Bounds for the first non-zero Steklov eigenvalue

TL;DR

This paper establishes sharp lower bounds for the first non-zero Steklov eigenvalue in star-shaped domains on spheres and provides two-sided bounds for eigenvalues on certain Riemannian balls, generalizing previous Euclidean results.

Contribution

It generalizes known Euclidean bounds to spherical and Riemannian settings, offering new sharp bounds for Steklov eigenvalues in these geometries.

Findings

Sharp lower bound for Steklov eigenvalues in star-shaped domains on spheres.

Two-sided bounds for Steklov eigenvalues on Riemannian balls with rotational symmetry.

Extension of Euclidean results to curved geometries.

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 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.