Unique continuation from a generalized impedance edge-corner for Maxwell's system and applications to inverse problems

TL;DR

This paper investigates the unique continuation properties of Maxwell's equations at impedance edge-corners, establishing a relationship between vanishing order and dihedral angle, with applications to inverse scattering problems.

Contribution

It provides new quantitative results on Maxwell's system near impedance edge-corners and applies these to prove unique identifiability of obstacles and surface impedance from limited measurements.

Findings

Strong unique continuation holds at irrational dihedral angles.

Vanishing order of solutions relates to the dihedral angle.

New identifiability results for obstacles and impedance from single measurements.

Abstract

We consider the time-harmonic Maxwell system in a domain with a generalized impedance edge-corner, namely the presence of two generalized impedance planes that intersect at an edge. The impedance parameter can be $0, \infty$ or a finite non-identically vanishing variable function. We establish an accurate relationship between the vanishing order of the solutions to the Maxwell system and the dihedral angle of the edge-corner. In particular, if the angle is irrational, the vanishing order is infinity, i.e. strong unique continuation holds from the edge-corner. The establishment of those new quantitative results involve a highly intricate and subtle algebraic argument. The unique continuation study is strongly motivated by our study of a longstanding inverse electromagnetic scattering problem. As a significant application, we derive several novel unique identifiability results in determining a polyhedral obstacle as well as it surface impedance by a single far-field measurement. We also discuss another potential and interesting application of our result in the inverse scattering theory related to the information encoding.