Simplicity of eigenvalues for elliptic problems with mixed Steklov-Robin boundary condition

TL;DR

This paper proves that for a generic domain, all eigenvalues of certain elliptic problems with mixed Steklov-Robin boundary conditions are simple, using domain perturbation and operator transversality techniques.

Contribution

It establishes the generic simplicity of eigenvalues for elliptic problems with mixed Steklov-Robin boundary conditions, a novel result in spectral theory.

Findings

Eigenvalues are simple for a generic domain.

Domain perturbation techniques are effective in spectral analysis.

Transversality of operators is key to proving simplicity.

Abstract

This paper investigates the spectral properties of two classes of elliptic problems characterized by mixed Steklov-Robin boundary conditions. Our main objective is to prove that, for a generic domain, all the eigenvalues are simple. This result is established by employing domain perturbation techniques and analyzing the transversality of the associated operators.