On corners scattering stably and stable shape determination by a single far-field pattern

TL;DR

This paper provides sharp quantitative results for the stability of inverse and direct acoustic wave scattering, including support recovery of inhomogeneous media and corner scattering stability, with implications for cloaking and a new quantitative Rellich's theorem.

Contribution

It introduces new stability estimates for inverse support recovery and corner scattering, and establishes a novel quantitative Rellich's theorem for wave fields.

Findings

Support of inhomogeneous media can be stably recovered from a single far-field measurement.

Corner scattering energy has a positive lower bound depending on geometry and refractive index.

Impossibility of approximate cloaking with devices containing corners made of isotropic material.

Abstract

In this paper, we establish two sharp quantitative results for the direct and inverse time-harmonic acoustic wave scattering. The first one is concerned with the recovery of the support of an inhomogeneous medium, independent of its contents, by a single far-field measurement. For this challenging inverse scattering problem, we establish a sharp stability estimate of logarithmic type when the medium support is a polyhedral domain in $\mathbb{R}^n$, $n=2,3$. The second one is concerned with the stability for corner scattering. More precisely if an inhomogeneous scatterer, whose support has a corner, is probed by an incident plane-wave, we show that the energy of the scattered far-field possesses a positive lower bound depending only on the geometry of the corner and bounds on the refractive index of the medium there. This implies the impossibility of approximate invisibility cloaking by a device containing a corner and made of isotropic material. Our results sharply quantify the qualitative corner scattering results in the literature, and the corresponding proofs involve much more subtle analysis and technical arguments. As a significant byproduct of this study, we establish a quantitative Rellich's theorem that continues smallness of the wave field from the far-field up to the interior of the inhomogeneity. The result is of significant mathematical interest for its own sake and is surprisingly not yet known in the literature.