On a novel UCP result and its application to inverse conductive scattering

TL;DR

This paper introduces a new Unique Continuation Principle for elliptic PDE systems, enabling improved inverse conductive scattering results with fewer assumptions, by combining CGO techniques and Fourier analysis.

Contribution

It develops a novel UCP for elliptic PDE systems and applies it to establish new unique identifiability results in inverse conductive scattering problems.

Findings

Established a new UCP for elliptic PDE systems.

Proved unique identifiability of conductive scatterers with a single far-field pattern.

Combined CGO and Fourier methods to analyze corner singularities.

Abstract

In this paper, we derive a novel Unique Continuation Principle (UCP) for a system of second-order elliptic PDEs system and apply it to investigate inverse problems in conductive scattering. The UCP relaxes the typical assumptions imposed on the domain or boundary with certain interior transmission conditions. This is motivated by the study of the associated inverse scattering problem and enables us to establish several novel unique identifiability results for the determination of generalized conductive scatterers using a single far-field pattern, significantly extending the results in [15,23]. A key technical advancement in our work is the combination of Complex Geometric Optics (CGO) techniques from [15,23] with the Fourier expansion method to microlocally analyze corner singularities and their implications for inverse problems. We believe that the methods developed can have broader applications in other contexts.