Discreteness of interior transmission eigenvalues revisited

TL;DR

This paper investigates the conditions under which interior transmission eigenvalues are discrete, employing multiplier and Fourier analysis techniques to extend known results and derive new conditions with minimal boundary information.

Contribution

It introduces new methods to establish discreteness of transmission eigenvalues, including results that require only boundary information and no contrast of coefficients.

Findings

Established discreteness under minimal boundary conditions

Revisited and extended known results on eigenvalue discreteness

Developed techniques based on multiplier and Fourier analysis

Abstract

This paper is devoted to the discreteness of the transmission eigenvalue problems. It is known that this problem is not self-adjoint and a priori estimates are non-standard and do not hold in general. Two approaches are used. The first one is based on the multiplier technique and the second one is based on the Fourier analysis. The key point of the analysis is to establish the compactness and the uniqueness for Cauchy problems under various conditions. Using these approaches, we are able to rediscover quite a few known discreteness results in the literature and obtain various new results for which only the information near the boundary are required and there might be no contrast of the coefficients on the boundary.