Nous démontrons deux bornes explicites pour les multiplicités des valeurs propres de Steklov sur les surfaces compactes avec bord. Une de ces bornes ne dépend que du genre de la surface et de l’indice de la valeur propre, tandis que l’autre dépend également du nombre de composantes connexes du bord. Nous montrons aussi que pour toute surface riemannienne lisse donnée, les multiplicités des valeurs propres de Steklov sont uniformément bornées en .
We prove two explicit bounds for the multiplicities of Steklov eigenvalues on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index of an eigenvalue, while the other depends as well on the number of boundary components. We also show that on any given Riemannian surface with smooth boundary the multiplicities of Steklov eigenvalues are uniformly bounded in .
@article{AIF_2014__64_6_2481_0,
author = {Karpukhin, Mikhail and Kokarev, Gerasim and Polterovich, Iosif},
title = {Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces},
journal = {Annales de l'Institut Fourier},
volume = {64},
year = {2014},
pages = {2481-2502},
doi = {10.5802/aif.2918},
zbl = {06387345},
mrnumber = {3331172},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2014__64_6_2481_0}
}
Karpukhin, Mikhail; Kokarev, Gerasim; Polterovich, Iosif. Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces. Annales de l'Institut Fourier, Tome 64 (2014) pp. 2481-2502. doi : 10.5802/aif.2918. http://gdmltest.u-ga.fr/item/AIF_2014__64_6_2481_0/
[1] Critical points of solutions of elliptic equations in two variables, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Tome 14 (1987) no. 2, p. 229-256 (1988) | Numdam | MR 939628 | Zbl 0649.35026
[2] Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions, SIAM J. Math. Anal., Tome 25 (1994) no. 5, pp. 1259-1268 | Article | MR 1289138 | Zbl 0809.35070
[3] Isoperimetric inequalities and applications, Pitman (Advanced Publishing Program), Boston, Mass.-London, Monographs and Studies in Mathematics, Tome 7 (1980), pp. x+228 | MR 572958 | Zbl 0436.35063
[4] Local behavior of solutions of general linear elliptic equations, Comm. Pure Appl. Math., Tome 8 (1955), pp. 473-496 | Article | MR 75416 | Zbl 0066.08101
[5] Sur la multiplicité de la première valeur propre des surfaces riemanniennes, Ann. Inst. Fourier (Grenoble), Tome 30 (1980) no. 1, pp. x, 109-128 | Article | Numdam | MR 576075 | Zbl 0417.30033
[6] À propos de la multiplicité de la première valeur propre du laplacien d’une surface de Riemann, C. R. Acad. Sci. Paris Sér. I Math., Tome 300 (1985) no. 8, pp. 247-249 | MR 785061 | Zbl 0574.53029
[7] Eigenfunctions and nodal sets, Comment. Math. Helv., Tome 51 (1976) no. 1, pp. 43-55 | Article | MR 397805 | Zbl 0334.35022
[8] Sur la multiplicité de la première valeur propre d’une surface de Riemann à courbure constante, Comment. Math. Helv., Tome 63 (1988) no. 2, pp. 194-208 | Article | MR 948777 | Zbl 0656.53043
[9] Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y. (1953), pp. xv+561 | MR 65391 | Zbl 0051.28802
[10] An inverse spectral result for the Neumann operator on planar domains, J. Funct. Anal., Tome 111 (1993) no. 2, pp. 312-322 | Article | MR 1203456 | Zbl 0813.47003
[11] Eigenvalue bounds and minimal surfaces in the ball (arXiv:1209.3789v2)
[12] The first Steklov eigenvalue, conformal geometry, and minimal surfaces, Adv. Math., Tome 226 (2011) no. 5, pp. 4011-4030 | Article | MR 2770439 | Zbl 1215.53052
[13] Graphs, surfaces and homology, Cambridge University Press, Cambridge (2010), pp. xx+251 | Article | MR 2722281 | Zbl 1201.55001
[14] Elliptic partial differential equations of second order, Springer-Verlag, Berlin, Classics in Mathematics (2001), pp. xiv+517 (Reprint of the 1998 edition) | MR 1814364 | Zbl 1042.35002
[15] (2009) (Private communication)
[16] Shape optimization for low Neumann and Steklov eigenvalues, Math. Methods Appl. Sci., Tome 33 (2010) no. 4, pp. 501-516 | Article | MR 2641628 | Zbl 1186.35121
[17] Upper bounds for Steklov eigenvalues on surfaces, Electron. Res. Announc. Math. Sci., Tome 19 (2012), pp. 77-85 | Article | MR 2970718 | Zbl 1257.58019
[18] Nodal sets for groundstates of Schrödinger operators with zero magnetic field in non-simply connected domains, Comm. Math. Phys., Tome 202 (1999) no. 3, pp. 629-649 | Article | MR 1690957 | Zbl 1042.81012
[19] On the multiplicity of eigenvalues of the Laplacian on surfaces, Ann. Global Anal. Geom., Tome 17 (1999) no. 1, pp. 43-48 | Article | MR 1674331 | Zbl 0923.35109
[20] Bounds on the multiplicity of eigenvalues for fixed membranes, Geom. Funct. Anal., Tome 9 (1999) no. 6, pp. 1169-1188 | Article | MR 1736932 | Zbl 0949.35102
[21] The spectral function of an elliptic operator, Acta Math., Tome 121 (1968), pp. 193-218 | Article | MR 609014 | Zbl 0164.13201
[22] Prescription du spectre de Steklov dans une classe conforme, Anal. PDE, Tome 7 (2014) no. 3, pp. 529-550 | Article | MR 3227426
[23] An inequality of a Stekloff eigenvalue by the method of defect, Proc. Amer. Math. Soc., Tome 20 (1969), pp. 357-360 | MR 235323 | Zbl 0176.09901
[24] Multiple eigenvalues of the Laplace operator, Mat. Sb. (N.S.), Tome 133(175) (1987) no. 2, p. 223-237, 272 | MR 905007 | Zbl 0672.35049
[25] Asymptotic behavior of the eigenvalues for some two-dimensional spectral problems, Boundary value problems. Spectral theory (Russian), Leningrad. Univ., Leningrad (Probl. Mat. Anal.) Tome 7 (1979), p. 188-203, 245 | MR 559110
[26] Pseudodifferential operators and spectral theory, Springer-Verlag, Berlin (2001), pp. xii+288 (Translated from the 1978 Russian original by Stig I. Andersson) | Article | MR 1852334 | Zbl 0616.47040
[27] Partial differential equations. II, Springer-Verlag, New York, Applied Mathematical Sciences, Tome 116 (1996), pp. xxii+528 (Qualitative studies of linear equations) | Article | MR 1395149 | Zbl 0869.35003