On uniqueness of inverse conductive scattering problem with unknown embedded obstacles

TL;DR

This paper establishes a new uniqueness theorem for inverse conductive scattering, showing that the shape and boundary properties of an inhomogeneous object with embedded obstacles can be uniquely identified from far-field data.

Contribution

It introduces a novel uniqueness result for the inverse problem, including embedded obstacles, using singularity analysis and interior transmission problem well-posedness.

Findings

Unique determination of the conductive object from far-field measurements

Boundary information of physical coefficients can be uniquely recovered

Conditions for well-posedness of the interior transmission problem provided

Abstract

This paper is concerning the inverse conductive scattering of acoustic waves by a bounded inhomogeneous object with possibly embedded obstacles inside. A new uniqueness theorem is proved that the conductive object is uniquely determined by the fixed frequency far-field measurements, ignoring its contents. Meanwhile, the boundary informations of several related physical coefficients are also uniquely determined. The proof is mainly based on a detailed singularity analysis of solutions near the interface associated with a family of point sources or hypersingular point sources, which is deduced by the potential theory. Moreover, the other key ingredient in the proof is the well-posedness of the interior transmission problem with the conductivity boundary condition in the L^2 sense, where several sufficient conditions depending on the domain and physical coefficients are provided.