The Weyl law of transmission eigenvalues and the completeness of generalized transmission eigenfunctions

TL;DR

This paper proves the Weyl law for transmission eigenvalues and the completeness of generalized eigenfunctions for a broad class of anisotropic elliptic problems, extending classical results to less regular coefficients.

Contribution

It establishes the Weyl law and eigenfunction completeness for anisotropic transmission problems with minimal regularity assumptions, using new $L^p$ regularity and spectral theory techniques.

Findings

Weyl law for transmission eigenvalues proved

Completeness of generalized eigenfunctions established

Extension to anisotropic coefficients with minimal regularity

Abstract

The transmission problem is a system of two second-order elliptic equations of two unknowns equipped with the Cauchy data on the boundary. After four decades of research motivated by scattering theory, the spectral properties of this problem are now known to depend on a type of contrast between coefficients near the boundary. Previously, we established the discreteness of eigenvalues for a large class of anisotropic coefficients which is related to the celebrated complementing conditions due to Agmon, Douglis, and Nirenberg. In this work, we establish the Weyl law for the eigenvalues and the completeness of the generalized eigenfunctions for this class of coefficients under an additional mild assumption on the continuity of the coefficients. The analysis is new and based on the $L^p$ regularity theory for the transmission problem established here. It also involves a subtle application of the spectral theory for the Hilbert Schmidt operators. Our work extends largely known results in the literature which are mainly devoted to the isotropic case with $C^\infty$-coefficients.