On an inequality between Dirichlet and Neumann eigenvalues for the Laplace operator

Abstract

A simple proof of the inequality µ k+1 < λ k is given. Here the λ k (re-spectively, µ k) are the eigenvalues of the Dirichlet (respectively, Neumann) problem for the Laplace operator in an arbitrary domain of finite measure in R d , d > 1. Let Ω be a domain in R d such that the Sobolev space W 1 2 (Ω) is compactly embedded in L 2 (Ω). Then the spectra of the Dirichlet problem and the Neumann problem for the Laplace operator are both discrete. We denote the respective operators by −∆ D and −∆ N , and enumerate their eigenvalues in increasing order (with multiplicity taken into account): σ(−∆ D) = {λ k } ∞ k=1 , σ(−∆ N) = {µ k } ∞ k=1. Note that 1) µ 1 = 0 if the measure of the domain Ω is finite; 2) µ k+1 = λ k if d = 1; 3) the inequality µ k ≤ λ k is deduced immediately by variational arguments. For d = 2 and for a domain bounded by an analytic curve, Pólya and Szeg˝ o proved (see [P, S]) that µ 2 ≤ γλ 1 , where γ is an absolute constant less than one (expressed in terms of zeros of Bessel functions). Since their proofs involve conformal mappings, they do not work in higher dimensions. Developing an idea used in [Pa], Levine and Weinberger established (see [LW]), for an arbitrary dimension, a series of inequalities of the form µ k+r < λ k , r = 1,. .. , d, under some conditions on the principal curvatures of the C 2+α-smooth boundary ∂Ω of a bounded domain Ω. In particular, µ k+1 < λ k if the mean curvature is nonnegative, and µ k+d ≤ λ k for all convex domains. Friedlander [F] proved the inequality µ k+1 ≤ λ k for the bounded domains Ω with ∂Ω ∈ C 1. He used the "Dirichlet-to-Neumann" operator R(λ) that maps a function ϕ defined on ∂Ω to the normal derivative on ∂Ω of the solution u of the problem (−∆ − λ)u = 0 in Ω, u = ϕ on ∂Ω. Friedlander obtained the following formula: n(λ) = N N (λ) − N D (λ) for λ ∈ σ(−∆ D) ∪ σ(−∆ N), where n(λ) is the number of negative eigenvalues of the operator R(λ), and N N , N D are the counting functions of the Laplace operator (see (1)), and he showed that n(λ) ≥ 1, which implies the desired inequality. We have succeeded in finding a simple proof of the inequality µ k+1 < λ k in a more general situation.