On $T$-coercive interior transmission eigenvalue problems on compact manifolds with smooth boundary

TL;DR

This paper investigates interior transmission eigenvalues on two compact Riemannian manifolds with smooth boundary, establishing their discreteness and existence of regions free of eigenvalues using the T-coercivity method.

Contribution

It extends the analysis of interior transmission eigenvalues to non-diffeomorphic manifolds with anisotropic metrics and boundary conditions, introducing new existence and discreteness results.

Findings

Set of interior transmission eigenvalues is infinite and discrete.

Existence of eigenvalue-free regions established.

Method employs T-coercivity for non-diffeomorphic manifolds with anisotropic metrics.

Abstract

In this paper, we consider an interior transmission eigenvalue problem on two compact Riemannian manifolds with common smooth boundary. We suppose that a couple of these manifolds is equipped with locally anisotropic type Riemannian metric tensors, i.e., these two tensors are not equivalent in a neighborhood of common boundary. Here we note that we do not assume that these manifolds are diffeomorphic. In addition, we impose some conditions of the refractive indices in a neighborhood of common boundary. Then we prove that the set of ITEs form infinite discrete set and the existence of ITE-free region. In order to prove our results, we employ so-called the $T$-coercivity method.