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
HTML articles powered by AMS MathViewer

by Mark L. Agranovsky and Eric Todd Quinto PDF

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

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

Mark Agranovsky, Carlos Berenstein, and Peter Kuchment, Approximation by spherical waves in $L^p$-spaces, J. Geom. Anal. 6 (1996), no. 3, 365–383 (1997). MR 1471897, DOI 10.1007/BF02921656
Mark L. Agranovsky and Eric Todd Quinto, Injectivity sets for the Radon transform over circles and complete systems of radial functions, J. Funct. Anal. 139 (1996), no. 2, 383–414. MR 1402770, DOI 10.1006/jfan.1996.0090
Mark L. Agranovsky and Eric Todd Quinto, Injectivity of the spherical mean operator and related problems, Complex analysis, harmonic analysis and applications (Bordeaux, 1995) Pitman Res. Notes Math. Ser., vol. 347, Longman, Harlow, 1996, pp. 12–36. MR 1402020
Mark L. Agranovsky and Eric Todd Quinto, Geometry of stationary sets for the wave equation in $\Bbb R^n$: the case of finitely supported initial data, Duke Math. J. 107 (2001), no. 1, 57–84. MR 1815250, DOI 10.1215/S0012-7094-01-10714-X
M. L. Agranovsky, V. V. Volchkov, and L. A. Zalcman, Conical uniqueness sets for the spherical Radon transform, Bull. London Math. Soc. 31 (1999), no. 2, 231–236. MR 1664137, DOI 10.1112/S0024609398005396
Christian Bär, On nodal sets for Dirac and Laplace operators, Comm. Math. Phys. 188 (1997), no. 3, 709–721. MR 1473317, DOI 10.1007/s002200050184
Pierre Bérard and Daniel Meyer, Inégalités isopérimétriques et applications, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 3, 513–541 (French). MR 690651, DOI 10.24033/asens.1435
Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
Jochen Brüning, Über Membranen mit speziellen Knotenlinien, Comment. Math. Helv. 55 (1980), no. 1, 13–19 (German). MR 569243, DOI 10.1007/BF02566672
Leonard S. Charlap, Bieberbach groups and flat manifolds, Universitext, Springer-Verlag, New York, 1986. MR 862114, DOI 10.1007/978-1-4613-8687-2
Shiu Yuen Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv. 51 (1976), no. 1, 43–55. MR 397805, DOI 10.1007/BF02568142
R. Courant and D. Hilbert, Methods of mathematical physics. Vol. II: Partial differential equations, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. (Vol. II by R. Courant.). MR 0140802
Harold Donnelly and Charles Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math. 93 (1988), no. 1, 161–183. MR 943927, DOI 10.1007/BF01393691
Harold Donnelly and Charles Fefferman, Nodal sets for eigenfunctions of the Laplacian on surfaces, J. Amer. Math. Soc. 3 (1990), no. 2, 333–353. MR 1035413, DOI 10.1090/S0894-0347-1990-1035413-2
Robert Hardt and Leon Simon, Nodal sets for solutions of elliptic equations, J. Differential Geom. 30 (1989), no. 2, 505–522. MR 1010169
M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and N. Nadirashvili, The nodal line of the second eigenfunction of the Laplacian in $\textbf {R}^2$ can be closed, Duke Math. J. 90 (1997), no. 3, 631–640. MR 1480548, DOI 10.1215/S0012-7094-97-09017-7
Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035, DOI 10.1007/978-3-642-96750-4