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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Stationary sets for the wave equation in crystallographic domains


Authors: Mark L. Agranovsky and Eric Todd Quinto
Journal: Trans. Amer. Math. Soc. 355 (2003), 2439-2451
MSC (2000): Primary 35L05, 44A12; Secondary 35B05, 35S30
Published electronically: January 14, 2003
MathSciNet review: 1973997
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Abstract | References | Similar Articles | Additional Information

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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Additional Information

Mark L. Agranovsky
Affiliation: Bar Ilan University, Ramat Gan, Israel
Email: agranovs@macs.biu.ac.il

Eric Todd Quinto
Affiliation: Tufts University, Medford, Massachusetts
Email: todd.quinto@tufts.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03228-8
PII: S 0002-9947(03)03228-8
Received by editor(s): September 4, 2002
Published electronically: January 14, 2003
Additional Notes: The first author was supported by the Israel Science Foundation (grant No. 279/02-1)
The second author was partially supported by NSF grants 9877155 and 0200788
Article copyright: © Copyright 2003 American Mathematical Society