On an electromagnetic problem in a corner and its applications

Abstract

Let ${\mathcal{K}}_{x_0}^{r_0}$ be a (non-degenerate) truncated corner in ${\mathbb{R}}^3$ with $x_0\in {\mathbb{R}}^3$ being its apex, and ${\mathbf{F}}_j\in C^{\alpha }(\overline{{\mathcal{K}}_{x_0}^{r_0}};{\mathbb{C}}^3)$, $j=1,2$, where $\alpha$ is the positive Hölder index. Consider the following electromagnetic problem $$\left\{\begin{split} & \nabla\wedge \mathbf{E}-\mathrm{i}ωμ_0 \mathbf{H}=\mathbf{F}_{1} \quad \mbox{in $\mathcal{K}^{r_0}_{x_0}$},\\ & \, \nabla\wedge \mathbf{H}+\mathrm{i}ω\varepsilon_0 \mathbf{E}=\mathbf{F}_{2} \quad \mbox{in $\mathcal{K}^{r_0}_{x_0}$}, \\ &\, ν\wedge \mathbf{E}=ν\wedge\mathbf{H}=0 \qquad\mbox{on $\partial \mathcal{K}^{r_0}_{x_0}\setminus \partial B_{r_0}(x_0)$}, \end{split}\right.$$ where $\nu$ denotes the exterior unit normal vector of $\partial {\mathcal{K}}_{x_0}^{r_0}$. We prove that ${\mathbf{F}}_1$ and ${\mathbf{F}}_2$ must vanish at the apex $x_0$. There are a series of interesting consequences of this vanishing property in several separate but intriguingly connected topics in electromagnetism. First, we can geometrically characterize non-radiating 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.