4.5 Article

The Steklov Problem on Triangle-Tiling Graphs in the Hyperbolic Plane

Journal

JOURNAL OF GEOMETRIC ANALYSIS
Volume 33, Issue 5, Pages -

Publisher

SPRINGER
DOI: 10.1007/s12220-023-01208-x

Keywords

Spectral geometry; Steklov problem; Graphs with boundary; Discrete Steklov problem

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We introduce a graph gamma that is approximately isometric to the hyperbolic plane and investigate the Steklov eigenvalues of a subgraph with boundary omega of gamma. For a sequence of subgraphs (omega(l))(l >= 1) of gamma such that the size of omega(l) approaches infinity, we prove that the k(th) eigenvalue tends to 0 in proportion to 1/|B-l| for each k is an element of N. The proof idea involves finding a bounded domain N in the hyperbolic plane that is approximately isometric to omega, establishing an upper bound for the Steklov eigenvalues of N, and transferring this bound to omega through a process called discretization.
We introduce a graph gamma which is roughly isometric to the hyperbolic plane, and we study the Steklov eigenvalues of a subgraph with boundary omega of gamma. For (omega(l))(l >= 1) a sequence of subgraphs of gamma such that |omega(l)| --> infinity, we prove that for each k is an element of N, the k(th) eigenvalue tends to 0 proportionally to 1/|B-l|. The idea of the proof consists in finding a bounded domain N of the hyperbolic plane which is roughly isometric to omega, giving an upper bound for the Steklov eigenvalues of N and transferring this bound to omega via a process called discretization.

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