Bounds for the Steklov eigenvalues

Abstract

Let \(\Omega \) be a star-shaped bounded domain in \((\mathbb {S}^{n}, ds^{2})\) with smooth boundary. In this article, we give a sharp lower bound for the first non-zero eigenvalue of the Steklov eigenvalue problem in \(\Omega .\) This result extends a result given by Kuttler and Sigillito (SIAM Rev 10:368–370, 1968) for a star-shaped bounded domain in \(\mathbb {R}^2\). Further we also obtain a two sided bound for the eigenvalues of the Steklov problem on a ball in \(\mathbb {R}^n\) with rotationally invariant metric and with bounded radial curvature.