Stationary sets for the wave equation and integral geometry
Mark Agranovsky and Eric Todd Quinto*
ABSTRACT:
We will study uniqueness for the spherical Radon transform in
Rn
with centers on a surface, S. This transform is a spread Radon transform
of Ehrenpreis. We put Courant and Hilbert's classical theorem in a
microlocal context and explain our own uniqueness theorems [J. Functional
Analysis, 139(1996), Duke Math J. 107(2001), submitted]. This transform
relates to problems in approximation theory, harmonic analysis, and PDE.
We will apply our theorems to characterize stationary sets for the wave
equation. A stationary set is a set in Rn on which some
non-trivial solution to the equation is identically zero.
We will outline the proof of our newest theorem. Let W be a
crystallographic group in Rn generated by reflections
and let O be the fundamental domain of W. We characterize stationary sets
for the Dirchlet problem for the wave equation in O when the initial velocity
is supported in the interior of O. We show that, for these initial data, the
(n-1)-dimensional part of the stationary sets consists of hyperplanes that
are mirrors of a crystallographic group TW, W < TW. This part comes from a
corresponding odd symmetry of the initial data. In physical language, the
result is that if the initial data is localized strictly inside of the
crystalline O, then unmovable interference hypersurfaces can be only faces
of a crystalline substructure subordinate to the original one.