The spectral gap of graphs and Steklov eigenvalues on surfaces

Bruno Colbois; Alexandre Girouard

doi:10.3934/era.2014.21.19

Article Contents

2014, Volume 21: 19-27. Doi: 10.3934/era.2014.21.19

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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
2.
Département de mathématiques et de statistique, Université Laval, Pavillon Alexandre- Vachon, 1045, av. de la Médecine, Quebec Qc G1V 0A6

Received: September 30, 2013

Revised: December 31, 2013

Published: December 2013

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Abstract

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.

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Mathematics Subject Classification: Primary 58J50, Secondary 35P15.

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References

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