Inequalities for the Steklov Eigenvalues

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 $n(\ge 2)$-dimensional Hadamard manifold an let $0={\lambda }_0<{\lambda }_1\le {\lambda }_2\le ...$ denote the eigenvalues of the Steklov problem: $\Delta u=0$ in $\Omega$ and $(\partial u)/(\partial \nu )=\lambda u$ on $\partial \Omega$. Then ${\sum }_{i=1}^n{\lambda }_i^{-1}\ge (n^2|\Omega |)/(|\partial \Omega |)$ with equality holding if and only if $\Omega$ is isometric to an $n$-dimensional Euclidean ball. Let $M$ be an $n(\ge 2)$-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 $q_1\ge c$ with equality holding if and only if $M$ is isometric to a ball of radius $1/c$ in ${\mathbf{R}}^n$.

17 pages