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- Title
Isoperimetric Upper Bound for the First Eigenvalue of Discrete Steklov Problems.
- Authors
Perrin, Hélène
- Abstract
We study upper bounds for the first non-zero eigenvalue of the Steklov problem defined on finite graphs with boundary. For finite graphs with boundary included in a Cayley graph associated to a group of polynomial growth, we give an upper bound for the first non-zero Steklov eigenvalue depending on the number of vertices of the graph and of its boundary. As a corollary, if the graph with boundary also satisfies a discrete isoperimetric inequality, we show that the first non-zero Steklov eigenvalue tends to zero as the number of vertices of the graph tends to infinity. This extends recent results of Han and Hua, who obtained a similar result in the case of Z n . We obtain the result using metric properties of Cayley graphs associated to groups of polynomial growth.
- Publication
Journal of Geometric Analysis, 2021, Vol 31, Issue 8, p8144
- ISSN
1050-6926
- Publication type
Article
- DOI
10.1007/s12220-020-00572-2