On an electromagnetic problem in a corner and its applications. (English) Zbl 1480.78011

Summary: Let \(\mathcal{K}^{r_0}_{x_0}\) be a (nondegenerate) truncated corner in \(\mathbb{R}^3\), with \(x_0\in\mathbb{R}^3\) being its apex, and \(\boldsymbol{F}_{j}\in C^{\alpha} (\overline{\mathcal{K}^{r_0}_{x_0}}; \mathbb{C}^3),\, j=1,2\), where \(\alpha\) is the positive Hölder index. Consider the electromagnetic problem \[ \begin{cases} \nabla\wedge \boldsymbol{E} -\operatorname{i}\omega \mu_0 \boldsymbol{H}=\boldsymbol{F}_1 & \text{in } \mathcal{K}^{r_0}_{x_0}, \\ \nabla\wedge \boldsymbol{H}+\operatorname{i}\omega\varepsilon_0 \boldsymbol{E}=\boldsymbol{F}_2 & \text{in }\mathcal{K}^{r_0}_{x_0}, \\ \nu\wedge\boldsymbol{E}=\nu\wedge\boldsymbol{H}=0 & \text{on } \partial \mathcal{K}^{r_0}_{x_0}\setminus \partial B_{r_0}(x_0), \end{cases} \] where \(\nu\) denotes the exterior unit normal vector of \(\partial \mathcal{K}^{r_0}_{x_0}\). We prove that \(\boldsymbol{F}_1\) and \(\boldsymbol{F}_2\) 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.


78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
78A45 Diffraction, scattering
35Q61 Maxwell equations
35P25 Scattering theory for PDEs
35R30 Inverse problems for PDEs
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