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