Upper bounds for Steklov eigenvalues on surfaces

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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

Abstract
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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.
[2]	Catherine Bandle, "Isoperimetric Inequalities and Applications," Monographs and Studies in Mathematics, 7, Pitman, Boston, Mass., 1980.
[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-#.
[4]	Robert Brooks and Eran Makover, Riemann surfaces with large first eigenvalue, J. Anal. Math., 83 (2001), 243-258. doi: 10.1007/BF02790263.
[5]	Peter Buser, On the bipartition of graphs, Discrete Appl. Math., 9 (1984), 105-109.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	Joseph Hersch, Lawrence E. Payne and Menahem M. Schiffer, Some inequalities for Stekloff eigenvalues, Arch. Rational Mech. Anal., 57 (1975), 99-114.
[17]	Gerasim Kokarev, Variational aspects of Laplace eigenvalues on Riemannian surfaces, preprint, arXiv:1103.2448, 2011.
[18]	Nicholas Korevaar, Upper bounds for eigenvalues of conformal metrics, J. Differential Geom., 37 (1993), 73-93.
[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.
[20]	Michael E. Taylor, "Partial Differential Equations. II," Applied Mathematical Sciences, 116, Springer-Verlag, New York, 1996.
[21]	Robert Weinstock, Inequalities for a classical eigenvalue problem, J. Rational Mech. Anal., 3 (1954), 745-753.
[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.
[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.

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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.
[2]	Catherine Bandle, "Isoperimetric Inequalities and Applications," Monographs and Studies in Mathematics, 7, Pitman, Boston, Mass., 1980.
[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-#.
[4]	Robert Brooks and Eran Makover, Riemann surfaces with large first eigenvalue, J. Anal. Math., 83 (2001), 243-258. doi: 10.1007/BF02790263.
[5]	Peter Buser, On the bipartition of graphs, Discrete Appl. Math., 9 (1984), 105-109.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	Joseph Hersch, Lawrence E. Payne and Menahem M. Schiffer, Some inequalities for Stekloff eigenvalues, Arch. Rational Mech. Anal., 57 (1975), 99-114.
[17]	Gerasim Kokarev, Variational aspects of Laplace eigenvalues on Riemannian surfaces, preprint, arXiv:1103.2448, 2011.
[18]	Nicholas Korevaar, Upper bounds for eigenvalues of conformal metrics, J. Differential Geom., 37 (1993), 73-93.
[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.
[20]	Michael E. Taylor, "Partial Differential Equations. II," Applied Mathematical Sciences, 116, Springer-Verlag, New York, 1996.
[21]	Robert Weinstock, Inequalities for a classical eigenvalue problem, J. Rational Mech. Anal., 3 (1954), 745-753.
[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.
[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.

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