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

Eremenko, Alexandre; Jakobson, Dmitry; Nadirashvili, Nikolai

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 ; Jakobson, Dmitry ; Nadirashvili, Nikolai

Annales de l'Institut Fourier, Tome 57 (2007), p. 2345-2360 / Harvested from Numdam

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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.

Publié le : 2007-01-01

MR 2394544 | Zbl pre05249488

DOI : https://doi.org/10.5802/aif.2335
Classification: 58J50, 11J70, 35P20, 81Q50
Mots clés: Laplacien, ensemble nodaux, domaines nodaux, harmonique sphérique, configuration topologique

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     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},
     volume = {57},
     year = {2007},
     pages = {2345-2360},
     doi = {10.5802/aif.2335},
     zbl = {pre05249488},
     mrnumber = {2394544},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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) pp. 2345-2360. doi : 10.5802/aif.2335. http://gdmltest.u-ga.fr/item/AIF_2007__57_7_2345_0/

Bibliographie

[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, Tome 44 (1989), pp. 191-202 (transl. in Russian Math. Surveys 44 (1989), p.157–171) | MR 1023106 | Zbl 0703.35002

[2] Courant, R.; Hilbert, D. Methods of Mathematical Physics, I, Interscience Publishers, New York (1953) | MR 65391 | Zbl 0051.28802

[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., Tome 155 (2002), pp. 105-129 | Article | MR 1888795 | Zbl 0997.14015

[4] Gudkov, D. The topology of real projective algebraic varieties, Uspehi Mat. Nauk, Tome 178 (1974), pp. 3-79 | MR 399085 | Zbl 0316.14018

[5] Jakobson, D.; Nadirashvili, N.; Toth, J. Geometric properties of eigenfunctions, Russian Math. Surveys, Tome 56 (2001), pp. 67-88 | Article | MR 1886720 | Zbl 1060.58019

[6] Karpushkin, V. N. Topology of zeros of eigenfunctions, Funct. Anal. Appl., Tome 23 (1989), pp. 218-220 | Article | MR 1026990 | Zbl 0705.47042

[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., Tome 28 (1994), pp. 116-118 | Article | MR 1283253 | Zbl 0846.57028

[8] Karpushkin, V. N. On the number of components of the complement to some algebraic curves, Russian Math. Surveys, Tome 57 (2002), p. 1228-1229 | Article | MR 1991874 | Zbl 1049.14043

[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, Springer-Verlag, Berlin, Tome 141 (2004) | MR 2036721 | Zbl 1040.05001

[10] Lewy, H. On the minimum number of domains in which the nodal lines of spherical harmonics divide the sphere, Comm. PDE,, Tome 2 (1977), pp. 1233-1244 | Article | MR 477199 | Zbl 0377.31008

[11] Leydold, J. On the number of nodal domains of spherical harmonics, Topology, Tome 35 (1996), pp. 301-321 | Article | MR 1380499 | Zbl 0853.33012

[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, Tome 9 (1956), pp. 543-550 | Article | Zbl 0070.32604

[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, Tome 10 (1997), pp. 291-310 (Special Issue suppl.) | MR 1485306 | Zbl 0949.14036

[15] Segura, J. Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros, Math. Comp., Tome 70 (2001), pp. 1205-1220 | Article | MR 1710198 | Zbl 0983.33004

[16] Tikhonov, A.; Samarskii, A. Equations of Mathematical Physics, Moscow (1953) | Zbl 0044.09302

[17] Viro, O. Real algebraic plane curves: constructions with controlled topology, Leningrad Math. J., Tome 1 (1990), pp. 1059-1134 | MR 1036837 | Zbl 0732.14026