On an electromagnetic problem in a corner and its applications

Blåsten, Emilia; Liu, Hongyu; Xiao, Jingni

doi:10.2140/apde.2021.14.2207

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On an electromagnetic problem in a corner and its applications

Emilia Blåsten, Hongyu Liu and Jingni Xiao

Vol. 14 (2021), No. 7, 2207–2224

DOI: 10.2140/apde.2021.14.2207

Abstract

Let $𝒦_{x_{0}}^{r_{0}}$ be a (nondegenerate) truncated corner in $ℝ^{3}$ , with $x_{0} \in ℝ^{3}$ being its apex, and $F_{j} \in C^{α} (\bar{𝒦_{x_{0}}^{r_{0}}}; ℂ^{3})$ , $j = 1, 2$ , where $α$ is the positive Hölder index. Consider the electromagnetic problem

\{\begin{matrix} \nabla \land E - i ω μ_{0} H = F_{1} & in 𝒦_{x_{0}}^{r_{0}}, \\ \nabla \land H + i ω 𝜀_{0} E = F_{2} & in 𝒦_{x_{0}}^{r_{0}}, \\ ν \land E = ν \land H = 0 & on \partial 𝒦_{x_{0}}^{r_{0}} ∖ \partial B_{r_{0}} (x_{0}), \end{matrix}

where $ν$ denotes the exterior unit normal vector of $\partial 𝒦_{x_{0}}^{r_{0}}$ . We prove that $F_{1}$ and $F_{2}$ must vanish at the apex $x_{0}$ . There is a series of interesting consequences of this vanishing property in several separate but intriguingly connected topics in electromagnetism. First, we can geometrically characterize nonradiating sources in time-harmonic electromagnetic scattering. Secondly, we consider the inverse source scattering problem for time-harmonic electromagnetic waves and establish the uniqueness result in determining the polyhedral support of a source by a single far-field measurement. Thirdly, we derive a property of the geometric structure of electromagnetic interior transmission eigenfunctions near corners. Finally, we also discuss its implication to invisibility cloaking and inverse medium scattering.

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Keywords

Maxwell system, corner singularity, invisible, vanishing, interior transmission eigenfunction, inverse scattering, single far-field measurement

Mathematical Subject Classification 2010

Primary: 78A45, 35Q61, 35P25

Secondary: 78A46, 35R30

Milestones

Received: 10 September 2019

Revised: 16 March 2020

Accepted: 22 April 2020

Published: 10 November 2021

Authors

Emilia Blåsten
Department of Mathematics and Systems Analysis Aalto University Aalto Finland	Division of Mathematics Tallinn University of Technology Department of Cybernetics Tallinn Estonia

Hongyu Liu
Department of Mathematics City University of Hong Kong Hong Kong China

Jingni Xiao
Department of Mathematics Rutgers University Piscataway, NJ United States

Vol. 14, No. 7, 2021