Stationary sets for the wave equation in crystallographic domains

Abstract

Let W W be a crystallographic group in R n \mathbb R^n generated by reflections and let Ω \Omega be the fundamental domain of W . W. We characterize stationary sets for the wave equation in Ω \Omega when the initial data is supported in the interior of Ω . \Omega . The stationary sets are the sets of time-invariant zeros of nontrivial solutions that are identically zero at t = 0 t=0 . We show that, for these initial data, the ( n − 1 ) (n-1) -dimensional part of the stationary sets consists of hyperplanes that are mirrors of a crystallographic group W ~ \tilde W , W > W ~ . W>\tilde W. This part comes from a corresponding odd symmetry of the initial data. In physical language, the result is that if the initial source is localized strictly inside of the crystalline Ω \Omega , then unmovable interference hypersurfaces can only be faces of a crystalline substructure of the original one.