Direct and inverse scattering for an isotropic medium with a second-order boundary condition

TL;DR

This paper addresses the direct and inverse scattering problems for an isotropic obstacle with a second-order Robin boundary condition, introducing a numerical method for shape recovery and studying associated transmission eigenvalues.

Contribution

It develops a direct sampling method for inverse shape reconstruction and initiates the analysis of the transmission eigenvalue problem for this setting.

Findings

Numerical recovery of scatterer shapes in 2D demonstrated.

Transmission eigenvalues are at most a discrete set.

Numerical computation of eigenvalues for various scatterers performed.

Abstract

We consider the direct and inverse scattering problem for a penetrable, isotropic obstacle with a second-order Robin boundary condition, which asymptotically models the delamination of the boundary of the scatterer. We develop a direct sampling method to solve the inverse shape problem by numerically recovering the scatterer. Here, we assume that the corresponding Cauchy data is measured on the boundary of a region that fully contains the scatterer. Similar methods have been applied to other inverse shape problems, but they have not been studied for a penetrable, isotropic scatterer with a second-order Robin boundary condition. We also initiate the study of the corresponding transmission eigenvalue problem, which is derived from assuming zero Cauchy data is measured on the boundary of the region that fully contains the scatterer. We prove that the transmission eigenvalues for this problem are at most a discrete set. Numerical examples will be presented for the inverse shape problem in two dimensions for circular and non-circular scatterers. Further, transmission eigenvalues are computed numerically for various scatterers.