On an electromagnetic problem in a corner and its applications

Abstract: Let $\mathcal{K}^{r_0}_{x_0}$ be a (non-degenerate) truncated corner in $\mathbb{R}^3$ with $x_0\in\mathbb{R}^3$ being its apex, and $\mathbf{F}j\in C^\alpha(\overline{\mathcal{K}^{r_0}{x_0}}; \mathbb{C}^3)$, $j=1,2$, where $\alpha$ is the positive H\"older index. Consider the following electromagnetic problem $$\left{\begin{split} & \nabla\wedge \mathbf{E}-\mathrm{i}\omega \mu_0 \mathbf{H}=\mathbf{F}{1} \quad \mbox{in $\mathcal{K}^{r_0}{x_0}$},\ & \, \nabla\wedge \mathbf{H}+\mathrm{i}\omega\varepsilon_0 \mathbf{E}=\mathbf{F}{2} \quad \mbox{in $\mathcal{K}^{r_0}{x_0}$}, \ &\, \nu\wedge \mathbf{E}=\nu\wedge\mathbf{H}=0 \qquad\mbox{on $\partial \mathcal{K}^{r_0}_{x_0}\setminus \partial B_{r_0}(x_0)$}, \end{split}\right.$$ where $\nu$ denotes the exterior unit normal vector of $\partial \mathcal{K}^{r_0}_{x_0}$. 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.