no. 6
Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces
[Bornes pour les multiplicités des valeurs propres de Steklov sur les surfaces riemanniennes]
Karpukhin, Mikhail1 ; Kokarev, Gerasim2 ; Polterovich, Iosif3
1 Moscow State University Department of Geometry and Topology Leninskie Gory, GSP-1, 119991, Moscow (Russia) Independent University of Moscow Bolshoy Vlasyevskiy pereulok 11, 119002 Moscow (Russia)
2 Mathematisches Institut der Universität München Theresienstr. 39, D-80333 München (Germany)
3 Université de Montréal Département de mathématiques et de statistique CP 6128 succ Centre-Ville Montréal, QC H3C 3J7 (Canada)
Annales de l'Institut Fourier, Tome 64 (2014) no. 6, pp. 2481-2502.

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.

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.

DOI : 10.5802/aif.2918
Classification : 58J50, 35P15, 35J25
Keywords: Steklov problem, eigenvalue multiplicity, Riemannian surface
Mot clés : spectre de Steklov, multiplicité de valeurs propres, surface riemannienne

Karpukhin, Mikhail 1 ; Kokarev, Gerasim 2 ; Polterovich, Iosif 3

1 Moscow State University Department of Geometry and Topology Leninskie Gory, GSP-1, 119991, Moscow (Russia) Independent University of Moscow Bolshoy Vlasyevskiy pereulok 11, 119002 Moscow (Russia)
2 Mathematisches Institut der Universität München Theresienstr. 39, D-80333 München (Germany)
3 Université de Montréal Département de mathématiques et de statistique CP 6128 succ Centre-Ville Montréal, QC H3C 3J7 (Canada) 
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     title = {Multiplicity bounds for {Steklov} eigenvalues on {Riemannian} surfaces},
     journal = {Annales de l'Institut Fourier},
     pages = {2481--2502},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Karpukhin, Mikhail; Kokarev, Gerasim; Polterovich, Iosif. Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces. Annales de l'Institut Fourier, Tome 64 (2014) no. 6, pp. 2481-2502. doi : 10.5802/aif.2918. https://aif.centre-mersenne.org/articles/10.5802/aif.2918/

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