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Alexandre Eremenko; Dmitry Jakobson; Nikolai Nadirashvili
On nodal sets and nodal domains on $\mathbf{S^2}$ and ${\mathbb{R}}^{\mathbf{2}}$
(Sur les ensembles nodaux et les domaines nodaux sur $S^2$ et ${\mathbb{R}}^2$)
Annales de l'institut Fourier, 57 no. 7 (2007), p. 2345-2360, doi: 10.5802/aif.2335
Article PDF | Reviews MR 2394544 | Zbl pre05249488
Class. Math.: 58J50, 11J70, 35P20, 81Q50
Keywords: Laplacian, nodal sets, nodal domains, spherical harmonic, topological configuration

Résumé - Abstract

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 $\Delta u=u$ in ${\mathbb{R}}^2$ that has only two nodal domains. This equation arises in the study of high energy eigenfunctions.

Bibliography

[1] V. Arnold, M. Vishik, Y. Ilyashenko, A. Kalashnikovand V. Kondratyev, S. Kruzhkov, E. Landis, V. Millionshchikov, O. Oleinik, A. Filippov & M. Shubin, “Some unsolved problems in the theory of differential equations and mathematical physics”, Uspekhi Mat. Nauk 44 (1989), p. 191-202, transl. in Russian Math. Surveys 44 (1989), p.157–171  MR 1023106 |  Zbl 0703.35002
[2] R. Courant & D. Hilbert, Methods of Mathematical Physics, I, Interscience Publishers, New York, 1953  MR 65391 |  Zbl 0051.28802
[3] A. Eremenko & A. Gabrielov, “Rational functions with real critical points and the B. and M.Shapiro Conjecture in real algebraic geometry”, Annals of Math. 155 (2002), p. 105-129 Article |  MR 1888795 |  Zbl 0997.14015
[4] D. Gudkov, “The topology of real projective algebraic varieties”, Uspehi Mat. Nauk 178 (1974), p. 3-79  MR 399085 |  Zbl 0316.14018
[5] D. Jakobson, N. Nadirashvili & J. Toth, “Geometric properties of eigenfunctions”, Russian Math. Surveys 56 (2001), p. 67-88 Article |  MR 1886720 |  Zbl 1060.58019
[6] V. N. Karpushkin, “Topology of zeros of eigenfunctions”, Funct. Anal. Appl. 23 (1989), p. 218-220 Article |  MR 1026990 |  Zbl 0705.47042
[7] V. N. Karpushkin, “The number of components of the complement of the level surface of a harmonic polynomial in three variables”, Funct. Anal. Appl. 28 (1994), p. 116-118 Article |  MR 1283253 |  Zbl 0846.57028
[8] V. N. Karpushkin, “On the number of components of the complement to some algebraic curves”, Russian Math. Surveys 57 (2002), p. 1228-1229 Article |  MR 1991874 |  Zbl 1049.14043
[9] S. Lando & A. Zvonkin, Graphs on surfaces and their applications. With an appendix by Don B. Zagier, Low-Dimensional Topology, II, Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 2004  MR 2036721 |  Zbl 1040.05001
[10] H. Lewy, “On the minimum number of domains in which the nodal lines of spherical harmonics divide the sphere”, Comm. PDE, 2 (1977), p. 1233-1244 Article |  MR 477199 |  Zbl 0377.31008
[11] J. Leydold, “On the number of nodal domains of spherical harmonics”, Topology 35 (1996), p. 301-321 Article |  MR 1380499 |  Zbl 0853.33012
[12] J. Neuheisel, Asymptotic distribution of nodal sets on spheres, thesis, Johns Hopkins University, Baltimore, MD 2000, 1994
[13] A. Pleijel, “Remarks on Courant’s nodal line theorem”, Comm. Pure Appl. Math 9 (1956), p. 543-550 Article |  Zbl 0070.32604
[14] F. Santos, “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 10 (1997), p. 291-310, Special Issue suppl.  MR 1485306 |  Zbl 0949.14036
[15] J. Segura, “Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros”, Math. Comp. 70 (2001), p. 1205-1220 Article |  MR 1710198 |  Zbl 0983.33004
[16] A. Tikhonov & A. Samarskii, Equations of Mathematical Physics, Moscow, 1953  Zbl 0044.09302
[17] O. Viro, “Real algebraic plane curves: constructions with controlled topology”, Leningrad Math. J. 1 (1990), p. 1059-1134  MR 1036837 |  Zbl 0732.14026
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