Electronic Research Announcements in Mathematical Sciences (ERA-MS)

The spectral gap of graphs and Steklov eigenvalues on surfaces

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

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Bruno Colbois - Université de Neuchâtel, Institut de Mathématiques, Rue Emile-Argand 11, Case postale 158, 2009 Neuchâtel, Switzerland (email)
Alexandre Girouard - Département de mathématiques et de statistique, Université Laval, Pavillon Alexandre- Vachon, 1045, av. de la Médecine, Quebec Qc G1V 0A6, Canada (email)

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.

Keywords:  Steklov problem, Riemannian surface, eigenvalue inequalities, expander graphs.
Mathematics Subject Classification:  Primary 58J50, Secondary 35P15.

Received: October 2013;      Revised: January 2014;      Available Online: February 2014.