Analysis of the transmission eigenvalue problem with two conductivity parameters

TL;DR

This paper analytically studies the transmission eigenvalue problem with two conductivity parameters, proving existence, discreteness, and parameter dependence, supported by numerical validation.

Contribution

It extends previous analysis from one to two conductivity parameters, providing new theoretical insights and numerical validation.

Findings

Existence and discreteness of transmission eigenvalues

Monotonicity of the first eigenvalue with respect to parameters

Validation through extensive numerical experiments

Abstract

In this paper, we provide an analytical study of the transmission eigenvalue problem with two conductivity parameters. We will assume that the underlying physical model is given by the scattering of a plane wave for an isotropic scatterer. In previous studies, this eigenvalue problem was analyzed with one conductive boundary parameter whereas we will consider the case of two parameters. We will prove the existence and discreteness of the transmission eigenvalues as well as study the dependence on the physical parameters. We are able to prove monotonicity of the first transmission eigenvalue with respect to the parameters and consider the limiting procedure as the second boundary parameter vanishes. Lastly, we provide extensive numerical experiments to validate the theoretical work.