A note on Kuttler-Sigillito's inequalities

TL;DR

This paper extends Kuttler-Sigillito's eigenvalue inequalities from Euclidean domains to Riemannian manifolds using a generalized Rellich identity, broadening their applicability in geometric analysis.

Contribution

It introduces new eigenvalue inequalities on Riemannian manifolds, generalizing previous Euclidean results with a novel proof technique.

Findings

Eigenvalue inequalities established for domains with $C^2$ boundary on Riemannian manifolds.

Extension of Kuttler-Sigillito inequalities from $ ext{Euclidean}$ to manifold settings.

Use of generalized Rellich identity as a key analytical tool.

Abstract

We provide several inequalities between eigenvalues of some classical eigenvalue problems on domains with $C^2$ boundary in complete Riemannian manifolds. A key tool in the proof is the generalized Rellich identity on a Riemannian manifold. Our results in particular extend some inequalities due to Kutller and Sigillito from subsets of $\mathbb{R}^2$ to the manifold setting.