On uniqueness in the inverse obstacle problem via the positive supersolutions of the Helmholtz equation

TL;DR

This paper provides a new proof of the Colton-Sleeman theorem for inverse obstacle scattering, using positive supersolutions of the Helmholtz equation, extending the theorem to cases with known unbounded containing sets.

Contribution

It introduces a novel proof technique based on positive supersolutions, broadening the theorem's applicability beyond previous eigenvalue monotonicity methods.

Findings

New proof of Colton-Sleeman theorem using supersolutions

Extension to cases with known unbounded containing sets

Additional corollaries not covered by original theorem

Abstract

This paper is concerned with an inverse obstacle scattering problem of an acoustic wave for a single incident plane wave and a wave number. The Colton-Sleeman theorem states the unique recovery of sound-soft obstacles with a smooth boundary from the far-field pattern of the scattered wave for a single incident plane wave at a fixed wave number. The wave number has a bound given by the first Dirichlet eigenvalue of the negative Laplacian in an open ball that contains the obstacles. In this paper, another proof of the Colton-Sleeman theorem that works also for the case when we have a known {\it unbounded} set that contains obstacles is given. Unlike original one, the proof given here is not based on the monotonicity of the first Dirichlet eigenvalue of the negative Laplacian. Instead, it relies on a {\it positive supersolution} of the Helmholtz equation in a known domain that contains obstacles. Some corollaries which are new and not covered by the Colton-Sleeman Theorem are also given.