American Institute of Mathematical Sciences

Advanced
January  2014, 21: 19-27. doi: 10.3934/era.2014.21.19

The spectral gap of graphs and Steklov eigenvalues on surfaces

Bruno Colbois 1, and  Alexandre Girouard 2,

1. 

Université de Neuchâtel, Institut de Mathématiques, Rue Emile-Argand 11, Case postale 158, 2009 Neuchâtel, Switzerland
2. 

Département de mathématiques et de statistique, Université Laval, Pavillon Alexandre- Vachon, 1045, av. de la Médecine, Quebec Qc G1V 0A6, Canada

Received  October 2013 Revised  January 2014 Published  February 2014

Using expander graphs, we construct a sequence $\{\Omega_N\}_{N\in\mathbb{N}}$ of smooth compact surfaces with boundary of perimeter $N$, and with the first non-zero Steklov eigenvalue $\sigma_1(\Omega_N)$ uniformly bounded away from zero. This answers a question which was raised in [10]. The sequence $\sigma_1(\Omega_N) L(\partial\Omega_n)$ grows linearly with the genus of $\Omega_N$, which is the optimal growth rate.
Keywords: eigenvalue inequalities, expander graphs., Riemannian surface, Steklov problem.
Mathematics Subject Classification: Primary 58J50, Secondary 35P1.
Citation: Bruno Colbois, Alexandre Girouard. The spectral gap of graphs and Steklov eigenvalues on surfaces. Electronic Research Announcements, 2014, 21: 19-27. doi: 10.3934/era.2014.21.19
References:
[1]

R. Bañuelos, T. Kulczycki, I. Polterovich and B. Siudeja, Eigenvalue inequalities for mixed Steklov problems,, in \emph{Operator Theory and its Applications}, (2010), 19. Google Scholar

[2]

R. Brooks, The first eigenvalue in a tower of coverings,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 13 (1985), 137. doi: 10.1090/S0273-0979-1985-15397-2. Google Scholar

[3]

R. Brooks, The spectral geometry of a tower of coverings,, \emph{J. Differential Geom.}, 23 (1986), 97. Google Scholar

[4]

M. Burger, Estimation de petites valeurs propres du laplacien d'un revêtement de variétés riemanniennes compactes,, \emph{C. R. Acad. Sci. Paris Sér. I Math.}, 302 (1986), 191. Google Scholar

[5]

P. Buser, On the bipartition of graphs,, \emph{Discrete Appl. Math.}, 9 (1984), 105. doi: 10.1016/0166-218X(84)90093-3. Google Scholar

[6]

F. R. K. Chung, Spectral Graph Theory,, CBMS Regional Conference Series in Mathematics, (1997). Google Scholar

[7]

B. Colbois, A. El Soufi and A. Girouard, Isoperimetric control of the Steklov spectrum,, \emph{J. Funct. Anal.}, 261 (2011), 1384. doi: 10.1016/j.jfa.2011.05.006. Google Scholar

[8]

B. Colbois and A.-M. Matei, On the optimality of J. Cheeger and P. Buser inequalities,, \emph{Differential Geom. Appl.}, 19 (2003), 281. doi: 10.1016/S0926-2245(03)00035-4. Google Scholar

[9]

A. Fraser and R. Schoen, Sharp eigenvalue bounds and minimal surfaces in the ball,, \arXiv{1209.3789}, (2013). Google Scholar

[10]

A. Girouard and I. Polterovich, Upper bounds for Steklov eigenvalues on surfaces,, \emph{Electron. Res. Announc. Math. Sci.}, 19 (2012), 77. doi: 10.3934/era.2012.19.77. Google Scholar

[11]

S. Hoory, N. Linial and A. Wigderson, Expander graphs and their applications,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 43 (2006), 439. doi: 10.1090/S0273-0979-06-01126-8. Google Scholar

[12]

M. Karpukhin, Large Steklov and Laplace eigenvalues,, in preparation., (). Google Scholar

[13]

G. Kokarev, Variational aspects of Laplace eigenvalues on Riemannian surfaces,, \arXiv{1103.2448}, (2011). Google Scholar

[14]

N. N. Moiseev, Introduction to the theory of oscillations of liquid-containing bodies,, in \emph{Advances in Applied Mechanics, (1964), 233. Google Scholar

[15]

M. S. Pinsker, On the complexity of a concentrator,, in \emph{7th International Teletraffic Conference}, (1973), 1. Google Scholar

[16]

P. C. Yang and S. T. Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)}, 7 (1980), 55. Google Scholar

show all references

References:
[1]

R. Bañuelos, T. Kulczycki, I. Polterovich and B. Siudeja, Eigenvalue inequalities for mixed Steklov problems,, in \emph{Operator Theory and its Applications}, (2010), 19. Google Scholar

[2]

R. Brooks, The first eigenvalue in a tower of coverings,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 13 (1985), 137. doi: 10.1090/S0273-0979-1985-15397-2. Google Scholar

[3]

R. Brooks, The spectral geometry of a tower of coverings,, \emph{J. Differential Geom.}, 23 (1986), 97. Google Scholar

[4]

M. Burger, Estimation de petites valeurs propres du laplacien d'un revêtement de variétés riemanniennes compactes,, \emph{C. R. Acad. Sci. Paris Sér. I Math.}, 302 (1986), 191. Google Scholar

[5]

P. Buser, On the bipartition of graphs,, \emph{Discrete Appl. Math.}, 9 (1984), 105. doi: 10.1016/0166-218X(84)90093-3. Google Scholar

[6]

F. R. K. Chung, Spectral Graph Theory,, CBMS Regional Conference Series in Mathematics, (1997). Google Scholar

[7]

B. Colbois, A. El Soufi and A. Girouard, Isoperimetric control of the Steklov spectrum,, \emph{J. Funct. Anal.}, 261 (2011), 1384. doi: 10.1016/j.jfa.2011.05.006. Google Scholar

[8]

B. Colbois and A.-M. Matei, On the optimality of J. Cheeger and P. Buser inequalities,, \emph{Differential Geom. Appl.}, 19 (2003), 281. doi: 10.1016/S0926-2245(03)00035-4. Google Scholar

[9]

A. Fraser and R. Schoen, Sharp eigenvalue bounds and minimal surfaces in the ball,, \arXiv{1209.3789}, (2013). Google Scholar

[10]

A. Girouard and I. Polterovich, Upper bounds for Steklov eigenvalues on surfaces,, \emph{Electron. Res. Announc. Math. Sci.}, 19 (2012), 77. doi: 10.3934/era.2012.19.77. Google Scholar

[11]

S. Hoory, N. Linial and A. Wigderson, Expander graphs and their applications,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 43 (2006), 439. doi: 10.1090/S0273-0979-06-01126-8. Google Scholar

[12]

M. Karpukhin, Large Steklov and Laplace eigenvalues,, in preparation., (). Google Scholar

[13]

G. Kokarev, Variational aspects of Laplace eigenvalues on Riemannian surfaces,, \arXiv{1103.2448}, (2011). Google Scholar

[14]

N. N. Moiseev, Introduction to the theory of oscillations of liquid-containing bodies,, in \emph{Advances in Applied Mechanics, (1964), 233. Google Scholar

[15]

M. S. Pinsker, On the complexity of a concentrator,, in \emph{7th International Teletraffic Conference}, (1973), 1. Google Scholar

[16]

P. C. Yang and S. T. Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)}, 7 (1980), 55. Google Scholar

[1]

Monika Laskawy. Optimality conditions of the first eigenvalue of a fourth order Steklov problem. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1843-1859. doi: 10.3934/cpaa.2017089
[2]

Christine A. Kelley, Deepak Sridhara. Eigenvalue bounds on the pseudocodeword weight of expander codes. Advances in Mathematics of Communications, 2007, 1 (3) : 287-306. doi: 10.3934/amc.2007.1.287
[3]

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

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

[5]

Joachim von Below, José A. Lubary. Isospectral infinite graphs and networks and infinite eigenvalue multiplicities. Networks & Heterogeneous Media, 2009, 4 (3) : 453-468. doi: 10.3934/nhm.2009.4.453
[6]

Vincenzo Ferone, Carlo Nitsch, Cristina Trombetti. On a conjectured reverse Faber-Krahn inequality for a Steklov--type Laplacian eigenvalue. Communications on Pure & Applied Analysis, 2015, 14 (1) : 63-82. doi: 10.3934/cpaa.2015.14.63
[7]

Erwann Delay, Pieralberto Sicbaldi. Extremal domains for the first eigenvalue in a general compact Riemannian manifold. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5799-5825. doi: 10.3934/dcds.2015.35.5799
[8]

Yuhua Sun. On the uniqueness of nonnegative solutions of differential inequalities with gradient terms on Riemannian manifolds. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1743-1757. doi: 10.3934/cpaa.2015.14.1743
[9]

Wolfgang Arendt, Rafe Mazzeo. Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2201-2212. doi: 10.3934/cpaa.2012.11.2201
[10]

Jun He, Guangjun Xu, Yanmin Liu. Some inequalities for the minimum M-eigenvalue of elasticity M-tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-11. doi: 10.3934/jimo.2019092
[11]

Mihai Mihăilescu. An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue. Communications on Pure & Applied Analysis, 2011, 10 (2) : 701-708. doi: 10.3934/cpaa.2011.10.701
[12]

Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure & Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012
[13]

Giacomo Bocerani, Dimitri Mugnai. A fractional eigenvalue problem in $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 619-629. doi: 10.3934/dcdss.2016016
[14]

David Colton, Yuk-J. Leung. On a transmission eigenvalue problem for a spherically stratified coated dielectric. Inverse Problems & Imaging, 2016, 10 (2) : 369-378. doi: 10.3934/ipi.2016004
[15]

Huan Gao, Zhibao Li, Haibin Zhang. A fast continuous method for the extreme eigenvalue problem. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1587-1599. doi: 10.3934/jimo.2017008
[16]

Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075
[17]

Isabeau Birindelli, Stefania Patrizi. A Neumann eigenvalue problem for fully nonlinear operators. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 845-863. doi: 10.3934/dcds.2010.28.845
[18]

Jean-Michel Rakotoson. Generalized eigenvalue problem for totally discontinuous operators. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 343-373. doi: 10.3934/dcds.2010.28.343
[19]

Fioralba Cakoni, Houssem Haddar, Isaac Harris. Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem. Inverse Problems & Imaging, 2015, 9 (4) : 1025-1049. doi: 10.3934/ipi.2015.9.1025
[20]

Gang Wang, Yiju Wang, Yuan Zhang. Brualdi-type inequalities on the minimum eigenvalue for the Fan product of M-tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-12. doi: 10.3934/jimo.2019069

2018 Impact Factor: 0.263

Tools

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences