Mathematics – Spectral Theory
Scientific paper
2010-06-07
Mathematics
Spectral Theory
17 pages
Scientific paper
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 ${\bf R}^n$.
Wang Qiaoling
Xia Changyu
No associations
LandOfFree
Inequalities for the Steklov Eigenvalues does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Inequalities for the Steklov Eigenvalues, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Inequalities for the Steklov Eigenvalues will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-366288