Kuttler–Sigillito 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.