Stable determination of an elastic medium scatterer by a single far-field measurement and beyond

TL;DR

This paper proves a logarithmic stability estimate for identifying elastic scatterers with convex polyhedral support using a single far-field measurement, and shows corners prevent invisibility in elastic scattering.

Contribution

It establishes a sharp stability estimate for elastic scatterer support determination and links the presence of corners to stable scattering properties.

Findings

Stability estimate is independent of material content.

Corners ensure the scatterer always produces a detectable far-field pattern.

Invisibility does not occur if the scatterer has a corner.

Abstract

We are concerned with the time-harmonic elastic scattering due to an inhomogeneous elastic material inclusion located inside a uniformly homogeneous isotropic medium. We establish a sharp stability estimate of logarithmic type in determining the support of the elastic scatterer, independent of its material content, by a single far-field measurement when the support is a convex polyhedral domain in $\mathbb{R}^n$, $n=2,3$. Our argument in establishing the stability result is localized around a corner of the medium scatterer. This enables us to further establish a byproduct result by proving that if a generic medium scatterer, not necessary to be a polyhedral shape, possesses a corner, then there exists a positive lower bound of the scattered far-field patterns. The latter result indicates that if an elastic material object possesses a corner on its support, then it scatters every incident wave stably and invisibility phenomenon does not occur.