Abstract

In this talk, we are concerned with a coupled-physics spectral problem arising in the coupled propagation of acoustic and elastic waves, which is referred to as the acoustic-elastic transmission eigenvalue problem. There are two major contributions in this work which are new to the literature. First, under a mild condition on the medium parameters, we prove the existence of an acoustic-elastic transmission eigenvalue. Second, we establish a geometric rigidity result of the transmission eigenfunctions by showing that they tend to localize on the boundary of the underlying domain. Moreover, we also consider the vanishing property of the underlying transmission eigenfunctions near a 2D corner or 3D edge corner under generic regularity assumptions on the transmission eigenfunctions. The geometrical characterization of transmission eigenfunctions can be used to established unique results for determining the polygonal scatterer by finite far field pattern.