On vanishing and localizing of transmission eigenfunctions near singular points: a numerical study

Abstract

This paper is concerned with the intrinsic geometric structure of interior transmission eigenfunctions arising in wave scattering theory. We numerically show that the aforementioned geometric structure can be very delicate and intriguing. The major findings can be roughly summarized as follows. We say that a point on the boundary of the inhomogeneity is singular if the surface tangent is discontinuous there. The interior transmission eigenfunction then vanishes near a singular point where the interior angle is less than π, whereas the interior transmission eigenfunction localizes near a singular point if its interior angle is bigger than π. Furthermore, we show that the vanishing and blowup orders are inversely proportional to the interior angle of the singular point: the sharper the corner, the higher the convergence order. Our results are first of its type in the spectral theory for transmission eigenvalue problems, and the existing studies in the literature concentrate more on the intrinsic properties of the transmission eigenvalues instead of the transmission eigenfunctions. Due to the finiteness of computing resources, our study is by no means exclusive and complete. We consider our study only in a certain geometric setup including corner, curved corner and edge singularities. Nevertheless, we believe that similar results hold for more general singularities and rigorous theoretical justifications are much desirable. Our study enriches the spectral theory for transmission eigenvalue problems. We also discuss its implication to inverse scattering theory.