no. 7
On nodal sets and nodal domains on S 2 and ℝ 2
[Sur les ensembles nodaux et les domaines nodaux sur S 2 et ℝ 2 ]
Eremenko, Alexandre1 ; Jakobson, Dmitry2 ; Nadirashvili, Nikolai3
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.

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

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) 
@article{AIF_2007__57_7_2345_0,
     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/}
}
TY  - JOUR
AU  - Eremenko, Alexandre
AU  - Jakobson, Dmitry
AU  - Nadirashvili, Nikolai
TI  - On nodal sets and nodal domains on $\mathbf{S^2}$ and ${\mathbb{R}}^{\mathbf{2}}$
JO  - Annales de l'Institut Fourier
PY  - 2007
SP  - 2345
EP  - 2360
VL  - 57
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2335/
DO  - 10.5802/aif.2335
LA  - en
ID  - AIF_2007__57_7_2345_0
ER  - 
%0 Journal Article
%A Eremenko, Alexandre
%A Jakobson, Dmitry
%A Nadirashvili, Nikolai
%T On nodal sets and nodal domains on $\mathbf{S^2}$ and ${\mathbb{R}}^{\mathbf{2}}$
%J Annales de l'Institut Fourier
%D 2007
%P 2345-2360
%V 57
%N 7
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2335/
%R 10.5802/aif.2335
%G en
%F AIF_2007__57_7_2345_0
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. https://aif.centre-mersenne.org/articles/10.5802/aif.2335/

[1] Arnold, V.; Vishik, M.; Ilyashenko, Y.; Kondratyev, A. Kalashnikovand V.; Kruzhkov, S.; Landis, E.; Millionshchikov, V.; Oleinik, O.; Filippov, A.; Shubin, M. Some unsolved problems in the theory of differential equations and mathematical physics, Uspekhi Mat. Nauk, Volume 44 (1989), pp. 191-202 transl. in Russian Math. Surveys 44 (1989), p.157–171 | MR | Zbl

[2] Courant, R.; Hilbert, D. Methods of Mathematical Physics, I, Interscience Publishers, New York, 1953 | MR | Zbl

[3] Eremenko, A.; Gabrielov, A. Rational functions with real critical points and the B. and M.Shapiro Conjecture in real algebraic geometry, Annals of Math., Volume 155 (2002), pp. 105-129 | DOI | MR | Zbl

[4] Gudkov, D. The topology of real projective algebraic varieties, Uspehi Mat. Nauk, Volume 178 (1974), pp. 3-79 | MR | Zbl

[5] Jakobson, D.; Nadirashvili, N.; Toth, J. Geometric properties of eigenfunctions, Russian Math. Surveys, Volume 56 (2001), pp. 67-88 | DOI | MR | Zbl

[6] Karpushkin, V. N. Topology of zeros of eigenfunctions, Funct. Anal. Appl., Volume 23 (1989), pp. 218-220 | DOI | MR | Zbl

[7] Karpushkin, V. N. The number of components of the complement of the level surface of a harmonic polynomial in three variables, Funct. Anal. Appl., Volume 28 (1994), pp. 116-118 | DOI | MR | Zbl

[8] Karpushkin, V. N. On the number of components of the complement to some algebraic curves, Russian Math. Surveys, Volume 57 (2002), pp. 1228-1229 | DOI | MR | Zbl

[9] Lando, S.; Zvonkin, A. Graphs on surfaces and their applications. With an appendix by Don B. Zagier, Low-Dimensional Topology, II, Encyclopaedia of Mathematical Sciences, Volume 141, Springer-Verlag, Berlin, 2004 | MR | Zbl

[10] Lewy, H. On the minimum number of domains in which the nodal lines of spherical harmonics divide the sphere, Comm. PDE,, Volume 2 (1977), pp. 1233-1244 | DOI | MR | Zbl

[11] Leydold, J. On the number of nodal domains of spherical harmonics, Topology, Volume 35 (1996), pp. 301-321 | DOI | MR | Zbl

[12] Neuheisel, J. Asymptotic distribution of nodal sets on spheres, Johns Hopkins University, Baltimore, MD 2000 (1994) (Ph. D. Thesis)

[13] Pleijel, A. Remarks on Courant’s nodal line theorem, Comm. Pure Appl. Math, Volume 9 (1956), pp. 543-550 | DOI | Zbl

[14] Santos, F. Optimal degree construction of real algebraic plane nodal curves with prescribed topology. I. The orientable case. Real algebraic and analytic geometry (Segovia, 1995), Rev. Mat. Univ. Complut. Madrid, Volume 10 (1997), pp. 291-310 (Special Issue suppl.) | MR | Zbl

[15] Segura, J. Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros, Math. Comp., Volume 70 (2001), pp. 1205-1220 | DOI | MR | Zbl

[16] Tikhonov, A.; Samarskii, A. Equations of Mathematical Physics, Moscow, 1953 | Zbl

[17] Viro, O. Real algebraic plane curves: constructions with controlled topology, Leningrad Math. J., Volume 1 (1990), pp. 1059-1134 | MR | Zbl

Cité par Sources :