Stationary sets for the wave equation in crystallographic domains

Agranovsky, Mark; Quinto, Eric

doi:10.1090/S0002-9947-03-03228-8

Stationary sets for the wave equation in crystallographic domains
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by Mark L. Agranovsky and Eric Todd Quinto

Trans. Amer. Math. Soc. 355 (2003), 2439-2451

DOI: https://doi.org/10.1090/S0002-9947-03-03228-8

Published electronically: January 14, 2003

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Abstract:

Let $W$ be a crystallographic group in $\mathbb R^n$ generated by reflections and let $\Omega$ be the fundamental domain of $W.$ We characterize stationary sets for the wave equation in $\Omega$ when the initial data is supported in the interior of $\Omega .$ The stationary sets are the sets of time-invariant zeros of nontrivial solutions that are identically zero at $t=0$. 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 $\tilde W$, $W<\tilde W.$ 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 $\Omega$, then unmovable interference hypersurfaces can only be faces of a crystalline substructure of the original one.

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