M$.

Proposition 3.6**.**

Let $\Omega$ a bounded open set of $\mathbb{R}^{n}$, for any $k\in\mathbb{N}$, the value

[TABLE]

is an eigenvalue of $-\mathcal{Q}_{p}$. Moreover,

[TABLE]

and

[TABLE]

Hence, we have at least a sequence of eigenvalues of $-\operatorname{\mathcal{Q}_{p}}$. Furthermore, the following proposition holds.

Proposition 3.7**.**

Let $\Omega$ a bounded open set of $\mathbb{R}^{n}$. The spectrum of $-\operatorname{\mathcal{Q}_{p}}$ is a closed set.

Proof.

Let $\lambda_{k}$ be a sequence of eigenvalues converging to $\mu<+\infty$ and let $u_{k}$ be the corresponding normalized eigenfunctions, that is such that $\|u_{k}\|_{L^{p}(\Omega)}=1$. We have to show that $\mu$ is an eigenvalue of $-\operatorname{\mathcal{Q}_{p}}$.

We have that

[TABLE]

for any test function $\varphi\in W_{0}^{1,p}(\Omega)$. Since

[TABLE]

and being $\lambda_{k}$ a convergent sequence, up to a subsequence we have that there exists a function $u\in W_{0}^{1,p}(\Omega)$ such that $u_{k}\rightarrow u$ strongly in $L^{p}(\Omega)$ and $\nabla u_{k}\rightharpoonup\nabla u$ weakly in $L^{p}(\Omega)$. Our aim is to prove that $u$ is an eigenfunction relative to $\lambda$.

Choosing $\varphi=u_{k}-u$ as test function in the equation solved by $u_{k}$, we have

[TABLE]

By the strong convergence of $u_{k}$ and the weak one of $\nabla u_{k}$, the right-hand side of the above identity goes to zero as $k$ diverges. Hence

[TABLE]

By nowadays standard arguments, this limit implies the strong convergence of the gradient, hence we can pass to the limit under the integral sign in (13) to obtain

[TABLE]

This shows that $\lambda$ is an eigenvalue and the proof is completed. ∎

Finally, we list some properties of the eigenfunctions, well-known in the Euclidean case (see for example [L2, AFT]). Recall that a nodal domain of an eigenfunction $u$ is a connected component of $\{u>0\}$ or $\{u<0\}$.

Proposition 3.8**.**

Let $p>1$, and let $\Omega$ be a bounded open set in $\mathbb{R}^{n}$. Then the following facts hold.

1. (i)

Any eigenfunction of $-\operatorname{\mathcal{Q}_{p}}$ has only a finite number of nodal domains. 2. (ii)

Let $\lambda$ be an eigenvalue of $-\operatorname{\mathcal{Q}_{p}}$, and $u$ be a corresponding eigenfunction. The following estimate holds:

[TABLE]

where $C_{n,p,F}$ is a constant depending only on $n$, $p$ and $F$. 3. (iii)

All the eigenfunctions of (1) are in $C^{1,\alpha}(\Omega)$, for some $\alpha\in(0,1)$.

Proof.

Let $\lambda$ be an eigenvalue of $-\operatorname{\mathcal{Q}_{p}}$, and $u$ a corresponding eigenfunction.

In order to prove $(i)$, let us denote by $\Omega^{+}_{j}$ a connected component of the set $\Omega^{+}:=\{u>0\}$. Being $\lambda=\lambda_{1}(\Omega^{+}_{j})$, then by (10)

[TABLE]

Then, the thesis follows observing that

[TABLE]

In order to prove $(ii)$, let $k>0$, and choose $\varphi(x)=\max\{u(x)-k,0\}$ as test function in (8). Then

[TABLE]

where $A_{k}=\{x\in\Omega\colon u(x)>k\}$. Being $k|A_{k}|\leq||u||_{L^{1}(\Omega)}$, then $|A_{k}|\rightarrow 0$ as $k\rightarrow\infty$. By the inequality $a^{p-1}\leq 2^{p-1}(a-k)^{p-1}+2^{p-1}k^{p-1}$, we have

[TABLE]

By Poincaré inequality and property (5), then (15) and (16) give that

[TABLE]

By choosing $k$ sufficiently large, the Hölder inequality implies

[TABLE]

This estimate allows to apply [LU, Lemma 5.1, p. 71] in order to get the boundedness of $\operatorname{ess\,sup}u$. Similar argument gives that $\operatorname{ess\,inf}u$ is bounded.

Since (14) holds, by standard elliptic regularity theory (see e.g. [LU]) the eigenfunction is $C^{1,\alpha}(\Omega)$. ∎

4 The second Dirichlet eigenvalue of $-\mathcal{Q}_{p}$

If $\Omega$ is a bounded domain, Proposition 3.5 assures that the first eigenvalue $\lambda_{1}(p,\Omega)$ of (1) is isolated. This suggests the following definition.

Definition 3**.**

Let $\Omega$ be a bounded open set of $\mathbb{R}^{n}$. Then the second eigenvalue of $-\operatorname{\mathcal{Q}_{p}}$ is

[TABLE]

Remark 4.1**.**

If $\Omega$ is connected, by theorems 3.1 and 3.2 we deduce the following characterization of the second eigenvalue:

[TABLE]

We point out that in [F] it is proved that in a bounded open set it holds

[TABLE]

where $\tilde{\lambda}_{2}(p,\Omega)$ is given in Proposition 3.6, and

[TABLE]

with $M$ as in (12). As immediate consequence of (18) we get

Proposition 4.2**.**

If $\Omega_{1}\subseteq\Omega_{2}\subseteq\Omega$, then $\lambda_{2}(p,\Omega_{1})\geq\lambda_{2}(p,\Omega_{2})$.

By adapting the method contained in [CDG1, CDG2], it is possible to prove the following result.

Proposition 4.3**.**

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$. The eigenfunctions associated to $\lambda_{2}(p,\Omega)$ admit exactly two nodal domains.

Proof.

We will proceed as in the proof of [CDG2, Th. 2.1]. In such a case, $\lambda_{2}(p,\Omega)$ is characterized as in (17). Then any eigenfunction $u_{2}$ has to change sign, and it admits at least two nodal domains $\Omega_{1}\subset\Omega^{+}$ and $\Omega_{2}\subset\Omega^{-}$. Let us assume, by contradiction, the existence of a third nodal domain $\Omega_{3}$ and let us suppose, without loss of generality, that $\Omega_{3}\subset\Omega^{+}$.

Claim. There exists a connected open set $\widetilde{\Omega}_{2}$, with $\Omega_{2}\subset\widetilde{\Omega}_{2}\subset\Omega$ such that $\widetilde{\Omega}_{2}\cap\Omega_{1}=\emptyset$ or $\widetilde{\Omega}_{2}\cap\Omega_{3}=\emptyset$.

The proof of the claim follows line by line as in [CDG2, Th. 2.1]. One of the main tool is the Hopf maximum principle, that for the operator $-\operatorname{\mathcal{Q}_{p}}$ is proved for example in [CT, Th. 2.1].

Now, without loss of generality, we assume that $\widetilde{\Omega}_{2}$ is disjoint of $\Omega_{1}$ and from this fact a contradiction is derived.

By the fact that $u_{2}$ does not change sign on the nodal domains and by Proposition 4.2, we have that $\lambda_{1}(p,\Omega_{1})=\lambda_{2}(p,\Omega)$ and that $\lambda_{1}(p,\widetilde{\Omega}_{2})<\lambda_{1}(p,\Omega_{2})=\lambda_{2}(p,\Omega)$. Now, we may construct the disjoint sets $\widetilde{\widetilde{\Omega}}_{2}$ and $\widetilde{\Omega}_{1}$ such that $\Omega_{2}\subset\widetilde{\widetilde{\Omega}}_{2}\subset\widetilde{\Omega}_{2}$ and $\Omega_{1}\subset\widetilde{\Omega}_{1}$, in order to have

[TABLE]

Now let $v_{1}$ and $v_{2}$ be the extension by zero outside $\widetilde{\Omega}_{1}$ and $\widetilde{\widetilde{\Omega}}_{2}$, respectively, of the positive normalized eigenfunctions associated to $\lambda_{1}(p,\widetilde{\Omega}_{1})$ and $\lambda(p,\widetilde{\widetilde{\Omega}}_{2})$. Hence we easily verify that the function $v=v_{1}-v_{2}$ belongs to $W^{1,p}_{0}(\Omega)$, it changes sign and satisfies

[TABLE]

The final aim is to construct a path $\gamma([0,1])$ such

[TABLE]

obtaining a contradiction from (18). The construction of this path follows adapting the method contained in [CDG1, CDG2].

∎

Remark 4.4**.**

In order to better understand the behavior of $\lambda_{1}(p,\Omega)$ and $\lambda_{2}(p,\Omega)$ on disconnected sets, an meaningful model is given when

[TABLE]

We distinguish two cases.

Case $r_{1}<r_{2}$.

We have

[TABLE]

Hence $\lambda_{1}(p,\Omega)$ is simple, and any eigenfunction is identically zero on $\mathcal{W}_{1}$ and has constant sign in $\mathcal{W}_{2}$. Moreover,

[TABLE]

Hence, if $r_{1}$ is not too small, then the second eigenvalue is $\lambda_{1}(\mathcal{W}_{r_{1}})$, and the second eigenfunctions of $\Omega$ coincide with the first eigenfunctions of $\mathcal{W}_{r_{1}}$, that do not change sign in $\mathcal{W}_{r_{1}}$, and vanish on $\mathcal{W}_{r_{2}}$.

Case $r_{1}=r_{2}$.

We have

[TABLE]

The first eigenvalue $\lambda_{1}(p,\Omega)$ is not simple: choosing, for example, the function $U=u_{1}\chi_{\mathcal{W}_{r_{1}}}-u_{2}\chi_{\mathcal{W}_{r_{2}}}$, where $u_{i}$, $i=1,2$, is the first normalized eigenfunction of $\lambda_{1}(p,\mathcal{W}_{r_{i}})$, and $V=u_{1}\chi_{\mathcal{W}_{r_{1}}}$, then $U$ and $V$ are two nonproportional eigenfunctions relative to $\lambda_{1}(p,\Omega)$. Hence, in this case, by definition,

[TABLE]

In order to prove the Hong-Krahn-Szego inequality, we need the following key lemma.

Proposition 4.5**.**

Let $\Omega$ be an open bounded set of $\mathbb{R}^{n}$. Then there exists two disjoint domains $\Omega_{1},\Omega_{2}$ of $\Omega$ such that

[TABLE]

Proof.

Let $u_{2}\in W^{1,p}_{0}(\Omega)$ be a second normalized eigenfunction. First of all, suppose that $u_{2}$ changes sign in $\Omega$. Then, consider two nodal domains $\Omega_{1}\subseteq\Omega_{+}$ and $\Omega_{2}\subseteq\Omega_{-}$. By definition, $\Omega_{1}$ and $\Omega_{2}$ are connected sets. The restriction of $u_{2}$ to $\Omega_{1}$ is, by Theorem 3.2, a first eigenfunction for $\Omega_{1}$ and hence $\lambda_{2}(p,\Omega)=\lambda_{1}(p,\Omega_{1})$. Analogously for $\Omega_{2}$, hence

[TABLE]

and the proof of the proposition is completed, in the case $u_{2}$ changes sign.

In the case that $u_{2}$ has constant sign in $\Omega$, for example $u_{2}\geq 0$, then by Theorem 3.2 $\Omega$ must be disconnected. If $\lambda_{1}(p,\Omega)$ is simple, by definition $\lambda_{2}(p,\Omega)>\lambda_{1}(p,\Omega)$. Otherwise, $\lambda_{1}(p,\Omega)=\lambda_{2}(p,\Omega)$. Hence in both cases, we can consider a first nonnegative normalized eigenfunction $u_{1}$ not proportional to $u_{2}$.

Observe that in any connected component of $\Omega$, by the Harnack inequality, $u_{i}$, $i=1,2$, must be positive or identically zero. Hence we can choose two disjoint connected open sets $\Omega_{1}$ and $\Omega_{2}$, contained respectively in $\{x\in\Omega\colon u_{1}(x)>0\}$ and $\{x\in\Omega\colon u_{2}(x)>0\}$. Then, $u_{1}$ and $u_{2}$ are first Dirichlet eigenfunctions in $\Omega_{1}$ and $\Omega_{2}$, respectively, and

[TABLE]

and the proof is completed. ∎

Now we are in position to prove the Hong-Krahn-Szego inequality for $\lambda_{2}(p,\Omega)$.

Theorem 4.6**.**

Let $\Omega$ be a bounded open set of $\mathbb{R}^{n}$. Then

[TABLE]

where $\widetilde{\mathcal{W}}$ is the union of two disjoint Wulff shapes, each one of measure $\frac{|\Omega|}{2}$. Moreover equality sign in (19) occurs if $\Omega$ is the disjoint union of two Wulff shapes of the same measure.

Proof.

Let $\Omega_{1}$ and $\Omega_{2}$ given by Proposition 4.5. By the Faber-Krahn inequality we have

[TABLE]

with $|\mathcal{W}_{r_{i}}|=|\Omega_{i}|$. By the rescaling property of $\lambda_{1}(p,\,\cdot\,)$, and observing that, being $\Omega_{1}$ and $\Omega_{2}$ disjoint subsets of $\Omega$, $|\Omega_{1}|+|\Omega_{2}|\leq|\Omega|$, we have that

[TABLE]

∎

5 The limit case $p\to\infty$

In this section we derive some information on $\lambda_{2}(p,\Omega)$ as $p$ goes to infinity. First of all we recall some known result about the limit of the first eigenvalue. Let us consider a bounded open set $\Omega$.

The anisotropic distance of $x\in\overline{\Omega}$ to the boundary of $\Omega$ is the function

[TABLE]

We stress that when $F=|\cdot|$ then $d_{F}=d_{\mathcal{E}}$, the Euclidean distance function from the boundary.

It is not difficult to prove that $d_{F}$ is a uniform Lipschitz function in $\overline{\Omega}$ and

[TABLE]

Obviously, $d_{F}\in W_{0}^{1,\infty}(\Omega)$. Let us consider the quantity

[TABLE]

If $\Omega$ is connected, $\rho_{F}$ is called the anisotropic inradius of $\Omega$. If not, $\rho_{F}$ is the maximum of the inradii of the connected components of $\Omega$.

For further properties of the anisotropic distance function we refer the reader to [CM].

Remark 5.1**.**

It is easy to prove (see also [JLM, BKJ]) that the distance function satisfies

[TABLE]

Indeed it is sufficient to observe that if $\varphi\in C_{0}^{1}(\Omega)\cap C(\overline{\Omega})$, then $\varphi\in C_{0}^{1}(\Omega_{i})\cap C(\overline{\Omega}_{i})$, for any connected component $\Omega_{i}$ of $\Omega$. Then for a.e. $x\in\Omega_{i}$, for $y\in\partial\Omega_{i}$ which achieves $F^{o}(x-y)=d_{F}(x)$, it holds

[TABLE]

Passing to the supremum and by density we get (20).

The following result holds (see [BKJ, JLM]).

Theorem 5.2**.**

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$, and let $\lambda_{1}(p,\Omega)$ be the first eigenvalue of (7). Then

[TABLE]

Now let us define

[TABLE]

The value $\Lambda_{1}(\infty,\Omega)$ is related to the so-called anisotropic infinity Laplacian operator defined in [BKJ], that is

[TABLE]

where $J(\xi)=\frac{1}{2}\nabla_{\xi}\left[F^{2}\right](\xi)$. Note that we mean, by continuous extension, $J(0)=0$. This is possible being $F$ $1$-homogeneous and $F(0)=0$.

Indeed, in [BKJ] the following result is proved.

Theorem 5.3**.**

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$. Then, there exists a positive solution $u_{\infty}\in W_{0}^{1,\infty}(\Omega)\cap C(\bar{\Omega})$ which satisfies, in the viscosity sense, the following problem:

[TABLE]

with $\Lambda=\Lambda_{1}(\infty,\Omega)$. Moreover, any positive solution $v\in W_{0}^{1,\infty}(\Omega)$ to (21) with $\Lambda=\Lambda_{1}(\infty,\Omega)$ satisfies

[TABLE]

Finally, if problem (21) admits a positive viscosity solution in $\Omega$, then $\Lambda=\Lambda_{1}(\infty,\Omega)$.

Proposition 5.4**.**

Theorem 5.2 holds also when $\Omega$ is a bounded open set of $\mathbb{R}^{n}$.

Proof.

Suppose that $\Omega$ is not connected, and consider a connected component $\Omega_{0}$ of $\Omega$ with anisotropic inradius $\rho_{F}(\Omega)$. By the monotonicity property of $\lambda_{1}(p,\Omega)$ given in Proposition 3.3, we have

[TABLE]

Then up to a subsequence, passing to the limit as $p\to+\infty$ and using Theorem 5.2 we have

[TABLE]

In order to prove that $\tilde{\Lambda}=\rho_{F}(\Omega)^{-1}$, let $u_{p_{j}}$ the first nonnegative normalized eigenfunction associated to $\lambda_{1}(p_{j},\Omega)$. Reasoning as in [BKJ], the sequence $u_{p_{j}}$ converges to a function $u_{\infty}$ in $C^{0}(\Omega)$ which is a viscosity solution of (21) associated to $\tilde{\Lambda}$. Then by the maximum principle contained in [BB, Lemma 3.2], in each connected component of $\Omega$, $u_{\infty}$ is either positive or identically zero. Denoting by $\tilde{\Omega}$ a connected component of $\{u_{\infty}>0\}$, by the uniform convergence, for $p_{j}$ large, also $u_{p_{j}}$ is positive in $\tilde{\Omega}$. Then by Theorem 3.2 we have

[TABLE]

By (22) and by definition of $\rho_{F}$, $\tilde{\Lambda}\leq\rho_{F}(\Omega)^{-1}\leq\rho_{F}(\tilde{\Omega})^{-1}=\tilde{\Lambda}$; then necessarily $\tilde{\Lambda}=\rho_{F}(\Omega)^{-1}$. ∎

In order to define the eigenvalue problem for $\mathcal{Q}_{\infty}$, let us consider the following operator

[TABLE]

with $(s,\xi,X)\in\mathbb{R}\times\mathbb{R}^{n}\times S^{n\times n}$, where $S^{n\times n}$ denotes the space of real, symmetric matrices of order $n$. Clearly $\mathcal{A}_{\Lambda}$ is not continuous in $s=0$.

For completeness we recall the definition of viscosity solution for the operator $\mathcal{A}_{\Lambda}$.

Definition 4**.**

Let $\Omega\subset\mathbb{R}^{n}$ a bounded open set. A function $u\in C(\Omega)$ is a viscosity subsolution (resp. supersolution) of $\mathcal{A}_{\Lambda}(x,u,\nabla u)=0$ if

[TABLE]

for every $\phi\in C^{2}(\Omega)$ such that $u-\phi$ has a local maximum (resp. minimum) zero at $x$. A function $u\in C(\Omega)$ is a viscosity solution of $\mathcal{A}_{\Lambda}=0$ if it is both a viscosity subsolution and a viscosity supersolution and in this case the number $\Lambda$ is called an eigenvalue for $\mathcal{Q}_{\infty}$.

Definition 5**.**

We say that $u\in C(\bar{\Omega})$, $u|_{\partial\Omega}=0$, $u\not\equiv 0$ is an eigenfunction for the anisotropic $\infty-$Laplacian if there exists $\Lambda\in\mathbb{R}$ such that

[TABLE]

in the viscosity sense. Such value $\Lambda$ will be called an eigenvalue for the anisotropic $\infty-$Laplacian.

In order to define the second eigenvalue for $\mathcal{Q}_{\infty}$ we introduce the following number:

[TABLE]

and let us define

[TABLE]

Clearly

[TABLE]

Remark 5.5**.**

It is easy to construct open sets $\Omega$ such that $\Lambda_{1}(\infty,\Omega)=\Lambda_{2}(\infty,\Omega)$. For example, this holds when $\Omega$ coincides with the union of two disjoint Wulff shapes with same measure, or their convex envelope.

Remark 5.6**.**

A simple example of $\rho_{2,F}(\Omega)$ is given when $\Omega$ is the union of two disjoint Wulff sets, $\Omega=\mathcal{W}_{r_{1}}\cup W_{r_{2}}$, with $r_{2}\leq r_{1}$. In this case, $\Lambda_{1}(\infty,\Omega)=\frac{1}{r_{1}}$ and, if $r_{2}$ is not too small, then $\Lambda_{2}(\infty,\Omega)=\frac{1}{r_{2}}$.

Theorem 5.7**.**

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded open set and let $\lambda_{2}(p,\Omega)$ be the second Dirichlet eigenvalue of $-\mathcal{Q}_{p}$ in $\Omega$. Then

[TABLE]

Moreover $\Lambda_{2}(\infty,\Omega)$ is an eigenvalue of $\mathcal{Q}_{\infty}$, that is $\Lambda_{2}(\infty,\Omega)$ is an eigenvalue for the anisotropic infinity Laplacian in the sense of Definition 5.

Proof.

First we observe that $\lambda_{2}(p,\Omega)^{\frac{1}{p}}$ is bounded from above with respect to $p$. More precisely we have

[TABLE]

Indeed if we consider two disjoint Wulff shapes $\mathcal{W}_{1}$ and $\mathcal{W}_{2}$ of radius $\rho_{2,F}(\Omega)$, clearly $\mathcal{W}_{1}\cup\mathcal{W}_{2}\subset\Omega$ and then by monotonicity property (Proposition 4.2) of $\lambda_{2}(p,\Omega)$ we have

[TABLE]

where last equality follows from Remark 4.4. Then passing to the limit as $p\to\infty$ in the right hand side, by Theorem 5.2 we have (24). Hence there exists a sequence $p_{j}$ such that $p_{j}\to+\infty$ as $j\to\infty$, and

[TABLE]

In order to conclude the proof we have to show that $\overline{\Lambda}$ is an eigenvalue for $\mathcal{Q}_{\infty}$ and that $\overline{\Lambda}=\Lambda_{2}(\infty,\Omega)$.

Let us consider $u_{j}\in W_{0}^{1,p}(\Omega)$ eigenfunction of $\lambda_{2}(p_{j},\Omega)$ such that $\|u_{j}\|_{L^{p_{j}}(\Omega)}=1$. Then by standard arguments $u_{j}$, converges, up to a subsequence of $p_{j}$, uniformly to a function $u\in W_{0}^{1,\infty}(\Omega)\cap C(\bar{\Omega})$. The function $u$ is a viscosity solution of (23) with $\Lambda=\bar{\Lambda}$. Indeed, let $x_{0}\in\Omega$. If $u(x_{0})>0$, being $u$ continuous, it is positive in a sufficiently small ball centered at $x_{0}$. Then it is possible to proceed exactly as in [BKJ] in order to obtain that, in the viscosity sense,

[TABLE]

Similarly, if $u(x_{0})<0$ then

[TABLE]

It remains to consider the case $u(x_{0})=0$. We will show that $u$ is a subsolution of (23).

Let $\varphi$ a $C^{2}(\Omega)$ function such that $u-\varphi$ has a strict maximum point at $x_{0}$. By the definition of $\mathcal{A}_{\bar{\Lambda}}$, we have to show that $-\mathcal{Q}_{\infty}\varphi(x_{0})\leq 0$.

For any $j$, let $x_{j}$ be a maximum point of $u_{j}-\varphi$, so that $x_{j}\to x_{0}$ as $j\to\infty$. Such sequence exists by the uniform convergence of $u_{j}$. By [BKJ, Lemma 2.3] $u_{j}$ verifies in the viscosity sense $-\mathcal{Q}_{p}u_{j}=\lambda_{2}(p_{j},\Omega)|u_{j}|^{p_{j}-2}u_{j}$. Then

[TABLE]

here $A\otimes B:=\sum_{i,k}A_{ik}B_{ik}$, for two $n\times n$ matrices $A,B$. If $\nabla\varphi(x_{0})\neq 0$, then dividing the above inequality by $(p_{j}-2)F^{p_{j}-4}(\nabla\varphi)$ we have

[TABLE]

Passing to the limit as $j\to\infty$, recalling that $\varphi\in C^{2}(\Omega)$, $F\in C^{2}(\mathbb{R}^{n}\setminus\{0\})$, $\lambda_{2}(p_{j},\Omega)^{\frac{1}{p_{j}}}\to\bar{\Lambda}$, $\nabla\varphi(x_{0})\neq 0$ and $u_{j}(x_{j})\to 0$ we get

[TABLE]

Finally, we note that if $\nabla\varphi(x_{0})=0$, the above inequality is trivially true. Hence, we can conclude that $u$ is a viscosity subsolution.

The proof that $u$ is also a viscosity supersolution can be done by repeating the same argument than before, considering $-u$.

Last step of the proof of the Theorem consists in showing that $\bar{\Lambda}=\Lambda_{2}(\infty,\Omega)$. We distinguish two cases.

Case 1: The function $u$ changes sign in $\Omega$.

Let us consider the following sets

[TABLE]

Being $u\in C^{0}(\Omega)$ then $\Omega^{+},\Omega^{-}$ are two disjoint open sets of $\mathbb{R}^{n}$ and $|\Omega^{+}|>0$ and $|\Omega^{-}|>0$.

By Theorem 5.3 we have

[TABLE]

Then by definition of $\rho_{2,F}$ we get

[TABLE]

that implies, by (25) that

[TABLE]

Case 2: The function $u$ does not change sign in $\Omega$.

We first observe that in this case $\Omega$ cannot be connected. Indeed since $u_{j}$ converges to $u$ in $C^{0}(\overline{\Omega})$, for sufficiently large $p$ we have that there exist second eigenfunctions relative to $\lambda_{2}(p,\Omega)$ with constant sign in $\Omega$ and this cannot happen if $\Omega$ is connected.

Then in this case, we have to replace the sequence $u_{j}$ (and then the function $u$) in order to find two disjoint connected open subsets $\Omega_{1},\Omega_{2}$ of $\Omega$, such that

[TABLE]

and

[TABLE]

Once we prove that such subsets exist, by (25) and the definition of $\rho_{2,F}$ we obtain

[TABLE]

that implies, again by (25),

[TABLE]

In order to prove (26) and (27), we consider $u_{1,\infty}$, an eigenfunction associated to $\Lambda_{1}(\infty,\Omega)$, obtained as limit in $C^{0}(\Omega)$ of a sequence $u_{1,p}$ of first normalized eigenfunctions associated to $\lambda_{1}(p,\Omega)$, and consider a connected component of $\Omega$, say $\Omega_{1}$, where $u_{1,\infty}>0$ and such that $\Lambda_{1}(\infty,\Omega)=\Lambda_{1}(\infty,\Omega_{1})$. The argument of the proof of Proposition 5.4 gives that such $u_{1,\infty}$ and $\Omega_{1}$ exist. Then, let $u_{2,p}\geq 0$ be a normalized eigenfunction associated to $\lambda_{2}(p,\Omega)$ such that for any $p$ sufficiently large, $\operatorname{spt}(u_{2,p})\cap\Omega_{1}=\emptyset$.

The existence of such a sequence is guaranteed from this three observations:

• •

if $u_{2,p}$ changes sign for a divergent sequence of $p$’s, then we come back to the case 1;

• •

by the maximum principle, in each connected component of $\Omega$ $u_{2,p}$ is either positive or identically zero;

• •

the condition $\operatorname{spt}(u_{2,p})\cap\Omega_{1}=\emptyset$ depends from the fact that $u_{2,p}$ can be chosen not proportional to $u_{1,p}$.

Hence, there exists $\Omega_{2}$ connected component of $\Omega$ disjoint from $\Omega_{1}$, such that $u_{2,p}$ converges to $u_{2,\infty}$ (up to a subsequence) in $C^{0}(\Omega_{2})$, and where $u_{2,\infty}>0$. By Theorem 5.3, (27) holds. ∎

Theorem 5.8**.**

Given $\Omega$ bounded open set of $\mathbb{R}^{n}$, let $\Lambda>\Lambda_{1}(\infty,\Omega)$ be an eigenvalue for $\mathcal{Q}_{\infty}$. Then $\Lambda\geq\Lambda_{2}(\infty,\Omega)$ and $\Lambda_{2}(\infty,\Omega)$ is the second eigenvalue of $\mathcal{Q}_{\infty}$, in the sense that there are no eigenvalues of $\mathcal{Q}_{\infty}$ between $\Lambda_{1}(\infty,\Omega)$ and $\Lambda_{2}(\infty,\Omega)$.

Proof.

Let $u_{\Lambda}$ be an eigenfunction corresponding to $\Lambda$. We distinguish two cases.

Case 1: The function $u_{\Lambda}$ changes sign in $\Omega$.

Let us consider the following sets

[TABLE]

Being $u_{\Lambda}\in C^{0}(\Omega)$ then $\Omega^{+},\Omega^{-}$ are two disjoint open sets of $\mathbb{R}^{n}$ and $|\Omega^{+}|>0$ and $|\Omega^{-}|>0$.

By Theorem 5.3 we have

[TABLE]

Then by definition of $\rho_{2,F}$ we get

[TABLE]

that implies, by (25) that

[TABLE]

Case 2: The function $u_{\Lambda}$ does not change sign in $\Omega$.

By Theorem 5.3 $\Omega$ cannot be connected being $\Lambda>\Lambda_{1}(\infty,\Omega)$.

In this case, again by By Theorem 5.3 we can find two disjoint connected open subsets $\Omega_{1},\Omega_{2}$ of $\Omega$, such that

[TABLE]

and

[TABLE]

Being $\Lambda>\Lambda_{1}(\infty,\Omega),$ we obtain

[TABLE]

that by the definition of $\rho_{2,F}$ implies,

[TABLE]

∎

Remark 5.9**.**

We observe that if $\Omega$ is a bounded open set and $\widetilde{\mathcal{W}}$ is the union of two disjoint Wulff sets with the same measure $|\Omega|/2$, it holds that

[TABLE]

that is,

[TABLE]

that is the Hong-Krahn-Szego inequality for the second eigenvalue of $-\mathcal{Q}_{\infty}$.

Acnowledgements

This work has been partially supported by the FIRB 2013 project “Geometrical and qualitative aspects of PDE’s” and by GNAMPA of INdAM.