Ellipticity in the interior transmission problem in anisotropic media

TL;DR

This paper investigates the conditions under which the interior transmission problem in anisotropic media has discrete eigenvalues and solutions, providing explicit criteria and showing how small perturbations can ensure these properties.

Contribution

It explicitly formulates ellipticity conditions for the problem and identifies simple sufficient conditions for eigenvalue discreteness and solvability based on anisotropy and boundary data.

Findings

Ellipticity conditions are explicitly written.

Discreteness and solvability are not guaranteed by ellipticity alone.

Small perturbations in the refraction index can ensure eigenvalue discreteness and solvability.

Abstract

The paper concerns the discreteness of the eigenvalues and the solvability of the interior transmission problem for anisotropic media. Conditions for the ellipticity of the problem are written explicitly, and it is shown that they do not guarantee the discreteness of the eigenvalues. Some simple sufficient conditions for the discreteness and solvability are found. They are expressed in terms of the values of the anisotropy matrix and the refraction index at the boundary of the domain. The discreteness of the eigenvalues and the solvability of the interior transmission problem are shown if a small perturbation is applied to the refraction index.