Inequalities for the Steklov Eigenvalues

TL;DR

This paper establishes sharp inequalities for Steklov eigenvalues on smooth domains in Hadamard manifolds and on Riemannian manifolds with boundary, characterizing equality cases as Euclidean balls.

Contribution

It provides new sharp estimates for Steklov eigenvalues on manifolds, extending classical results to more general geometric settings and identifying conditions for equality.

Findings

Sum of inverse eigenvalues bounded below by domain volume and boundary area ratio.

First eigenvalue of a biharmonic Steklov problem bounded below by boundary mean curvature.

Equality cases characterized as Euclidean balls in respective geometries.

Abstract

This paper studies eigenvalues of some Steklov problems. Among other things, we show the following sharp estimtes. Let $\Omega$ be a bounded smooth domain in an $n(\geq 2)$-dimensional Hadamard manifold an let $0=\lambda_0 < \lambda_1\leq \lambda_2\leq ... $ denote the eigenvalues of the Steklov problem: $\Delta u=0$ in $\Omega$ and $(\partial u)/(\partial \nu)=\lambda u$ on $\partial \Omega$. Then $\sum_{i=1}^{n} \lambda^{-1}_i \geq (n^2|\Omega|)/(|\partial\Omega|) $ with equality holding if and only if $\Omega$ is isometric to an $n$-dimensional Euclidean ball. Let $M$ be an $n(\geq 2)$-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 $q_1$ be the first eigenvalue of the Steklov problem: $ \Delta^2 u= 0$ in $ M$ and $u= (\partial^2 u)/(\partial \nu^2) -q(\partial u)/(\partial \nu) =0$ on $ \partial M$. Then $q_1\geq c$ with equality holding if and only if $M $ is isometric to a ball of radius $1/c$ in ${\bf R}^n$.