American Institute of Mathematical Sciences

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  2012, 19: 77-85. doi: 10.3934/era.2012.19.77

Upper bounds for Steklov eigenvalues on surfaces

Alexandre Girouard 1, and  Iosif Polterovich 2,

1. 

Laboratoire de Mathématiques (LAMA), Université de Savoie campus scientifique, 73376 Le Bourget-du-Lac, France
2. 

Département de Mathématiques et de Statistique, Université de Montréal, C. P. 6128, Succ. Centre-ville, Montréal, Québec, H3C 3J7, Canada

Received  March 2012 Published  August 2012

We give explicit isoperimetric upper bounds for all Steklov eigenvalues of a compact orientable surface with boundary, in terms of the genus, the length of the boundary, and the number of boundary components. Our estimates generalize a recent result of Fraser-Schoen, as well as the classical inequalites obtained by Hersch-Payne-Schiffer, whose approach is used in the present paper.
Keywords: eigenvalue inequalities., Steklov problem, Riemannian surface.
Mathematics Subject Classification: Primary: 58J50; Secondary: 35P1.
Citation: Alexandre Girouard, Iosif Polterovich. Upper bounds for Steklov eigenvalues on surfaces. Electronic Research Announcements, 2012, 19: 77-85. doi: 10.3934/era.2012.19.77
References:
[1]

Lars L. Ahlfors, Open Riemann surfaces and extremal problems on compact subregions,, Comment. Math. Helv., 24 (1950), 100.  doi: 10.1007/BF02567028.  Google Scholar

[2]

Catherine Bandle, "Isoperimetric Inequalities and Applications," Monographs and Studies in Mathematics, 7,, Pitman, (1980).   Google Scholar

[3]

Friedemann Brock, An isoperimetric inequality for eigenvalues of the Stekloff problem,, Z. Angew. Math. Mech., 81 (2001), 69.  doi: 10.1002/1521-4001(200101)81:1<69::AID-ZAMM69>3.0.CO;2-#.  Google Scholar

[4]

Robert Brooks and Eran Makover, Riemann surfaces with large first eigenvalue,, J. Anal. Math., 83 (2001), 243.  doi: 10.1007/BF02790263.  Google Scholar

[5]

Peter Buser, On the bipartition of graphs,, Discrete Appl. Math., 9 (1984), 105.   Google Scholar

[6]

Alberto P. Calderón, On an inverse boundary value problem,, in, (1980).   Google Scholar

[7]

Bruno Colbois, Ahmad El Soufi and Alexandre Girouard, Isoperimetric control of the Steklov spectrum,, J. Funct. Anal., 261 (2011), 1384.  doi: 10.1016/j.jfa.2011.05.006.  Google Scholar

[8]

Ahmad El Soufi and Saïd Ilias, Le volume conforme et ses applications d'après Li et Yau,, in, (1984), 1983.   Google Scholar

[9]

José F. Escobar, An isoperimetric inequality and the first Steklov eigenvalue,, J. Funct. Anal., 165 (1999), 101.  doi: 10.1006/jfan.1999.3402.  Google Scholar

[10]

Ailana Fraser and Richard Schoen, The first Steklov eigenvalue, conformal geometry, and minimal surfaces,, Adv. Math., 226 (2011), 4011.  doi: 10.1016/j.aim.2010.11.007.  Google Scholar

[11]

Alexandre Gabard, Sur la représentation conforme des surfaces de Riemann à bord et une caractérisation des courbes séparantes,, Comment. Math. Helv., 81 (2006), 945.  doi: 10.4171/CMH/82.  Google Scholar

[12]

Alexandre Girouard and Iosif Polterovich, On the Hersch-Payne-Schiffer estimates for the eigenvalues of the Steklov problem,, Funktsional. Anal. i Prilozhen., 44 (2010), 33.   Google Scholar

[13]

Alexander Grigor'yan, Yuri Netrusov and Shing-Tung Yau, Eigenvalues of elliptic operators and geometric applications,, in, (2004).   Google Scholar

[14]

Asma Hassannezhad, Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem,, Journal of Functional Analysis, 261 (2011), 3419.  doi: 10.1016/j.jfa.2011.08.003.  Google Scholar

[15]

Antoine Henrot, Gérard A. Philippin and Abdessamad Safoui, Some isoperimetric inequalities with application to the Stekloff problem,, J. Convex Anal., 15 (2008), 581.   Google Scholar

[16]

Joseph Hersch, Lawrence E. Payne and Menahem M. Schiffer, Some inequalities for Stekloff eigenvalues,, Arch. Rational Mech. Anal., 57 (1975), 99.   Google Scholar

[17]

Gerasim Kokarev, Variational aspects of Laplace eigenvalues on Riemannian surfaces,, preprint, (2011).   Google Scholar

[18]

Nicholas Korevaar, Upper bounds for eigenvalues of conformal metrics,, J. Differential Geom., 37 (1993), 73.   Google Scholar

[19]

Matti Lassas, Michael Taylor and Gunther Uhlmann, The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary,, Comm. Anal. Geom., 11 (2003), 207.   Google Scholar

[20]

Michael E. Taylor, "Partial Differential Equations. II," Applied Mathematical Sciences, 116,, Springer-Verlag, (1996).   Google Scholar

[21]

Robert Weinstock, Inequalities for a classical eigenvalue problem,, J. Rational Mech. Anal., 3 (1954), 745.   Google Scholar

[22]

Lewis Wheeler and Cornelius O. Horgan, Isoperimetric bounds on the lowest nonzero Stekloff eigenvalue for plane strip domains,, SIAM J. Appl. Math., 31 (1976), 385.  doi: 10.1137/0131032.  Google Scholar

[23]

Paul C. Yang and Shing-Tung Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 7 (1980), 55.   Google Scholar

show all references

References:
[1]

Lars L. Ahlfors, Open Riemann surfaces and extremal problems on compact subregions,, Comment. Math. Helv., 24 (1950), 100.  doi: 10.1007/BF02567028.  Google Scholar

[2]

Catherine Bandle, "Isoperimetric Inequalities and Applications," Monographs and Studies in Mathematics, 7,, Pitman, (1980).   Google Scholar

[3]

Friedemann Brock, An isoperimetric inequality for eigenvalues of the Stekloff problem,, Z. Angew. Math. Mech., 81 (2001), 69.  doi: 10.1002/1521-4001(200101)81:1<69::AID-ZAMM69>3.0.CO;2-#.  Google Scholar

[4]

Robert Brooks and Eran Makover, Riemann surfaces with large first eigenvalue,, J. Anal. Math., 83 (2001), 243.  doi: 10.1007/BF02790263.  Google Scholar

[5]

Peter Buser, On the bipartition of graphs,, Discrete Appl. Math., 9 (1984), 105.   Google Scholar

[6]

Alberto P. Calderón, On an inverse boundary value problem,, in, (1980).   Google Scholar

[7]

Bruno Colbois, Ahmad El Soufi and Alexandre Girouard, Isoperimetric control of the Steklov spectrum,, J. Funct. Anal., 261 (2011), 1384.  doi: 10.1016/j.jfa.2011.05.006.  Google Scholar

[8]

Ahmad El Soufi and Saïd Ilias, Le volume conforme et ses applications d'après Li et Yau,, in, (1984), 1983.   Google Scholar

[9]

José F. Escobar, An isoperimetric inequality and the first Steklov eigenvalue,, J. Funct. Anal., 165 (1999), 101.  doi: 10.1006/jfan.1999.3402.  Google Scholar

[10]

Ailana Fraser and Richard Schoen, The first Steklov eigenvalue, conformal geometry, and minimal surfaces,, Adv. Math., 226 (2011), 4011.  doi: 10.1016/j.aim.2010.11.007.  Google Scholar

[11]

Alexandre Gabard, Sur la représentation conforme des surfaces de Riemann à bord et une caractérisation des courbes séparantes,, Comment. Math. Helv., 81 (2006), 945.  doi: 10.4171/CMH/82.  Google Scholar

[12]

Alexandre Girouard and Iosif Polterovich, On the Hersch-Payne-Schiffer estimates for the eigenvalues of the Steklov problem,, Funktsional. Anal. i Prilozhen., 44 (2010), 33.   Google Scholar

[13]

Alexander Grigor'yan, Yuri Netrusov and Shing-Tung Yau, Eigenvalues of elliptic operators and geometric applications,, in, (2004).   Google Scholar

[14]

Asma Hassannezhad, Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem,, Journal of Functional Analysis, 261 (2011), 3419.  doi: 10.1016/j.jfa.2011.08.003.  Google Scholar

[15]

Antoine Henrot, Gérard A. Philippin and Abdessamad Safoui, Some isoperimetric inequalities with application to the Stekloff problem,, J. Convex Anal., 15 (2008), 581.   Google Scholar

[16]

Joseph Hersch, Lawrence E. Payne and Menahem M. Schiffer, Some inequalities for Stekloff eigenvalues,, Arch. Rational Mech. Anal., 57 (1975), 99.   Google Scholar

[17]

Gerasim Kokarev, Variational aspects of Laplace eigenvalues on Riemannian surfaces,, preprint, (2011).   Google Scholar

[18]

Nicholas Korevaar, Upper bounds for eigenvalues of conformal metrics,, J. Differential Geom., 37 (1993), 73.   Google Scholar

[19]

Matti Lassas, Michael Taylor and Gunther Uhlmann, The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary,, Comm. Anal. Geom., 11 (2003), 207.   Google Scholar

[20]

Michael E. Taylor, "Partial Differential Equations. II," Applied Mathematical Sciences, 116,, Springer-Verlag, (1996).   Google Scholar

[21]

Robert Weinstock, Inequalities for a classical eigenvalue problem,, J. Rational Mech. Anal., 3 (1954), 745.   Google Scholar

[22]

Lewis Wheeler and Cornelius O. Horgan, Isoperimetric bounds on the lowest nonzero Stekloff eigenvalue for plane strip domains,, SIAM J. Appl. Math., 31 (1976), 385.  doi: 10.1137/0131032.  Google Scholar

[23]

Paul C. Yang and Shing-Tung Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 7 (1980), 55.   Google Scholar

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