Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces
Annales de l'Institut Fourier, Volume 64 (2014) no. 6, p. 2481-2502

We prove two explicit bounds for the multiplicities of Steklov eigenvalues σ k on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index k 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 σ k are uniformly bounded in k.

Nous démontrons deux bornes explicites pour les multiplicités des valeurs propres de Steklov σ k sur les surfaces compactes avec bord. Une de ces bornes ne dépend que du genre de la surface et de l’indice k 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 σ k sont uniformément bornées en k.

DOI : https://doi.org/10.5802/aif.2918
Classification:  58J50,  35P15,  35J25
Keywords: Steklov problem, eigenvalue multiplicity, Riemannian surface
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {64},
     number = {6},
     year = {2014},
     pages = {2481-2502},
     doi = {10.5802/aif.2918},
     mrnumber = {3331172},
     zbl = {06387345},
     language = {en},
     url = {http://www.numdam.org/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, Volume 64 (2014) no. 6, pp. 2481-2502. doi : 10.5802/aif.2918. http://www.numdam.org/item/AIF_2014__64_6_2481_0/

[1] Alessandrini, Giovanni 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] Alessandrini, Giovanni; Magnanini, R. 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] Bandle, Catherine 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] Bers, Lipman 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] Besson, Gérard 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] Burger, Marc; Colbois, Bruno 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] Cheng, Shiu Yuen Eigenfunctions and nodal sets, Comment. Math. Helv., Tome 51 (1976) no. 1, pp. 43-55 | Article | MR 397805 | Zbl 0334.35022

[8] Colbois, B.; Colin De Verdière, Y. 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] Courant, R.; Hilbert, D. Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y. (1953), pp. xv+561 | MR 65391 | Zbl 0051.28802

[10] Edward, Julian 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] Fraser, Ailana; Schoen, Richard Eigenvalue bounds and minimal surfaces in the ball (arXiv:1209.3789v2)

[12] Fraser, Ailana; Schoen, Richard 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] Giblin, Peter Graphs, surfaces and homology, Cambridge University Press, Cambridge (2010), pp. xx+251 | Article | MR 2722281 | Zbl 1201.55001

[14] Gilbarg, David; Trudinger, Neil S. 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] Girouard, Alexandre (2009) (Private communication)

[16] Girouard, Alexandre; Polterovich, Iosif 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] Girouard, Alexandre; Polterovich, Iosif 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] Helffer, B.; Hoffmann-Ostenhof, M.; Hoffmann-Ostenhof, T.; Owen, M. P. 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] Hoffmann-Ostenhof, M.; Hoffmann-Ostenhof, T.; Nadirashvili, N. 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] Hoffmann-Ostenhof, T.; Michor, P. W.; Nadirashvili, N. 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] Hörmander, Lars The spectral function of an elliptic operator, Acta Math., Tome 121 (1968), pp. 193-218 | Article | MR 609014 | Zbl 0164.13201

[22] Jammes, Pierre Prescription du spectre de Steklov dans une classe conforme, Anal. PDE, Tome 7 (2014) no. 3, pp. 529-550 | Article | MR 3227426

[23] Kuttler, J. R.; Sigillito, V. G. 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] Nadirashvili, N. S. 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] Rozenbljum, G. V. 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] Shubin, M. A. 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] Taylor, Michael E. 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