Kuttler-Sigillito's Inequalities and Rellich-Christianson Identity

This article has two main objectives: The first one is to show Kuttler-Sigillito type inequalities involving the mixed Neumann-Dirichlet, mixed Steklov-Dirichlet, and mixed Robin-Dirichlet eigenvalue problems for the Laplacian. We will demonstrate these inequalities on different domains e.g., squares and balls. The second objective is to 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.