The Weyl law of transmission eigenvalues and the completeness of generalized transmission eigenfunctions without complementing conditions

TL;DR

This paper proves the Weyl law and the completeness of generalized transmission eigenfunctions for a degenerate elliptic system without complementing conditions, extending previous results to anisotropic, less regular coefficients.

Contribution

It establishes the Weyl law and eigenfunction completeness for a degenerate elliptic system with anisotropic, $C^2$ coefficients, without requiring complementing conditions or isotropy.

Findings

Established Weyl law for transmission eigenvalues

Proved completeness of generalized eigenfunctions

Extended known results to less regular, anisotropic coefficients

Abstract

The transmission eigenvalue problem is a system of two second-order elliptic equations of two unknowns equipped with the Cauchy data on the boundary. In this work, we establish the Weyl law for the eigenvalues and the completeness of the generalized eigenfunctions for a system without complementing conditions, i.e., the two equations of the system have the same coefficients for the second order terms, and thus being degenerate. These coefficients are allowed to be anisotropic and are assumed to be of class $C^2$. One of the keys of the analysis is to establish the well-posedness and the regularity in $L^p$-scale for such a system. As a result, we largely extend and rediscover known results for which the coefficients for the second order terms are required to be isotropic and of class $C^\infty$ using a new approach.