Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces
Annales de l'Institut Fourier, Volume 64 (2014) no. 6, pp. 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: 10.5802/aif.2918
Classification: 58J50,  35P15,  35J25
Keywords: Steklov problem, eigenvalue multiplicity, Riemannian surface
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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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/articles/10.5802/aif.2918/

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