Isoperimetric inequalities of the fourth order Neumann eigenvalues

In this paper, we obtain some isoperimetric inequalities for the first (n−1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(n-1)$\end{document} 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 R n , we consider the Neumann problem of the Laplacian as follows: 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 = μ 0 < μ 1 ≤ μ 2 ≤ · · · → +∞.
When Ω is a bounded domain in R 2 , Szegö [6] proved the following classical isoperimetric inequality: where B Ω is the ball of same volume as Ω. Weinberger [11] generalized this result to ndimensions. 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 , with equality if and only if Ω is a ball, (1.3) where μ i (Ω) is the ith Neumann eigenvalue on Ω, μ 1 (B Ω ) is the first nonzero Neumann eigenvalue on B Ω . 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 R n as follows: , with equality if and only if Ω 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 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][8][9]. Let and 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: where τ ≥ 0 and σ are two constants, div ∂Ω denotes the tangential divergence operator on ∂Ω, and ∇ 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: By the Rayleigh-Ritz characterization, the (k + 1)th eigenvalue of (1.5) can be given as follows (see, e.g., [5]): Letting B Ω be the ball of same volume as Ω, Chasman [5] proved the following isoperimetric inequality: , with equality if and only if Ω is a ball.
Chasman [5] also conjectured that , with equality if and only if Ω 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 , with equality if and only if Ω 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: where η n,τ ,|Ω| > 0, and A(Ω) is the so-called Fraenkel asymmetry of the domain Ω ∈ R n , which is defined by: where B Ω is the ball of same volume as Ω and Ω B Ω is the symmetric difference of Ω and B Ω . In what follows, we generalize (1.9) to the sum of the first (n -1) eigenvalues, and we get (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 ν , I ν as follows: Let B be the unit ball in R n centered at the origin and ω n be the Lebesgue measure |B| of B, and let Λ 1 (B) be the first eigenvalue of problem (1.5) on unit ball B. For τ > 0, a, b are positive constants satisfying the conditions Then we define the function ρ : [0, +∞) → [0, +∞) as Let u i : R n → R be defined by The functions u i | B are, in fact, the eigenfunctions associated with the eigenvalues λ 1 (B) of problem (1.5) on unit ball B. We know that λ 1 (B) has multiplicity and u i satisfy . Then ρ and N[ρ] satisfy the following properties which given in [4,5].
We introduce the notation of a partially monotonic function. A function F is partially monotonic on Ω if it satisfies F(x) > F(y), for all x ∈ Ω and y / ∈ Ω. (2.5) It is seen that N[ρ(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
with equality if and only if Ω = B Ω . For any strictly increasing radial function F(r(x)), with equality if and only if Ω = B Ω .
Proof For any u ∈ H 2 (Ω) with letũ(x) = u(x/s), thenũ is a valid trial function on sΩ and so 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 R n and let B Ω be the ball of same volume as Ω, then the first (n -1) eigenvalues of (1.5) in Ω satisfy 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 ϕ i = ρ(r)x i r , we know that Ω ϕ i (r) dx = 0, for i = 1, . . . , n, which means ϕ i is perpendicular to u 0 = 1/ √ |Ω|, which is the first eigenfunction of (1.5).
On the other hand, , (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 R n , we get . (3.10) This completes the proof of Theorem 3.1.