Nature of the transmission eigenvalue spectrum for elastic bodies

This study develops a spectral theory of the interior transmission problem (ITP) for heterogeneous and anisotropic elastic solids. The importance of this subject stems from its central role in a certain class of inverse scattering theories (the so-called qualitative methods) involving penetrable scatterers. Although simply stated as a coupled pair of elastodynamic wave equations, the ITP for elastic bodies is neither selfadjoint 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 parameters between the scatterer and the background solid. Due to the lack of problem self-adjointness, however, these studies were successful in substantiating only 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 heterogeneous and anisotropic elastic bodies that transcends the limitations of earlier treatments. Considering a broad range of material-contrast configurations (both in terms of elastic tensors and mass densities), this paper investigates the questions of the solvability of the ITP, the discreteness of its eigenvalues and, for the first time, of the actual existence of such eigenvalue spectrum. Necessitated by the breadth of material configurations studied, the relevant claims are established through the development of a suite of variational formulations, each customized to meet the needs of a particular subclass of eigenvalue problems. As a secondary result, the lower and upper bounds on the first transmission eigenvalue are obtained in terms of the elasticity and mass density contrasts between the obstacle and the background. Given the fact that the transmission eigenvalues are computable from experimental observations of the scattered field, such estimates may have significant potential toward exposing the nature (e.g. compliance) of penetrable scatterers in elasticity.