Kuttler-Sigillito's Inequalities and Rellich-Christianson Identity

TL;DR

This paper establishes inequalities and identities related to mixed boundary eigenvalue problems, including examples for specific shapes, and introduces a Rellich-Christianson identity for the Neumann-Dirichlet problem.

Contribution

It presents new inequalities and identities for mixed boundary eigenvalue problems, extending classical results to these cases.

Findings

Kuttler-Sigillito's inequalities for mixed problems

Examples on squares and balls

Rellich-Christianson identity for Neumann-Dirichlet problem

Abstract

This article has two main objectives. The first one is to show Kuttler-Sigillito's type inequalities involving the mixed Neumann-Dirichlet, mixed Steklov-Dirichlet, and mixed Robin-Dirichlet eigenvalue problems. We will provide examples on e.g. squares and balls for the inequalities presented. Next we will show a Rellich identity for the mixed Neumann-Dirichlet problem. This identity will be used to prove a Rellich-Christianson identity for the Neumann-Dirichlet problem.