American Institute of Mathematical Sciences

Advanced
  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), 100-134. doi: 10.1007/BF02567028.  Google Scholar

[2]

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

[3]

Friedemann Brock, An isoperimetric inequality for eigenvalues of the Stekloff problem, Z. Angew. Math. Mech., 81 (2001), 69-71. 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), 243-258. doi: 10.1007/BF02790263.  Google Scholar

[5]

Peter Buser, On the bipartition of graphs, Discrete Appl. Math., 9 (1984), 105-109.  Google Scholar

[6]

Alberto P. Calderón, On an inverse boundary value problem, in "Seminar on Numerical Analysis and its Applications to Continuum Physics" (Rio de Janeiro), Soc. Brasil. Mat., Rio de Janeiro, 1980.  Google Scholar

[7]

Bruno Colbois, Ahmad El Soufi and Alexandre Girouard, Isoperimetric control of the Steklov spectrum, J. Funct. Anal., 261 (2011), 1384-1399. 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 "Séminaire de Théorie Spectrale et Géométrie, Année 1983-1984," VII.1-VII.15, Univ. Grenoble I, Saint-Martin-d'Héres, 1984.  Google Scholar

[9]

José F. Escobar, An isoperimetric inequality and the first Steklov eigenvalue, J. Funct. Anal., 165 (1999), 101-116. 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), 4011-4030. 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), 945-964. 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), 33-47.  Google Scholar

[13]

Alexander Grigor'yan, Yuri Netrusov and Shing-Tung Yau, Eigenvalues of elliptic operators and geometric applications, in "Surveys in Differential Geometry," Vol. IX, Surv. Differ. Geom., Int. Press, Somerville, MA, 2004.  Google Scholar

[14]

Asma Hassannezhad, Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem, Journal of Functional Analysis, 261 (2011), 3419-3436. 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), 581-592.  Google Scholar

[16]

Joseph Hersch, Lawrence E. Payne and Menahem M. Schiffer, Some inequalities for Stekloff eigenvalues, Arch. Rational Mech. Anal., 57 (1975), 99-114.  Google Scholar

[17]

Gerasim Kokarev, Variational aspects of Laplace eigenvalues on Riemannian surfaces, preprint, arXiv:1103.2448, 2011. Google Scholar

[18]

Nicholas Korevaar, Upper bounds for eigenvalues of conformal metrics, J. Differential Geom., 37 (1993), 73-93.  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), 207-221.  Google Scholar

[20]

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

[21]

Robert Weinstock, Inequalities for a classical eigenvalue problem, J. Rational Mech. Anal., 3 (1954), 745-753.  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), 385-391. 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), 55-63.  Google Scholar

show all references

References:
[1]

Lars L. Ahlfors, Open Riemann surfaces and extremal problems on compact subregions, Comment. Math. Helv., 24 (1950), 100-134. doi: 10.1007/BF02567028.  Google Scholar

[2]

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

[3]

Friedemann Brock, An isoperimetric inequality for eigenvalues of the Stekloff problem, Z. Angew. Math. Mech., 81 (2001), 69-71. 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), 243-258. doi: 10.1007/BF02790263.  Google Scholar

[5]

Peter Buser, On the bipartition of graphs, Discrete Appl. Math., 9 (1984), 105-109.  Google Scholar

[6]

Alberto P. Calderón, On an inverse boundary value problem, in "Seminar on Numerical Analysis and its Applications to Continuum Physics" (Rio de Janeiro), Soc. Brasil. Mat., Rio de Janeiro, 1980.  Google Scholar

[7]

Bruno Colbois, Ahmad El Soufi and Alexandre Girouard, Isoperimetric control of the Steklov spectrum, J. Funct. Anal., 261 (2011), 1384-1399. 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 "Séminaire de Théorie Spectrale et Géométrie, Année 1983-1984," VII.1-VII.15, Univ. Grenoble I, Saint-Martin-d'Héres, 1984.  Google Scholar

[9]

José F. Escobar, An isoperimetric inequality and the first Steklov eigenvalue, J. Funct. Anal., 165 (1999), 101-116. 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), 4011-4030. 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), 945-964. 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), 33-47.  Google Scholar

[13]

Alexander Grigor'yan, Yuri Netrusov and Shing-Tung Yau, Eigenvalues of elliptic operators and geometric applications, in "Surveys in Differential Geometry," Vol. IX, Surv. Differ. Geom., Int. Press, Somerville, MA, 2004.  Google Scholar

[14]

Asma Hassannezhad, Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem, Journal of Functional Analysis, 261 (2011), 3419-3436. 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), 581-592.  Google Scholar

[16]

Joseph Hersch, Lawrence E. Payne and Menahem M. Schiffer, Some inequalities for Stekloff eigenvalues, Arch. Rational Mech. Anal., 57 (1975), 99-114.  Google Scholar

[17]

Gerasim Kokarev, Variational aspects of Laplace eigenvalues on Riemannian surfaces, preprint, arXiv:1103.2448, 2011. Google Scholar

[18]

Nicholas Korevaar, Upper bounds for eigenvalues of conformal metrics, J. Differential Geom., 37 (1993), 73-93.  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), 207-221.  Google Scholar

[20]

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

[21]

Robert Weinstock, Inequalities for a classical eigenvalue problem, J. Rational Mech. Anal., 3 (1954), 745-753.  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), 385-391. 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), 55-63.  Google Scholar

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