On an electromagnetic problem in a corner and its applications

Let (Formula Presented) be a (nondegenerate) truncated corner in ℝ³, with x₀ ∈ ℝ³ being its apex, and (Formula Presented) where α is the positive Hölder index. Consider the electromagnetic problem (Formula Presented) where ν denotes the exterior unit normal vector of (Formula Presented) We prove that F₁ and F₂ must vanish at the apex x₀. 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.