Piecewise-analytic interfaces with weakly singular points of arbitrary order always scatter

TL;DR

This paper proves that inhomogeneous media with boundaries containing weakly singular points of any order always scatter incoming waves, leading to implications for inverse scattering problems and the non-existence of non-radiating sources.

Contribution

It establishes that weakly singular boundary points of arbitrary order guarantee scattering and radiating behavior, extending previous results to less smooth domains.

Findings

Weakly singular boundary points always scatter waves.

Non-scattering energies and non-radiating sources do not exist in non-smooth domains.

Provides conditions for the absence of analytical continuation of solutions.

Abstract

It is proved that an inhomogeneous medium whose boundary contains a weakly singular point of arbitrary order scatters every incoming wave. Similarly, a compactly supported source term with weakly singular points on the boundary always radiates acoustic waves. These results imply the absence of non-scattering energies and non-radiating sources in a domain that is not $C^\infty$-smooth. Local uniqueness results with a single far-field pattern are obtained for inverse source and inverse medium scattering problems. Our arguments provide a sufficient condition of the surface under which solutions to the Helmholtz equation admits no analytical continuation.