On an electromagnetic problem in a corner and its applications

Abstract

Let ${\mathcal{𝒦}}_{x_0}^{r_0}$ be a (nondegenerate) truncated corner in ${ℝ}^3$, with $x_0\in {ℝ}^3$ being its apex, and ${\mathit{F}}_j\in C^{\alpha }\left(\overline{{\mathcal{𝒦}}_{x_0}^{r_0}};{ℂ}^3\right)$, $j=1,2$, where $\alpha$ is the positive Hölder index. Consider the electromagnetic problem

where $\nu$ denotes the exterior unit normal vector of $\partial {\mathcal{𝒦}}_{x_0}^{r_0}$. We prove that ${\mathit{F}}_1$ and ${\mathit{F}}_2$ 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.