Some sharp bounds for Steklov eigenvalues

Abstract

This work is an extension of a result given by Kuttler and Sigillito (SIAM Rev $10$:$368-370$, $1968$) on a star-shaped bounded domain in ${\mathbb{R}}^2$. Let $\Omega$ be a star-shaped bounded domain in a hypersurface of revolution, having smooth boundary. In this article, we obtain a sharp lower bound for all Steklov eigenvalues on $\Omega$ in terms of the Steklov eigenvalues of the largest geodesic ball contained in $\Omega$ with the same center as $\Omega$. We also obtain similar bounds for all Steklov eigenvalues on star-shaped bounded domain in paraboloid, $P=\left\{(x,y,z)\in {\mathbb{R}}^3:z=x^2+y^2\right\}$.

9 pages