Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-26T12:55:35.238Z Has data issue: false hasContentIssue false
  • English
  • Français

The Steklov spectrum of surfaces: asymptotics and invariants

Published online by Cambridge University Press:  19 August 2014

ALEXANDRE GIROUARD ,
LEONID PARNOVSKI ,
IOSIF POLTEROVICH  and
DAVID A. SHER
ALEXANDRE GIROUARD
Affiliation:
Département de Mathématiques et de Statistique, Université Laval, 1045, av. de la Médecine, Québec Qc G1V 0A6, Canada. e-mail: alexandre.girouard@mat.ulaval.ca
LEONID PARNOVSKI
Affiliation:
Department of Mathematics, University College London, Gower Street, LondonWC1E 6BT. e-mail: leonid@math.ucl.ac.uk
IOSIF POLTEROVICH
Affiliation:
Département de Mathématiques et de Statistique, Université de Montréal, CP 6128 succ Centre-Ville, Montréal QC H3C 3J7, Canada. e-mail: iossif@dms.umontreal.ca
DAVID A. SHER
Affiliation:
Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, MI 48109-1043, U.S.A. e-mail: dsher@umich.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We obtain precise asymptotics for the Steklov eigenvalues on a compact Riemannian surface with boundary. It is shown that the number of connected components of the boundary, as well as their lengths, are invariants of the Steklov spectrum. The proofs are based on pseudodifferential techniques for the Dirichlet-to-Neumann operator and on a number–theoretic argument.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 157 , Issue 3 , November 2014 , pp. 379 - 389
Copyright
Copyright © Cambridge Philosophical Society 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[Ag]Agranovich, M. S.On a mixed Poincaré-Steklov type spectral problem in a Lipschitz domain. Russ. J. Math. Phys. 13 (3), (2006), 281–290.CrossRefGoogle Scholar
[ABBG]Alvarez, S., Berend, D., Birbrair, L. and Girão, D.Resonance sequences and focal decomposition. Israel J. Math. 170, no. 1 (2009), 269–284.Google Scholar
[CEG]Colbois, B., El Soufi, A. and Girouard, A.Isoperimetric control of the Steklov spectrum. J. Funct. Anal. 261, no. 5 (2011), 1384–1399.Google Scholar
[DR]Doyle, P.G. and Rossetti, J.P. Laplace-isospectral hyperbolic 2-orbifolds are representation-equivalent. arXiv:1103.4372.Google Scholar
[DG]Duistermaat, H. and Guillemin, V.Spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29, no.1 (1975), 39–79.Google Scholar
[Ed]Edward, J.An inverse spectral result for the Neumann operator on planar domains. J. Funct. Anal. 111 (1993), 312–322.CrossRefGoogle Scholar
[FS1]Fraser, A. and Schoen, R. Sharp eigenvalue bounds and minimal surfaces in the ball. arXiv:1209.3789.Google Scholar
[FS2]Fraser, A. and Schoen, R.Minimal surfaces and eigenvalue problems. In Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations. Contemp. Math. 599 (Amer. Math. Soc., Providence, RI, 2013), pp. 105–121.CrossRefGoogle Scholar
[GP]Girouard, A. and Polterovich, I.Shape optimization for low Neumann and Steklov eigenvalues. Math. Meth. Appl. Sci. 33, no. 4 (2010), 501–516.Google Scholar
[GPS]Gordon, C., Perry, P. and Schueth, D.Isospectral and isoscattering manifolds: a survey of techniques and examples. In: Geometry, Spectral Theory, Groups and Dynamics (Entov, M., Pinchover, Y., Sageev, M., Editors), Contemp. Math. 387 (2005), 157–179.Google Scholar
[GW]Guillemin, V. and Weinstein, A.Eigenvalues associated with a closed geodesic. Bull. Amer. Math. Soc. 82, no. 1 (1976), 92–94.CrossRefGoogle Scholar
[HL]Hislop, P.D. and Lutzer, C.V.Spectral asymptotics of the Dirichlet-to-Neumannmap on multiply connected domains in Rd. Inverse Problems 17 (2001), 1717–1741.CrossRefGoogle Scholar
[Hö]Hörmander, L.The Analysis of Partial Differential Operators, Vol. IV. Grundlehren 275, (Springer-Verlag, New York, 1984).Google Scholar
[JS]Jollivet, A. and Sharafutdinov, V. On an inverse problem for the Steklov spectrum of a Riemannian surface. To appear in Contemp. Math.Google Scholar
[LU]Lee, J. and Uhlmann, G.Determining isotropic real-analytic conductivities by boundary measurements. Comm. Pure. Appl. Math. 42 (1989), 1097–1112.Google Scholar
[KKP]Karpukhin, M., Kokarev, G. and Polterovich, I. Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces. To appear in Ann. Inst. Fourier. arXiv:1209:4869.Google Scholar
[Pa]Parzanchevski, O. On G–sets and isospectrality. arXiv: 1104.0315.Google Scholar
[PS]Polterovich, I. and Sher, D.A. Heat invariants of the Steklov problem. arXiv:1304.7233. To appear in J. Geom. Anal. (published online in September 2013).Google Scholar
[Ro]Rozenblyum, G.V.On the asymptotics of the eigenvalues of certain two-dimensional spectral problems. Sel. Math. Sov. 5, (1986), 233–244.Google Scholar
[SH]Shubin, M. A.Pseudodifferential Operators and Spectral Theory. (Springer–Verlag, Berlin 1987).Google Scholar
[Tay]Taylor, M.Partial Differential Equations II. Qualitative Studies of Linear Equations. Applied Mathematical Sciences 116. (Springer-Verlag, New York, 1996).CrossRefGoogle Scholar
[Tr]Treves, F.Introduction to Pseudodifferential and Fourier Integral Operators, Vol. 1. (Plenum Press, 1982).Google Scholar
[We]Weinstock, R.Inequalities for a classical eigenvalue problem. J. Rat. Mech. Anal. 3 (1954), 343–356.Google Scholar