Bounds for the first non-zero Steklov eigenvalue

Abstract

Let $\Omega$ be a star-shaped bounded domain in $({\mathbb{S}}^n,d{s}^2)$ 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 is the generalization of a result given by Kuttler and Sigillito for a star-shaped bounded domain in ${\mathbb{R}}^2.$ Further we also obtain a two sided bound for the first non-zero eigenvalue of the Steklov problem on the ball in ${\mathbb{R}}^n$ with rotationally invariant metric and with bounded radial curvature.