On an electromagnetic problem in a corner and its applications

Let K-x0(r0) be a (nondegenerate) truncated corner in R-3, with x(0) is an element of R-3 being its apex, and F-j is an element of C-alpha(, C-3), j = 1, 2, where alpha is the positive Holder index. Consider the electromagnetic problem

{del boolean AND E - i omega mu H-0 = F-1 in K-x0(r0),

del boolean AND H - i omega epsilon E-0 = F-2 in K-x0(r0),

nu boolean AND E = nu boolean AND H = 0 on partial derivative K-x0(r0) \ partial derivative B-r0(x(0)),

where nu denotes the exterior unit normal vector of partial derivative K-x0(r0). We prove that F-1 and 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.