## 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.## References

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## Bibliographic Information

**Mark L. Agranovsky**- Affiliation: Bar Ilan University, Ramat Gan, Israel
- MR Author ID: 203078
- Email: agranovs@macs.biu.ac.il
**Eric Todd Quinto**- Affiliation: Tufts University, Medford, Massachusetts
- Email: todd.quinto@tufts.edu
- 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 - © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**355**(2003), 2439-2451 - MSC (2000): Primary 35L05, 44A12; Secondary 35B05, 35S30
- DOI: https://doi.org/10.1090/S0002-9947-03-03228-8
- MathSciNet review: 1973997