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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.

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 [79].

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)

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. (1)

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

  2. (2)

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

  3. (3)

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

  4. (4)

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

  5. (5)

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

  6. (6)

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

  7. (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. (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 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). □

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. □

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Acknowledgements

The authors would like to thank the referee for his or her careful reading and valuable comments such that the article appears in its present version.

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This work is supported by NSF of Hubei Provincial Department of Education (Grant Nos. B2019211, D20184301, B2016261), Research Team Project of Jingchu University of Technology (Grant No. TD202006), Research Project of Jingchu University of Technology (Grant No. QDB201608) and Hubei Key Laboratory of Applied Mathematics (Hubei University).

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Deng, Y., Du, F. Isoperimetric inequalities of the fourth order Neumann eigenvalues. J Inequal Appl 2020, 210 (2020). https://doi.org/10.1186/s13660-020-02476-9

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MSC

  • 35P15
  • 53C40
  • 58C40

Keywords

  • Eigenvalues
  • Neumann problem
  • Isoperimetric inequality