On nodal sets and nodal domains on S 2 and 2
Annales de l'Institut Fourier, Volume 57 (2007) no. 7, pp. 2345-2360.

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 Δu=u in 2 that has only two nodal domains. This equation arises in the study of high energy eigenfunctions.

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 Δu=u dans  2 qui possède seulement deux domaines nodaux. Cette équation est considérée dans l’étude des fonctions propres à haute énergie.

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
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)
     author = {Eremenko, Alexandre and Jakobson, Dmitry and Nadirashvili, Nikolai},
     title = {On nodal sets and nodal domains on $\mathbf{S^2}$ and ${\mathbb{R}}^{\mathbf{2}}$},
     journal = {Annales de l'Institut Fourier},
     pages = {2345--2360},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
     number = {7},
     year = {2007},
     doi = {10.5802/aif.2335},
     mrnumber = {2394544},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2335/}
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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, Volume 57 (2007) no. 7, pp. 2345-2360. doi : 10.5802/aif.2335. https://aif.centre-mersenne.org/articles/10.5802/aif.2335/

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