Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

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.

References [Enhancements On Off] (What's this?)

  • 1. M. Agranovsky, C. Berenstein, and P. Kuchment Approximation by spherical waves in $L^p$ spaces, J. Geom. Analysis 6 (1996), 365-383. MR 99c:41038
  • 2. M. Agranovsky and E. T. Quinto, Injectivity sets for the Radon transform over circles and complete systems of radial functions, J. Funct. Anal. 139 (2), (1996), 383-414. MR 98g:58171
  • 3. M. Agranovsky and E. T. Quinto, Injectivity of the spherical mean operator and related problems, in: Complex Analysis, Harmonic Analysis and Applications (R. Deville et al, eds.), Addison Wesley, London, 1996, pp. 12-36. MR 97m:44004
  • 4. M. Agranovsky and E. T. Quinto, Geometry of stationary sets for the wave equation in $\mathbb R^n$. The case of finitely supported initial data, Duke Math. J. 107 (2001), 57-84. MR 2001m:35192
  • 5. M. Agranovsky, V. V. Volchkov and L. Zalcman, Conical Uniqueness Sets for the Spherical Radon Transform, Bull. London Math. Soc., 31(1999), 231-236. MR 2000j:44002
  • 6. C. Bar, On nodal sets for Dirac and Laplace operators, Comm. Math. Phys., 188 (1997), 709-721. MR 98g:58179
  • 7. P. Bérard and D. Meyer Inégalités isopérimétriques et applications, Ann. Sci. École Norm. Sup. 15 (1982), 513-542. MR 84h:58147
  • 8. N. Bourbaki, Éléments de Mathématiques, Livre III, Topologie Générale, Actualités Sci. Ind., no 1092, Hermann et cie., Paris, 1947, MR 9:261a
  • 9. J. Brüning, Über Membranen mit speziellen Knotenlinien, Comment. Math. Helv. 55(1980), 13-19. MR 81e:53040
  • 10. L. S. Charlap, Bieberbach Groups and Flat Manifolds. Springer-Verlag, New York, 1986. MR 88j:57042
  • 11. S. Y. Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv., 51 (1976), 43-55. MR 53:1661
  • 12. Courant R. and Hilbert D., Methods of Mathematical Physics, vol. 2, Interscience, New York, 1961. MR 25:4216
  • 13. H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math. 93 (1988), 161-183. MR 89m:58207
  • 14. -, Nodal sets for eigenfunctions of the Laplacian on surfaces, J. Amer. Math. Soc., 3 (1990), 333-353. MR 92d:58209
  • 15. R. Hardt and L. Simon, Nodal sets for solutions of elliptic equations, J. Diff. Geom. 30(1989), 505-522. MR 90m:58031
  • 16. M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and N. Nadirashvili, The nodal line of the second eigenfunction of the Laplacian in $\mathbb R^2$ can be closed, Duke Math. J., 90 (1997), 3, 631-640. MR 98m:35146
  • 17. L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, New York, 1983. MR 85g:35002a

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35L05, 44A12, 35B05, 35S30

Retrieve articles in all journals with MSC (2000): 35L05, 44A12, 35B05, 35S30

Additional Information

Mark L. Agranovsky
Affiliation: Bar Ilan University, Ramat Gan, Israel

Eric Todd Quinto
Affiliation: Tufts University, Medford, Massachusetts

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

American Mathematical Society