Unique determination by a single far-field measurement for an inverse elastic problem

TL;DR

This paper proves that the shape and physical parameters of certain elastic scatterers with polygonal structures can be uniquely identified using only a single far-field measurement, advancing inverse scattering theory.

Contribution

It establishes the uniqueness of shape and parameter determination for elastic scatterers with polygonal structures from a single far-field pattern, using a microlocal analysis approach.

Findings

Unique shape determination from one measurement for polygonal structures.

Simultaneous identification of density and boundary impedance parameters.

Use of microlocal analysis near corners to prove uniqueness.

Abstract

This paper is concerned with the unique identification of the shape of a scatterer through a single far-field pattern in an inverse elastic medium scattering problem with a generalized transmission boundary condition. The uniqueness issue by a single far-field measurement is a challenging problem in inverse scattering theory, which has a long and colorful history. In this paper, we demonstrate the well-posedness of the direct problem by the variational approach. We establish the uniqueness results by a single far-field measurement under a generic scenario when dealing with underlying elastic scatterers exhibiting polygonal-nest or polygonal-cell structures. Furthermore, for a polygonal-nest or polygonal-cell structure scatterer associated with density and boundary impedance parameters as piecewise constants, we show that these physical quantities can be uniquely determined simultaneously by a single far-field measurement. The corresponding proof relies heavily on examining the singular behaviour of a coupled PDE system near a corner in a microlocal manner.