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

Upper bounds for Steklov eigenvalues on surfaces

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.
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.

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.

[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.

[1]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure and Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[2]

Monika Laskawy. Optimality conditions of the first eigenvalue of a fourth order Steklov problem. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1843-1859. doi: 10.3934/cpaa.2017089

[3]

Lujuan Yu. The asymptotic behaviour of the $ p(x) $-Laplacian Steklov eigenvalue problem. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2621-2637. doi: 10.3934/dcdsb.2020025

[4]

Eugenia Pérez. On periodic Steklov type eigenvalue problems on half-bands and the spectral homogenization problem. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 859-883. doi: 10.3934/dcdsb.2007.7.859

[5]

Robert Brooks and Eran Makover. The first eigenvalue of a Riemann surface. Electronic Research Announcements, 1999, 5: 76-81.

[6]

Germain Gendron. Uniqueness results in the inverse spectral Steklov problem. Inverse Problems and Imaging, 2020, 14 (4) : 631-664. doi: 10.3934/ipi.2020029

[7]

Vincenzo Ferone, Carlo Nitsch, Cristina Trombetti. On a conjectured reverse Faber-Krahn inequality for a Steklov--type Laplacian eigenvalue. Communications on Pure and Applied Analysis, 2015, 14 (1) : 63-82. doi: 10.3934/cpaa.2015.14.63

[8]

Jiantao Jiang, Jing An, Jianwei Zhou. A novel numerical method based on a high order polynomial approximation of the fourth order Steklov equation and its eigenvalue problems. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022066

[9]

Erwann Delay, Pieralberto Sicbaldi. Extremal domains for the first eigenvalue in a general compact Riemannian manifold. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5799-5825. doi: 10.3934/dcds.2015.35.5799

[10]

Yuhua Sun. On the uniqueness of nonnegative solutions of differential inequalities with gradient terms on Riemannian manifolds. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1743-1757. doi: 10.3934/cpaa.2015.14.1743

[11]

Mohamed Jleli, Bessem Samet. Nonexistence for time-fractional wave inequalities on Riemannian manifolds. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022115

[12]

Jun He, Guangjun Xu, Yanmin Liu. Some inequalities for the minimum M-eigenvalue of elasticity M-tensors. Journal of Industrial and Management Optimization, 2020, 16 (6) : 3035-3045. doi: 10.3934/jimo.2019092

[13]

Wolfgang Arendt, Rafe Mazzeo. Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2201-2212. doi: 10.3934/cpaa.2012.11.2201

[14]

Mihai Mihăilescu. An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue. Communications on Pure and Applied Analysis, 2011, 10 (2) : 701-708. doi: 10.3934/cpaa.2011.10.701

[15]

Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure and Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012

[16]

Giacomo Bocerani, Dimitri Mugnai. A fractional eigenvalue problem in $\mathbb{R}^N$. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 619-629. doi: 10.3934/dcdss.2016016

[17]

David Colton, Yuk-J. Leung. On a transmission eigenvalue problem for a spherically stratified coated dielectric. Inverse Problems and Imaging, 2016, 10 (2) : 369-378. doi: 10.3934/ipi.2016004

[18]

Huan Gao, Zhibao Li, Haibin Zhang. A fast continuous method for the extreme eigenvalue problem. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1587-1599. doi: 10.3934/jimo.2017008

[19]

Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075

[20]

Isabeau Birindelli, Stefania Patrizi. A Neumann eigenvalue problem for fully nonlinear operators. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 845-863. doi: 10.3934/dcds.2010.28.845

2020 Impact Factor: 0.929

Metrics

  • PDF downloads (140)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]