#### 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
##### Abstract

Let ${\mathsc{𝒦}}_{{x}_{0}}^{{r}_{0}}$ be a (nondegenerate) truncated corner in ${ℝ}^{3}$, with ${x}_{0}\in {ℝ}^{3}$ being its apex, and ${F}_{j}\in {C}^{\alpha }\left(\overline{{\mathsc{𝒦}}_{{x}_{0}}^{{r}_{0}}};{ℂ}^{3}\right)$, $j=1,2$, where $\alpha$ is the positive Hölder index. Consider the electromagnetic problem

where $\nu$ denotes the exterior unit normal vector of $\partial {\mathsc{𝒦}}_{{x}_{0}}^{{r}_{0}}$. 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.

##### 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