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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.

MSC:

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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