Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces

Mikhail Karpukhin[1]; Gerasim Kokarev[2]; Iosif Polterovich[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)

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 6, page 2481-2502
  • ISSN: 0373-0956

Abstract

top
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 .

How to cite

top

Karpukhin, Mikhail, Kokarev, Gerasim, and Polterovich, Iosif. "Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces." Annales de l’institut Fourier 64.6 (2014): 2481-2502. <http://eudml.org/doc/275456>.

@article{Karpukhin2014,
abstract = {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 $k$.},
affiliation = {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); Mathematisches Institut der Universität München Theresienstr. 39, D-80333 München (Germany); Université de Montréal Département de mathématiques et de statistique CP 6128 succ Centre-Ville Montréal, QC H3C 3J7 (Canada)},
author = {Karpukhin, Mikhail, Kokarev, Gerasim, Polterovich, Iosif},
journal = {Annales de l’institut Fourier},
keywords = {Steklov problem; eigenvalue multiplicity; Riemannian surface; nodal sets, nodal domains; nodal graphs},
language = {eng},
number = {6},
pages = {2481-2502},
publisher = {Association des Annales de l’institut Fourier},
title = {Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces},
url = {http://eudml.org/doc/275456},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Karpukhin, Mikhail
AU - Kokarev, Gerasim
AU - Polterovich, Iosif
TI - Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 6
SP - 2481
EP - 2502
AB - 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 $k$.
LA - eng
KW - Steklov problem; eigenvalue multiplicity; Riemannian surface; nodal sets, nodal domains; nodal graphs
UR - http://eudml.org/doc/275456
ER -

References

top
  1. Giovanni Alessandrini, Critical points of solutions of elliptic equations in two variables, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), 229-256 (1988) Zbl0649.35026MR939628
  2. Giovanni Alessandrini, R. Magnanini, Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions, SIAM J. Math. Anal. 25 (1994), 1259-1268 Zbl0809.35070MR1289138
  3. Catherine Bandle, Isoperimetric inequalities and applications, 7 (1980), Pitman (Advanced Publishing Program), Boston, Mass.-London Zbl0436.35063MR572958
  4. Lipman Bers, Local behavior of solutions of general linear elliptic equations, Comm. Pure Appl. Math. 8 (1955), 473-496 Zbl0066.08101MR75416
  5. Gérard Besson, Sur la multiplicité de la première valeur propre des surfaces riemanniennes, Ann. Inst. Fourier (Grenoble) 30 (1980), x, 109-128 Zbl0417.30033MR576075
  6. Marc Burger, Bruno Colbois, 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. 300 (1985), 247-249 Zbl0574.53029MR785061
  7. Shiu Yuen Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv. 51 (1976), 43-55 Zbl0334.35022MR397805
  8. B. Colbois, Y. Colin de Verdière, Sur la multiplicité de la première valeur propre d’une surface de Riemann à courbure constante, Comment. Math. Helv. 63 (1988), 194-208 Zbl0656.53043MR948777
  9. R. Courant, D. Hilbert, Methods of mathematical physics. Vol. I, (1953), Interscience Publishers, Inc., New York, N.Y. Zbl0051.28802MR65391
  10. Julian Edward, An inverse spectral result for the Neumann operator on planar domains, J. Funct. Anal. 111 (1993), 312-322 Zbl0813.47003MR1203456
  11. Ailana Fraser, Richard Schoen, Eigenvalue bounds and minimal surfaces in the ball Zbl1337.35099
  12. Ailana Fraser, Richard Schoen, The first Steklov eigenvalue, conformal geometry, and minimal surfaces, Adv. Math. 226 (2011), 4011-4030 Zbl1215.53052MR2770439
  13. Peter Giblin, Graphs, surfaces and homology, (2010), Cambridge University Press, Cambridge Zbl1201.55001MR2722281
  14. David Gilbarg, Neil S. Trudinger, Elliptic partial differential equations of second order, (2001), Springer-Verlag, Berlin Zbl1042.35002MR1814364
  15. Alexandre Girouard, (2009) 
  16. Alexandre Girouard, Iosif Polterovich, Shape optimization for low Neumann and Steklov eigenvalues, Math. Methods Appl. Sci. 33 (2010), 501-516 Zbl1186.35121MR2641628
  17. Alexandre Girouard, Iosif Polterovich, Upper bounds for Steklov eigenvalues on surfaces, Electron. Res. Announc. Math. Sci. 19 (2012), 77-85 Zbl1257.58019MR2970718
  18. B. Helffer, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, M. P. Owen, Nodal sets for groundstates of Schrödinger operators with zero magnetic field in non-simply connected domains, Comm. Math. Phys. 202 (1999), 629-649 Zbl1042.81012MR1690957
  19. M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, N. Nadirashvili, On the multiplicity of eigenvalues of the Laplacian on surfaces, Ann. Global Anal. Geom. 17 (1999), 43-48 Zbl0923.35109MR1674331
  20. T. Hoffmann-Ostenhof, P. W. Michor, N. Nadirashvili, Bounds on the multiplicity of eigenvalues for fixed membranes, Geom. Funct. Anal. 9 (1999), 1169-1188 Zbl0949.35102MR1736932
  21. Lars Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193-218 Zbl0164.13201MR609014
  22. Pierre Jammes, Prescription du spectre de Steklov dans une classe conforme, Anal. PDE 7 (2014), 529-550 Zbl1304.35452MR3227426
  23. J. R. Kuttler, V. G. Sigillito, An inequality of a Stekloff eigenvalue by the method of defect, Proc. Amer. Math. Soc. 20 (1969), 357-360 Zbl0176.09901MR235323
  24. N. S. Nadirashvili, Multiple eigenvalues of the Laplace operator, Mat. Sb. (N.S.) 133(175) (1987), 223-237, 272 Zbl0672.35049MR905007
  25. G. V. Rozenbljum, Asymptotic behavior of the eigenvalues for some two-dimensional spectral problems, Boundary value problems. Spectral theory (Russian) 7 (1979), 188-203, 245, Leningrad. Univ., Leningrad MR559110
  26. M. A. Shubin, Pseudodifferential operators and spectral theory, (2001), Springer-Verlag, Berlin Zbl0616.47040MR1852334
  27. Michael E. Taylor, Partial differential equations. II, 116 (1996), Springer-Verlag, New York Zbl0869.35003MR1395149

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.