Inequalities for the Steklov Eigenvalues

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 ${\bf R}^n$.