Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces

Karpukhin, Mikhail; Kokarev, Gerasim; Polterovich, Iosif

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 ${\sigma }_k$ 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 ${\sigma }_k$ sont uniformément bornées en $k$.

We prove two explicit bounds for the multiplicities of Steklov eigenvalues ${\sigma }_k$ 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 ${\sigma }_k$ are uniformly bounded in $k$.

Classification : 58J50, 35P15, 35J25
Keywords: Steklov problem, eigenvalue multiplicity, Riemannian surface
Mot clés : spectre de Steklov, multiplicité de valeurs propres, surface riemannienne

[1] Alessandrini, Giovanni Critical points of solutions of elliptic equations in two variables, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume 14 (1987) no. 2, p. 229-256 (1988) | Numdam | MR | Zbl

[2] Alessandrini, Giovanni; Magnanini, R. Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions, SIAM J. Math. Anal., Volume 25 (1994) no. 5, pp. 1259-1268 | DOI | MR | Zbl

[3] Bandle, Catherine Isoperimetric inequalities and applications, Monographs and Studies in Mathematics, 7, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1980, pp. x+228 | MR | Zbl

[4] Bers, Lipman Local behavior of solutions of general linear elliptic equations, Comm. Pure Appl. Math., Volume 8 (1955), pp. 473-496 | DOI | MR | Zbl

[5] Besson, Gérard Sur la multiplicité de la première valeur propre des surfaces riemanniennes, Ann. Inst. Fourier (Grenoble), Volume 30 (1980) no. 1, pp. x, 109-128 | DOI | Numdam | MR | Zbl

[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., Volume 300 (1985) no. 8, pp. 247-249 | MR | Zbl

[7] Cheng, Shiu Yuen Eigenfunctions and nodal sets, Comment. Math. Helv., Volume 51 (1976) no. 1, pp. 43-55 | DOI | MR | Zbl

[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., Volume 63 (1988) no. 2, pp. 194-208 | DOI | MR | Zbl

[9] Courant, R.; Hilbert, D. Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953, pp. xv+561 | MR | Zbl

[10] Edward, Julian An inverse spectral result for the Neumann operator on planar domains, J. Funct. Anal., Volume 111 (1993) no. 2, pp. 312-322 | DOI | MR | Zbl

[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., Volume 226 (2011) no. 5, pp. 4011-4030 | DOI | MR | Zbl

[13] Giblin, Peter Graphs, surfaces and homology, Cambridge University Press, Cambridge, 2010, pp. xx+251 | DOI | MR | Zbl

[14] Gilbarg, David; Trudinger, Neil S. Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, pp. xiv+517 (Reprint of the 1998 edition) | MR | Zbl

[15] Girouard, Alexandre, 2009 (Private communication)

[16] Girouard, Alexandre; Polterovich, Iosif Shape optimization for low Neumann and Steklov eigenvalues, Math. Methods Appl. Sci., Volume 33 (2010) no. 4, pp. 501-516 | DOI | MR | Zbl

[17] Girouard, Alexandre; Polterovich, Iosif Upper bounds for Steklov eigenvalues on surfaces, Electron. Res. Announc. Math. Sci., Volume 19 (2012), pp. 77-85 | DOI | MR | Zbl

[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., Volume 202 (1999) no. 3, pp. 629-649 | DOI | MR | Zbl

[19] Hoffmann-Ostenhof, M.; Hoffmann-Ostenhof, T.; Nadirashvili, N. On the multiplicity of eigenvalues of the Laplacian on surfaces, Ann. Global Anal. Geom., Volume 17 (1999) no. 1, pp. 43-48 | DOI | MR | Zbl

[20] Hoffmann-Ostenhof, T.; Michor, P. W.; Nadirashvili, N. Bounds on the multiplicity of eigenvalues for fixed membranes, Geom. Funct. Anal., Volume 9 (1999) no. 6, pp. 1169-1188 | DOI | MR | Zbl

[21] Hörmander, Lars The spectral function of an elliptic operator, Acta Math., Volume 121 (1968), pp. 193-218 | DOI | MR | Zbl

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

[23] Kuttler, J. R.; Sigillito, V. G. An inequality of a Stekloff eigenvalue by the method of defect, Proc. Amer. Math. Soc., Volume 20 (1969), pp. 357-360 | MR | Zbl

[24] Nadirashvili, N. S. Multiple eigenvalues of the Laplace operator, Mat. Sb. (N.S.), Volume 133(175) (1987) no. 2, p. 223-237, 272 | MR | Zbl

[25] Rozenbljum, G. V. Asymptotic behavior of the eigenvalues for some two-dimensional spectral problems, Boundary value problems. Spectral theory (Russian) (Probl. Mat. Anal.), Volume 7, Leningrad. Univ., Leningrad, 1979, p. 188-203, 245 | MR

[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) | DOI | MR | Zbl

[27] Taylor, Michael E. Partial differential equations. II, Applied Mathematical Sciences, 116, Springer-Verlag, New York, 1996, pp. xxii+528 (Qualitative studies of linear equations) | DOI | MR | Zbl