Isoperimetric inequalities of the fourth order Neumann eigenvalues

Abstract

In this paper, we obtain some isoperimetric inequalities for the first $(n-1)$ eigenvalues of the fourth order Neumann Laplacian on bounded domains in an n-dimensional Euclidean space. Our result supports strongly the conjecture of Chasman.

1 Introduction

Letting Ω be a bounded domain with a smooth boundary ∂Ω in the Euclidean space $\mathbb{R}^{n}$, we consider the Neumann problem of the Laplacian Δ as follows:

$$ \textstyle\begin{cases} \Delta u=\mu u,& \mbox{in } \varOmega , \\ \frac{\partial u}{\partial \nu }=0, & \mbox{on } \partial \varOmega , \end{cases} $$

(1.1)

where ν is the outward unit normal to the boundary. It is well known that the free membrane problem (1.1) has a discrete spectrum consisting of a sequence

$$ 0=\mu _{0}< \mu _{1}\leq \mu _{2}\leq \cdots \rightarrow +\infty . $$

When Ω is a bounded domain in $\mathbb{R}^{2}$, Szegö [6] proved the following classical isoperimetric inequality:

$$ \mu _{1}(\varOmega )\leq \mu _{1}(B_{\varOmega }), $$

(1.2)

where $B_{\varOmega }$ is the ball of same volume as Ω. Weinberger [11] generalized this result to n-dimensions. Ashbaugh and Benguria [2] extended the Szegö–Weinberger inequality (1.2) to the bounded domains in hyperbolic space and a hemisphere. On the other hand, Ashbaugh and Benguria [1] conjectured that

$$ \sum_{i=1}^{n} \frac{1}{\mu _{i}(\varOmega )}\geq \frac{n}{\mu _{1}(B_{\varOmega })}, \quad \text{with equality if and only if } \varOmega \text{ is a ball}, $$

(1.3)

where $\mu _{i}(\varOmega )$ is the ith Neumann eigenvalue on Ω, $\mu _{1}(B_{\varOmega })$ is the first nonzero Neumann eigenvalue on $B_{\varOmega }$. In [10], Wang and Xia proved an isoperimetric inequality for the sums of the reciprocals of the first $(n-1)$ nonzero eigenvalues of the Neumann Laplacian on bounded domains in $\mathbb{R}^{n}$ as follows:

$$ \sum_{i=1}^{n-1} \frac{1}{\mu _{i}(\varOmega )}\geq \frac{n-1}{\mu _{1}(B_{\varOmega })}, \quad \text{with equality if and only if } \varOmega \text{ is a ball}, $$

(1.4)

which means the Ashbaugh–Benguria’s conjecture is true for the first $(n-1)$ nonzero eigenvalues of the Neumann Laplacian on bounded domains in $\mathbb{R}^{n}$. So (1.4) supports the above conjectures of Ashbaugh and Benguria. On the other hand, Benguria, et al. [3] proved a result which is similar to (1.4) for the first $(n-1)$ nontrivial Neumann eigenvalues on domains in a hemisphere of $S^{n}$. Moreover, some works on eigenvalues are related to the spectra of matrix operators and can be seen in [7–9].

Let Δ and Δ̅ be the Laplace–Beltrami operators on Ω and ∂Ω, respectively. Let ∇ and ∇̅ be the gradient operators on Ω and ∂Ω, respectively. Consider the following Neumann eigenvalue problem of the bi-harmonic operator:

$$ \textstyle\begin{cases} \Delta ^{2} u-\tau \Delta u=\varLambda u & \mbox{in } \varOmega , \\ \frac{\partial ^{2}u}{\partial \nu ^{2}}=0, & \mbox{on } \partial \varOmega , \\ \tau \frac{\partial u}{\partial \nu }-\operatorname{div}_{\partial \varOmega } (\nabla ^{2} u(\nu ) )-\frac{\partial \Delta u}{\partial \nu }=0, & \mbox{on } \partial \varOmega , \end{cases} $$

(1.5)

where $\tau \geq 0$ and σ are two constants, $\operatorname{div}_{\partial \varOmega }$ denotes the tangential divergence operator on ∂Ω, and $\nabla ^{2} u$ is the Hessian of u, ν is the outward unit normal to the boundary. In this setting, problem (1.5) has a discrete spectrum, and all eigenvalues in the discrete spectrum can be listed nondecreasingly as follows:

$$ 0=\varLambda _{0}< \varLambda _{1}\leq \varLambda _{2}\leq \cdots \uparrow + \infty . $$

By the Rayleigh–Ritz characterization, the $(k+1)$th eigenvalue of (1.5) can be given as follows (see, e.g., [5]):

$$ \varLambda _{k+1}= \inf_{u\in H^{2}(\varOmega )} \biggl\{ Q[u]= \frac{\int _{\varOmega } [ \vert \nabla ^{2} u \vert ^{2}+\tau \vert \nabla u \vert ^{2} ]\,dx}{\int _{\varOmega }u^{2}\,dx} {\Big|} \int _{\varOmega }u u_{j} =0, j=1,\dots ,k \biggr\} . $$

(1.6)

Letting $B_{\varOmega }$ be the ball of same volume as Ω, Chasman [5] proved the following isoperimetric inequality:

$$ \varLambda _{1}(\varOmega )\leq \varLambda _{1}(B_{\varOmega }), \quad \text{with equality if and only if } \varOmega \text{ is a ball}. $$

Chasman [5] also conjectured that

$$ \sum_{i=1}^{n} \frac{1}{\varLambda _{i}(\varOmega )}\geq \frac{n}{\varLambda _{1}(B_{\varOmega })},\quad \text{with equality if and only if } \varOmega \text{ is a ball}. $$

(1.7)

In this paper, we prove an isoperimetric inequality for the sums of the reciprocals of the first $(n-1)$ nonzero eigenvalues of the fourth Neumann Laplacian which supports the Chasman’s conjecture, actually, we get

$$ \sum_{i=1}^{n-1} \frac{1}{\varLambda _{i}(\varOmega )}\geq \frac{n-1}{\varLambda _{1}(B_{\varOmega })},\quad \text{with equality if and only if } \varOmega \text{ is a ball}. $$

(1.8)

In [4], Buoso et al. proved a quantitative isoperimetric inequality for the fundamental tone of problem (1.5) as follows:

$$ \varLambda _{1}(\varOmega )\leq \bigl(1-\eta _{n,\tau ,|\varOmega |}A^{2}(\varOmega )\bigr) \varLambda _{1}(B_{\varOmega }),$$

(1.9)

where $\eta _{n,\tau ,|\varOmega |}>0$, and $A(\varOmega )$ is the so-called Fraenkel asymmetry of the domain $\varOmega \in \mathbb{R}^{n}$, which is defined by:

$$ A(\varOmega ):=\mathrm{inf} \biggl\{ \frac{ \vert \varOmega \Delta B_{\varOmega } \vert }{ \vert \varOmega \vert } \biggr\} , $$

where $B_{\varOmega }$ is the ball of same volume as Ω and $\varOmega \Delta B_{\varOmega }$ is the symmetric difference of Ω and $B_{\varOmega }$. In what follows, we generalize (1.9) to the sum of the first $(n-1)$ eigenvalues, and we get

$$ \frac{1}{n-1}\sum _{i=1}^{n-1}\varLambda _{i}(\varOmega )\leq \bigl(1-\eta _{n, \tau ,|\varOmega |}A^{2}(\varOmega )\bigr)\varLambda _{1}(B_{\varOmega }). $$

(1.10)

2 Preliminaries

In this section, we recall some notations and results, more details can be seen in [4, 5].

Let $j_{1}$, $i_{1}$ be the ultraspherical and modified ultraspherical Bessel functions of the first kind and order 1, respectively; $j_{1}$, $i_{1}$ can be expressed by the standard Bessel and modified Bessel functions of the first kind $J_{\nu }$, $I_{\nu }$ as follows:

$$ j_{1}(z)=z^{1-n/2}J_{n/2}(z),\qquad i_{1}(z)=z^{1-n/2}I_{n/2}(z). $$

Let B be the unit ball in $\mathbb{R}^{n}$ centered at the origin and $\omega _{n}$ be the Lebesgue measure $|B|$ of B, and let $\varLambda _{1}(B)$ be the first eigenvalue of problem (1.5) on unit ball B. For $\tau >0$, a, b are positive constants satisfying the conditions $a^{2}b^{2}=\lambda _{1}(B)$ and $b^{2}-a^{2}=\tau$. Set

$$ R(r)=j_{1}(ar)+\gamma i_{1}(br),\quad \gamma = \frac{-a^{2}j''_{1}(ar)}{b^{2}i''_{1}(b)}>0. $$

Then we define the function $\rho :[0,+\infty )\rightarrow [0,+\infty )$ as

$$ \rho (r)= \textstyle\begin{cases} R(r),& r\in [0,1), \\ R(1)+(r-1)R'(1),& r\in [1,+\infty ). \end{cases} $$

Let $u_{i}:\mathbb{R}^{n}\rightarrow \mathbb{R}$ be defined by

$$ u_{i}(x):=\rho \bigl( \vert x \vert \bigr)\frac{x_{i}}{ \vert x \vert },\quad \text{for } i=1,\dots , n. $$

(2.1)

The functions $u_{i}|_{B}$ are, in fact, the eigenfunctions associated with the eigenvalues $\lambda _{1}(B)$ of problem (1.5) on unit ball B. We know that $\lambda _{1}(B)$ has multiplicity and $u_{i}$ satisfy

$$\begin{aligned}& \sum_{i=1}^{n} \vert u_{i} \vert ^{2}=\rho ^{2}\bigl( \vert x \vert \bigr), \end{aligned}$$

(2.2)

$$\begin{aligned}& \sum_{i=1}^{n} \vert \nabla u_{i} \vert ^{2}=\frac{n-1}{ \vert x \vert ^{2}}\rho \bigl( \vert x \vert \bigr)^{2}+\bigl( \rho '\bigl( \vert x \vert \bigr)\bigr)^{2}, \end{aligned}$$

(2.3)

$$\begin{aligned}& \sum_{i=1}^{n} \bigl\vert \nabla ^{2} u_{i} \bigr\vert ^{2}=\bigl(\rho ''\bigl( \vert x \vert \bigr) \bigr)^{2}+ \frac{3(n-1)}{ \vert x \vert ^{4}}\bigl[\rho \bigl( \vert x \vert \bigr)- \vert x \vert \rho '\bigl( \vert x \vert \bigr) \bigr]^{2}. \end{aligned}$$

(2.4)

Define $N[\rho ]=\sum_{i=1}^{n} (|\nabla ^{2} u_{i}|^{2}+\tau |\nabla u_{i}|^{2} )$. Then ρ and $N[\rho ]$ satisfy the following properties which given in [4, 5].

Lemma 2.1

Function ρ and $N[\rho ]$satisfy the following properties:

• (1) $\rho ''(r)<0$for all $r\geq 0$, therefore $\rho '$is nonincreasing.

• (2) $\rho (r)-r\rho '(r)\geq 0$, with equality holding only for $r=0$.

• (3) The function $\rho ^{2}(r)$is strictly increasing.

• (4) The function $\rho ^{2}(r)/r^{2}$is decreasing.

• (5) The function $3(\rho (r)-r\rho '(r))^{2}/r^{4}+\tau \rho ^{2}(r)/r^{2}$is decreasing.

• (6) $N[\rho (r_{1})]>N[\rho (r_{2})]$for any $r_{1}\in [0,1)$, $r_{2}\in [1,+\infty )$.

• (7) For all $r\geq 0$, we have

  $$ N\bigl[\rho (r)\bigr]=\bigl(\rho ''(r) \bigr)^{2}+ \frac{3(n-1)(\rho (r)-r\rho '(r))^{2}}{r^{4}}+\tau (n-1) \frac{\rho ^{2}(r)}{r^{2}}+ \tau \bigl(\rho '(r)\bigr)^{2}. $$

• (8) For all $r\geq 1$, $N[\rho (r)]$is decreasing.

We introduce the notation of a partially monotonic function. A function F is partially monotonic on Ω if it satisfies

$$ F(x)>F(y), \quad \text{for all } x\in \varOmega \text{ and } y\notin \varOmega . $$

(2.5)

It is seen that $N[\rho (r)]$ is a partially monotonic function from Lemma 2.1.

Lemma 2.2

For any radial function $F(r(x))$that satisfies the partially monotonicity condition on $B_{\varOmega }$,

$$ \int _{\varOmega }F \,dx\leq \int _{B_{\varOmega }}F \,dx $$

(2.6)

with equality if and only if $\varOmega =B_{\varOmega }$. For any strictly increasing radial function $F(r(x))$,

$$ \int _{\varOmega }F \,dx\geq \int _{B_{\varOmega }}F \,dx $$

(2.7)

with equality if and only if $\varOmega =B_{\varOmega }$.

Lemma 2.3

For all $s>0$, we have

$$ \varLambda _{i}(\tau ,\varOmega )=s^{4}\varLambda _{i}\bigl(s^{-2}\tau ,s\varOmega \bigr),\quad i=1,\dots ,n,$$

(2.8)

where $s\varOmega =\{x\in \mathbb{R}^{n}: x/s\in \varOmega \}$for $s>0$.

Proof

For any $u\in H^{2}(\varOmega )$ with

$$\begin{aligned} u\neq 0 \quad \text{and}\quad \int _{\varOmega }u \,dx= \int _{\varOmega }u u_{1}\,dx= \cdots = \int _{\varOmega }u u_{i-1}\,dx=0,\quad i=1,\dots ,n, \end{aligned}$$

let $\tilde{u}(x)=u(x/s)$, then ũ is a valid trial function on sΩ and so

$$\begin{aligned} Q_{s^{-2}\tau ,s\varOmega }[\tilde{u}] =&\frac{\int _{s\varOmega } ( \vert \nabla ^{2} \tilde{u} \vert ^{2}+s^{-2}\tau \vert \nabla \tilde{u} \vert ^{2} )\,dx}{\int _{s\varOmega } u^{2}\,dx} \\ =&\frac{\int _{s\varOmega } ( \vert s^{-2}(\nabla ^{2} u)(x/s) \vert ^{2}+s^{-2}\tau \vert s^{-1}(\nabla u)(x/s) \vert ^{2} )\,dx}{\int _{s\varOmega } u(x/s)^{2}\,dx} \\ =&\frac{s^{-4+n}\int _{\varOmega } ( \vert (\nabla ^{2} u) \vert ^{2}+\tau \vert (\nabla u) \vert ^{2} )\,dy}{s^{n}\int _{\varOmega } u^{2}\,dy} \quad ( \text{substituting } y=x/s) \\ =&s^{-4}Q_{\tau ,\varOmega }[u]. \end{aligned}$$

(2.9)

The lemma follows from (1.6). □

3 Proofs of the main results

In this section, we give the proofs of the main results of this paper.

Theorem 3.1

Let Ω be a bounded domain in an n-dimensional Euclidean space $\mathbb{R}^{n}$and let $B_{\varOmega }$be the ball of same volume as Ω, then the first $(n-1)$eigenvalues of (1.5) in Ω satisfy

$$ \sum_{i=1}^{n-1} \frac{1}{\varLambda _{i}(\varOmega )}\geq \frac{n-1}{\varLambda _{1}(B_{\varOmega })}, $$

(3.1)

with equality if and only if Ω is a ball.

Proof

Assume that the volume of Ω is equal to that of the unit ball B. Letting $\varphi _{i}=\frac{\rho (r)x_{i}}{r}$, we know that

$$ \int _{\varOmega }\varphi _{i}(r)\,dx=0, \quad \text{for } i=1,\dots ,n, $$

which means $\varphi _{i}$ is perpendicular to $u_{0}=1/\sqrt{|\varOmega |}$, which is the first eigenfunction of (1.5). Letting $\{u_{j}\}_{j=0}^{\infty }$ be an orthonormal set of eigenfunctions of (1.5) on Ω, next we will show that there exists new coordinate functions $\{x_{i}'\}_{i=1}^{n}$ such that

$$ \int _{\varOmega }\frac{\rho (r)x_{i}'}{r}u_{j} \,dx=0, $$

(3.2)

for $j=1,\dots ,i-1$ and $i=2,\dots , n$. To see this, we define an $n \times n$ matrix $A= (a_{ij} )$, where $a_{ji}= \int _{\varOmega }\varphi _{i}u_{j}\,dx=\int _{\varOmega } \frac{\rho (r)}{r}x_{i}u_{j}\,dx$, for $i,j=1,2,\dots ,n$. Using the orthogonalization of Gram and Schmidt (QR-factorization theorem), we know that there exist an upper-triangular matrix $T=(T_{ji})$ and an orthogonal matrix $B=(b_{ji})$ such that $T=UQ$, i.e.,

$$ T_{ij}=\sum_{k=1}^{n} b_{ik}a_{kj}= \int _{\varOmega }\sum_{k=1}^{n} \frac{\rho (r)}{r}b_{ik}x_{k}u_{j}\,dx =0,\quad 1\leq j< i\leq n. $$

Letting $x_{i}'=\sum_{k=1}^{n} b_{ik}x_{k}$, $i=1,\dots ,n$, we get (3.2). Since $B=(b_{ji})$ is an orthogonal matrix, $\{x_{i}'\}_{i=1}^{n}$ is also a set of coordinate functions. Therefore, denoting $x_{i}'$, $i=1\dots ,n$ still by $x_{i}$, $i=1\dots ,n$, and $\varphi _{i}= \frac{\rho (r)}{r}x_{i}$, we have

$$ \varphi _{i}\neq 0 \quad \text{and}\quad \int _{\varOmega }\varphi _{i}\,dx= \int _{ \varOmega }\varphi _{i} u_{1}\,dx= \cdots = \int _{\varOmega }\varphi _{i} u_{i-1}\,dx=0,\quad i=1,\dots ,n. $$

It follows from the Rayleigh–Ritz inequality that

$$ \varLambda _{i}(\varOmega ) \int _{\varOmega }\varphi _{i}^{2} \,dx\leq \int _{ \varOmega } \bigl( \bigl\vert \nabla ^{2} \varphi _{i} \bigr\vert ^{2}+\tau \vert \nabla \varphi _{i} \vert ^{2} \bigr)\,dx,\quad i=1,\dots ,n, $$

(3.3)

which implies that

$$ \int _{\varOmega }\varphi _{i}^{2} \,dx\leq \frac{1}{\varLambda _{i}(\varOmega )} \int _{\varOmega } \bigl( \bigl\vert \nabla ^{2} \varphi _{i} \bigr\vert ^{2}+\tau \vert \nabla \varphi _{i} \vert ^{2} \bigr)\,dx, \quad i=1, \dots ,n. $$

(3.4)

Summing over i from 1 to n, we have

$$ \sum_{i=1}^{n} \int _{\varOmega }\varphi _{i}^{2} \,dx\leq \sum_{i=1}^{n} \frac{1}{\varLambda _{i}(\varOmega )} \int _{\varOmega } \bigl( \bigl\vert \nabla ^{2} \varphi _{i} \bigr\vert ^{2}+\tau \vert \nabla \varphi _{i} \vert ^{2} \bigr)\,dx. $$

(3.5)

Since $\sum_{i=1}^{n}|\nabla ^{2}\varphi _{i}|^{2}=(\rho '')^{2}+ \frac{3(n-1)}{r^{4}}(\rho -r\rho ')^{2}$, for any point $p\in \varOmega$, by a transformation of coordinates if necessary, we have $|\nabla ^{2}\varphi _{i}|^{2}\leq \frac{(\rho '')^{2}}{n-1}+ \frac{3}{r^{4}}(\rho -r\rho ')^{2}$, $i=1,\dots ,n$. Then

$$\begin{aligned}& \sum_{i=1}^{n} \frac{1}{\varLambda _{i}(\varOmega )} \bigl\vert \nabla ^{2}\varphi _{i} \bigr\vert ^{2} \\& \quad = \sum _{i=1}^{n-1}\frac{1}{\varLambda _{i}(\varOmega )} \bigl\vert \nabla ^{2}\varphi _{i} \bigr\vert ^{2} + \frac{1}{\varLambda _{n}(\varOmega )} \bigl\vert \nabla ^{2}\varphi _{i} \bigr\vert ^{2} \\& \quad = \sum_{i=1}^{n-1} \frac{1}{\varLambda _{i}(\varOmega )} \bigl\vert \nabla ^{2} \varphi _{i} \bigr\vert ^{2}+\frac{1}{\varLambda _{n}(\varOmega )} \Biggl( \bigl(\rho ''\bigr)^{2} + \frac{3(n-1)}{r^{4}}\bigl(\rho -r\rho '\bigr)^{2}-\sum _{j=1}^{n-1} \bigl\vert \nabla ^{2} \varphi _{j} \bigr\vert ^{2} \Biggr) \\& \quad \leq \sum_{i=1}^{n-1} \frac{1}{\varLambda _{i}(\varOmega )} \bigl\vert \nabla ^{2} \varphi _{i} \bigr\vert ^{2} +\sum _{j=1}^{n-1}\frac{1}{\varLambda _{j}(\varOmega )} \biggl( \frac{(\rho '')^{2}+\frac{3(n-1)}{r^{4}}(\rho -r\rho ')^{2}}{n-1}- \bigl\vert \nabla ^{2}\varphi _{j} \bigr\vert ^{2} \biggr) \\& \quad = \frac{1}{n-1} \biggl(\bigl(\rho '' \bigr)^{2}+\frac{3(n-1)}{r^{4}}\bigl(\rho -r \rho ' \bigr)^{2} \biggr)\sum_{i=1}^{n-1} \frac{1}{\varLambda _{i}(\varOmega )}. \end{aligned}$$

(3.6)

Similarly, we have

$$\begin{aligned} \sum_{i=1}^{n} \frac{1}{\varLambda _{i}(\varOmega )} \vert \nabla \varphi _{i} \vert ^{2} =&\sum_{i=1}^{n-1} \frac{1}{\varLambda _{i}(\varOmega )} \vert \nabla \varphi _{i} \vert ^{2}+ \frac{1}{\varLambda _{n}(\varOmega )} \vert \nabla \varphi _{i} \vert ^{2} \\ =&\sum_{i=1}^{n-1} \frac{1}{\varLambda _{i}(\varOmega )} \vert \nabla \varphi _{i} \vert ^{2} +\frac{1}{\varLambda _{n}(\varOmega )} \Biggl(\frac{n-1}{r^{2}}\rho ^{2}+\bigl( \rho '\bigr)^{2}-\sum _{j=1}^{n-1} \vert \nabla \varphi _{j} \vert ^{2} \Biggr) \\ \leq &\sum_{i=1}^{n-1} \frac{1}{\varLambda _{i}(\varOmega )} \vert \nabla \varphi _{i} \vert ^{2} +\sum_{j=1}^{n-1} \frac{1}{\varLambda _{j}(\varOmega )} \biggl(\frac{\frac{n-1}{r^{2}}\rho ^{2}+(\rho ')^{2}}{n-1}- \vert \nabla \varphi _{j} \vert ^{2} \biggr) \\ =&\frac{1}{n-1} \biggl(\frac{n-1}{r^{2}}\rho ^{2}+ \bigl(\rho '\bigr)^{2} \biggr) \sum _{i=1}^{n-1}\frac{1}{\varLambda _{i}(\varOmega )}. \end{aligned}$$

(3.7)

On the other hand,

$$ \sum_{i=1}^{n} \vert \varphi _{i} \vert ^{2}=\rho ^{2}. $$

(3.8)

Substituting (3.6)–(3.8) into (3.5), we have

$$\begin{aligned} \frac{1}{n-1}\sum _{i=1}^{n-1}\frac{1}{\varLambda _{i}(\varOmega )} \geq & \frac{\int _{\varOmega }\rho ^{2}\,dx}{\int _{\varOmega } ((\rho '')^{2}+\frac{3(n-1)}{r^{4}}(\rho -r\rho ')^{2}+\tau (\frac{n-1}{r^{2}}\rho ^{2}+(\rho ')^{2} ) )\,dx} \\ =&\frac{\int _{\varOmega }\rho ^{2}\,dx}{\int _{\varOmega }N[\rho ]\,dx}\geq \frac{\int _{B_{\varOmega }} \rho ^{2}\,dx}{\int _{B_{\varOmega }} N[\rho ]\,dx}= \frac{1}{\varLambda _{1}(B_{\varOmega })}, \end{aligned}$$

(3.9)

the last step is deduced by Lemma 2.2. If the equality holds, then equality holds in (3.9), which implies Ω must be a unit ball. By Lemma 2.3, for any domain Ω in $\mathbb{R}^{n}$, we get

$$ \frac{1}{n-1}\sum_{i=1}^{n-1} \frac{1}{\varLambda _{i}(\varOmega )}\geq \frac{1}{\varLambda _{1}(B_{\varOmega })}. $$

(3.10)

This completes the proof of Theorem 3.1. □

Theorem 3.2

Let Ω be a bounded domain in an n-dimensional Euclidean space $\mathbb{R}^{n}$and let $B_{\varOmega }$be the ball of same volume as Ω, then the first $(n-1)$eigenvalues of (1.5) in Ω satisfy

$$ \frac{1}{n-1}\sum _{i=1}^{n-1}\varLambda _{i}(\varOmega )\leq \bigl(1-\eta _{n, \tau ,|\varOmega |}A^{2}(\varOmega )\bigr)\varLambda _{1}(B_{\varOmega }). $$

(3.11)

Proof

Case 1. Ω is a bounded domain in $\mathbb{R}^{n}$ of class $C^{1}$ with the same measure as the unit ball B. By a similar argument as in the proof of Theorem 3.1, we have

$$ \varLambda _{i}(\varOmega ) \int _{\varOmega }\varphi _{i}^{2} \,dx\leq \int _{ \varOmega } \bigl( \bigl\vert \nabla ^{2} \varphi _{i} \bigr\vert ^{2}+\tau \vert \nabla \varphi _{i} \vert ^{2} \bigr)\,dx,\quad i=1,\dots ,n. $$

(3.12)

Summing over i from 1 to n, we have

$$ \sum_{i=1}^{n} \varLambda _{i}(\varOmega ) \int _{\varOmega }\varphi _{i}^{2} \,dx \leq \sum_{i=1}^{n} \int _{\varOmega } \bigl( \bigl\vert \nabla ^{2} \varphi _{i} \bigr\vert ^{2}+ \tau \vert \nabla \varphi _{i} \vert ^{2} \bigr)\,dx= \int _{\varOmega }N[\rho ]\,dx. $$

(3.13)

Since $\sum_{i=1}^{n}\varphi _{i}^{2}=\rho ^{2}$, for any point $p\in \varOmega$, by a transformation of coordinates if necessary, we have $\varphi _{i}^{2}\leq \frac{\rho ^{2}}{n-1}$, $i=1,\dots ,n$. Then

$$\begin{aligned} \sum _{i=1}^{n}\varLambda _{i}(\varOmega ) \varphi _{i}^{2} =&\sum_{i=1}^{n-1} \varLambda _{i}(\varOmega )\varphi _{i}^{2}+ \varLambda _{n}(\varOmega )\varphi _{n}^{2} \\ =&\sum_{i=1}^{n-1}\varLambda _{i}(\varOmega )\varphi _{i}^{2}+\varLambda _{n}( \varOmega ) \Biggl(\rho ^{2}-\sum _{j=1}^{n-1}\varphi _{j}^{2} \Biggr) \\ \geq &\sum_{i=1}^{n-1}\varLambda _{i}(\varOmega )\varphi _{i}^{2}+\sum _{j=1}^{n-1} \varLambda _{j} \biggl( \frac{\rho ^{2}}{n-1}-\varphi _{j}^{2} \biggr) \\ =&\sum_{i=1}^{n-1}\varLambda _{i}\frac{\rho ^{2}}{n-1}. \end{aligned}$$

(3.14)

Substituting (3.13) into (3.14), we have

$$ \frac{1}{n-1}\sum _{i=1}^{n-1}\varLambda _{i}(\varOmega )\leq \frac{\int _{\varOmega }N[\rho ]\,dx}{\int _{\varOmega }\rho ^{2}\,dx}. $$

(3.15)

On the other hand, we have

$$ \varLambda _{1}(B)=\frac{\int _{B} N[\rho ]\,dx}{\int _{B}\rho ^{2}\,dx}. $$

(3.16)

Combining (3.15) and (3.16), we have

$$ \varLambda _{1}(B) \int _{B}\rho ^{2}\,dx-\frac{1}{n-1}\sum _{i=1}^{n-1} \varLambda _{i}( \varOmega ) \int _{\varOmega }\rho ^{2}\,dx\geq \int _{B} N[\rho ]\,dx- \int _{\varOmega }N[\rho ]\,dx. $$

(3.17)

From equation (16) in [4], we know that

$$ \varLambda _{1}(B) \int _{B}\rho ^{2}\,dx-\varLambda _{1}( \varOmega ) \int _{ \varOmega }\rho ^{2}\,dx\leq C_{n,\tau }^{(1)} \bigl(\varLambda _{1}(B)-\varLambda _{1}( \varOmega )\bigr), $$

where $C_{n,\tau }^{(1)}=n\omega _{n}\int _{0}^{1}\rho ^{2}(r)r^{n-1}\,dr$. Then we have

$$ \varLambda _{1}(B) \int _{B}\rho ^{2}\,dx-\frac{1}{n-1}\sum _{i=1}^{n-1} \varLambda _{i}( \varOmega ) \int _{\varOmega }\rho ^{2}\,dx\leq C_{n,\tau }^{(1)} \Biggl(\varLambda _{1}(B)-\frac{1}{n-1}\sum _{i=1}^{n-1}\varLambda _{i}( \varOmega ) \Biggr). $$

(3.18)

From (15) and (20) in [4], we know that

$$ \varLambda _{1}(B) \int _{B}\rho ^{2}\,dx-\varLambda _{1}( \varOmega ) \int _{ \varOmega }\rho ^{2}\,dx\geq \int _{B/B_{1}}N(\rho )\,dx- \int _{B_{2}/B}N( \rho )\,dx, $$

and

$$ \int _{B/B_{1}}N(\rho )\,dx- \int _{B_{2}/B}N(\rho )\,dx=C_{n,\tau }^{(2)} \alpha ^{2}, $$

where $B_{1}$ and $B_{2}$ are two balls centered at the origin with radii $r_{1}$, $r_{2}$ such that $|\varOmega \cap B|=|B_{1}|=\omega _{n} r_{1}^{n}$ and $|\varOmega /B|=|B_{2}/B|=\omega _{n} (r_{2}^{n}-1)$. Then we have

$$ \int _{B} N[\rho ]\,dx- \int _{\varOmega }N[\rho ]\,dx\geq C_{n,\tau }^{(2)} \frac{ \vert \varOmega \Delta B \vert }{ \vert \varOmega \vert }, $$

(3.19)

where $C_{n,\tau }^{(2)}=n\omega _{n} ((3+\tau )(R(1)-R'(1))^{2}+2\tau R'(1)(R(1)-R'(1)) )c_{n}$.

Combining (3.18) and (3.19), we have

$$ \varLambda _{1}(B)-\frac{1}{n-1}\sum _{i=1}^{n-1}\varLambda _{i}( \varOmega )\geq \frac{C_{n,\tau }^{(2)}}{C_{n,\tau }^{(1)}}A^{2}(\varOmega ), $$

which implies that

$$ \frac{1}{n-1}\sum _{i=1}^{n-1}\varLambda _{i}(\varOmega )\leq \varLambda _{1}(B) \biggl(1-\frac{C_{n,\tau }^{(2)}}{\varLambda _{1}(B)C_{n,\tau }^{(1)}}A^{2}( \varOmega ) \biggr) .$$

(3.20)

Case 2. Ω is the generic domain in $\mathbb{R}^{n}$ of class $C^{1}$. Since

$$ \varLambda _{i}(\tau ,\varOmega )=s^{4}\varLambda _{i}\bigl(s^{-2}\tau ,s\varOmega \bigr),\quad i=1,\dots ,n, $$

(3.21)

for all $s>0$. Taking $s=(\omega _{n}/|\varOmega |)^{\frac{1}{n}}$, for any domain Ω in $\mathbb{R}^{n}$ of class $C^{1}$, we infer from (3.21) that

$$\begin{aligned} \frac{1}{n-1}\sum_{i=1}^{n-1} \varLambda _{i}(\tau ,\varOmega ) =&s^{4} \frac{1}{n-1} \sum_{i=1}^{n-1}\varLambda _{i} \bigl(s^{-2}\tau ,s\varOmega \bigr) \\ \leq &s^{4}\varLambda _{1}\bigl(s^{-2}\tau ,B \bigr) \biggl(1- \frac{C_{n,s^{-2}\tau }^{(2)}}{\varLambda _{1}(s^{-2}\tau ,B) C_{n,s^{-2}\tau }^{(1)}}A^{2}(s \varOmega ) \biggr) \\ =&\varLambda _{1}\bigl(s^{-2}\tau ,B\bigr) \biggl(1- \frac{C_{n,s^{-2}\tau }^{(2)}}{\varLambda _{1}(s^{-2}\tau ,B)C_{n,s^{-2}\tau }^{(1)}} A^{2}( \varOmega ) \biggr). \end{aligned}$$

Setting $\eta _{n,\tau ,|\varOmega |}= \frac{C_{n,s^{-2}\tau }^{(2)}}{\varLambda _{1}(s^{-2}\tau ,B)C_{n,s^{-2}\tau }^{(1)}}$, we have (1.10). This completes the proof of Theorem 3.2. □