On nodal sets and nodal domains on $\mathbf{S^2}$ and ${\mathbb{R}}^{\mathbf{2}}$

Eremenko, Alexandre; Jakobson, Dmitry; Nadirashvili, Nikolai

doi:10.5802/aif.2335

On nodal sets and nodal domains on

S^{2}

and

ℝ^{2}

[Sur les ensembles nodaux et les domaines nodaux sur

S^{2}

ℝ^{2}

]

Eremenko, Alexandre ¹ ; Jakobson, Dmitry ² ; Nadirashvili, Nikolai ³

¹ Purdue University Mathematics Department 150 N University Street West Lafayette, IN 47907-2067 (USA)
² McGill University Department of Mathematics and Statistics 805 Sherbrooke Str.West Montreal, QC H3A 2K6 (Canada)
³ Université de Provence Laboratoire d’Analyse, Topologie, Probabilités UMR 6632 Centre de Mathématiques et Informatique 39 rue F.Joliot-Curie 13453 Marseille Cedex 13 (France)

Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2345-2360.

Résumé
Abstract

On étudie les configurations topologiques possibles d’ensembles nodaux, en particulier, le nombre de leurs composantes, pour les harmoniques sphériques sur $S^{2}$ . Nous construisons aussi une solution de l’équation $Δ u = u$ dans $ℝ^{2}$ qui possède seulement deux domaines nodaux. Cette équation est considérée dans l’étude des fonctions propres à haute énergie.

We discuss possible topological configurations of nodal sets, in particular the number of their components, for spherical harmonics on $S^{2}$ . We also construct a solution of the equation $Δ u = u$ in $ℝ^{2}$ that has only two nodal domains. This equation arises in the study of high energy eigenfunctions.

MR | 1 citation dans Numdam

DOI : 10.5802/aif.2335

Classification : 58J50, 11J70, 35P20, 81Q50
Keywords: Laplacian, nodal sets, nodal domains, spherical harmonic, topological configuration
Mot clés : Laplacien, ensemble nodaux, domaines nodaux, harmonique sphérique, configuration topologique

Affiliations des auteurs :

Eremenko, Alexandre ¹ ; Jakobson, Dmitry ² ; Nadirashvili, Nikolai ³

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     journal = {Annales de l'Institut Fourier},
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     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Eremenko, Alexandre; Jakobson, Dmitry; Nadirashvili, Nikolai. On nodal sets and nodal domains on $\mathbf{S^2}$ and ${\mathbb{R}}^{\mathbf{2}}$. Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2345-2360. doi : 10.5802/aif.2335. http://www.numdam.org/articles/10.5802/aif.2335/

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