TY - JOUR

T1 - Stationary sets for the wave equation in crystallographic domains

AU - Agranovsky, Mark L.

AU - Quinto, Eric Todd

PY - 2003/6

Y1 - 2003/6

N2 - Let W be a crystallographic group in ℝn generated by reflections and let Ω be the fundamental domain of W. We characterize stationary sets for the wave equation in Ω when the initial data is supported in the interior of Ω. 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 W̃, W < W̃. 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, then unmovable interference hypersurfaces can only be faces of a crystalline substructure of the original one.

AB - Let W be a crystallographic group in ℝn generated by reflections and let Ω be the fundamental domain of W. We characterize stationary sets for the wave equation in Ω when the initial data is supported in the interior of Ω. 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 W̃, W < W̃. 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, then unmovable interference hypersurfaces can only be faces of a crystalline substructure of the original one.

UR - http://www.scopus.com/inward/record.url?scp=0038369074&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-03-03228-8

DO - 10.1090/S0002-9947-03-03228-8

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AN - SCOPUS:0038369074

SN - 0002-9947

VL - 355

SP - 2439

EP - 2451

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 6

ER -