Vol. 14, No. 7, 2021

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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

Let 𝒦x0r0 be a (nondegenerate) truncated corner in 3 , with x0 3 being its apex, and Fj Cα(𝒦x0r0 ¯; 3), j = 1,2, where α is the positive Hölder index. Consider the electromagnetic problem

E iωμ0H = F1 in 𝒦x0r0, H + iω𝜀0E = F2  in 𝒦x0r0, ν E = ν H = 0  on 𝒦x0r0 Br 0(x0),

where ν denotes the exterior unit normal vector of 𝒦x0r0. We prove that F1 and F2 must vanish at the apex x0. 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.

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
Received: 10 September 2019
Revised: 16 March 2020
Accepted: 22 April 2020
Published: 10 November 2021
Emilia Blåsten
Department of Mathematics and Systems Analysis
Aalto University
Division of Mathematics
Tallinn University of Technology Department of Cybernetics
Hongyu Liu
Department of Mathematics
City University of Hong Kong
Hong Kong
Jingni Xiao
Department of Mathematics
Rutgers University
Piscataway, NJ
United States