Nature of the transmission eigenvalue spectrum for elastic bodies

Abstract

This study develops a spectral theory of the interior transmission problem (ITP) for heterogeneous and anisotropic elastic solids. The subject is central to the so-called qualitative methods for inverse scattering involving penetrable obstacles. Although simply stated as a coupled pair of elastodynamic wave equations, the ITP for elastic bodies is neither self-adjoint nor elliptic. To help deal with such impediments, earlier studies have established the well-posedness of an elastodynamic ITP under notably restrictive assumptions on the contrast in elastic and mass density parameters between the scatterer and the background solid. Due to lack of self-adjointness of the problem, these analyses were further successful in substantiating the discreteness of the relevant eigenvalue spectrum but not its existence. The aim of this work is to provide a systematic treatment of the ITP for elastic bodies that transcends the limitations of earlier analyses. Considering a broad range of material-contrast configurations, this paper investigates the questions of the solvability of the ITP, the discreteness of its eigenvalues and, for the first time, of the existence of such eigenvalue spectrum. Necessitated by the breadth of material configurations studied, the relevant claims are established via a suite of variational formulations, each customized to meet the needs of a particular subclass of eigenvalue problems.