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Séminaire de Théorie spectrale et géométrie (Grenoble)

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Pierre Bérard; Bernard Helffer
Edited extracts from Antonie Stern’s thesis
Séminaire de Théorie spectrale et géométrie (Grenoble), 32 (2014-2015), p. 39-72, doi: 10.5802/tsg.303
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Class. Math.: 35P15, 49R50
Mots clés: Nodal domains, Nodal lines, Courant theorem

Résumé - Abstract

The main purpose of these notes is to provide a reproduction, with some editing, of the first part of Antonie Stern’s 1924 PhD thesis which deals with the construction of higher energy eigenfunctions with a prescribed number of nodal domains, in contrast with Courant’s nodal domain theorem. A. Stern considers both Dirichlet eigenfunctions for the square membrane – her main result in this framework is mentioned in the second edition of the classical book by R. Courant and D. Hilbert – and eigenfunctions of the spherical Laplacian, her spherical results seem to have been overlooked in the literature, until very recently.

Bibliographie

[1] Pierre Bérard & Bernard Helffer, Nodal sets of eigenfunctions, Antonie Stern’s results revisited, Séminaire de Théorie Spectrale et Géométrie, Institut Fourier, Grenoble, 2014-2015
[2] Pierre Bérard & Bernard Helffer, Dirichlet eigenfunctions of the square membrane: Courant’s property, and A. Stern’s and Å. Pleijel’s analyses, in Ali Baklouti, Aziz El Kacimi, Sadok Kallel, Nordine Mir, éd., Analysis and Geometry. MIMS-GGTM, Tunis, Tunisia, March 2014. In Honour of Mohammed Salah Baouendi, Springer Proceedings in Mathematics & Statistics, Springer International Publishing, 2015, p. 69-114  MR 3445517
[3] Pierre Bérard & Bernard Helffer, “A. Stern’s analysis of the nodal sets of some families of spherical harmonics revisited”, Monatshefte für Mathematik 180 (2016), p. 435-468 Article |  MR 3513215
[4] Antonie Stern, Bemerkungen über asymptotisches Verhalten von Eigenwerten und Eigenfunktionen, Ph. D. Thesis, Druck der Dieterichschen Universitäts-Buchdruckerei (W. Fr. Kaestner), Göttingen, Germany, 1925  JFM 51.0356.01
[5] Annette Vogt, Wissenschaftlerinnen in Kaiser-Wilhelm-Instituten. A-Z, Veröffentlichungen aus dem Archiv der Max-Planck-Gesellschaft 12, Archiv der Max-Planck-Gesellschaft, 2008
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