Inequalities for the Steklov Eigenvalues

2010/06/07 by Xia, Changyu, Wang, Qiaoling · 1 citation
#FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1006.1154

Abstract

This paper studies eigenvalues of some Steklov problems. Among other things, we show the following sharp estimtes. Let Ω be a bounded smooth domain in an n(≥ 2)-dimensional Hadamard manifold an let 0=λ₀ < λ₁≤ λ₂≤ ... denote the eigenvalues of the Steklov problem: Δu=0 in Ω and (∂ u)/(∂ ν)=λu on ∂ Ω. Then ∑_i=1ⁿ λ^-1_i ≥ (n²|Ω|)/(|∂Ω|) with equality holding if and only if Ω is isometric to an n-dimensional Euclidean ball. Let M be an n(≥ 2)-dimensional compact connected Riemannian manifold with boundary and non-negative Ricci curvature. Assume that the mean curvature of \pa M is bounded below by a positive constant c and let q₁ be the first eigenvalue of the Steklov problem: Δ² u= 0 in M and u= (∂² u)/(∂ ν²) -q(∂ u)/(∂ ν) =0 on ∂ M. Then q₁≥ c with equality holding if and only if M is isometric to a ball of radius 1/c in \bf Rⁿ.

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• Scaling inequalities for Steklov eigenvalues in space forms and sharp eigenvalue estimates on warped product manifolds

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