On Corner Scattering for Operators of Divergence Form and Applications to Inverse Scattering

TL;DR

This paper investigates how convex corners in inhomogeneous media influence wave scattering governed by the Helmholtz equation, establishing uniqueness in reconstructing the shape of the inhomogeneity from a single scattering measurement.

Contribution

It proves that convex corners always cause scattering under certain conditions and demonstrates unique determination of polygonal inhomogeneities from a single incident wave.

Findings

Convex corners guarantee scattering for the considered operators.

Single incident wave data suffices for unique shape reconstruction.

Results hold under minimal regularity assumptions near corners.

Abstract

We consider the scattering problem governed by the Helmholtz equation with inhomogeneity in both `conductivity' in the divergence form and `potential' in the lower order term. The support of the inhomogeneity is assumed to contain a convex corner. We prove that, due to the presence of such corner under appropriate assumptions on the potential and conductivity in the vicinity of the corner, any incident field scatters. Based on corner scattering analysis we present a uniqueness result on determination of the polygonal convex hull of the support of admissible inhomogeneities, from scattering data corresponding to one single incident wave. These results require only certain regularity around the corner for the coefficients modeling the inhomogeneity, whereas away from the corner they can be quite general. Our main results on scattering and inverse scattering are established for $\mathbb{R}^2$, while some analytic tools are developed in any dimension $n\geq 2$.