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.
Highlights
1 Introduction Letting Ω be a bounded domain with a smooth boundary ∂Ω in the Euclidean space Rn, we consider the Neumann problem of the Laplacian as follows:
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 Rn as follows: n–1
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, we get n–1
Summary
Letting Ω be a bounded domain with a smooth boundary ∂Ω in the Euclidean space Rn, we consider the Neumann problem of the Laplacian as follows:. 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 Rn as follows: n–1. Where τ ≥ 0 and σ are two constants, div∂Ω denotes the tangential divergence operator on ∂Ω, and ∇2u 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:.
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