The Steklov problem on triangle-tiling graphs in the hyperbolic plane

Abstract

We introduce a graph Γ which is roughly isometric to the hyperbolic plane and we study the Steklov eigenvalues of a subgraph with boundary (Ω, B) of Γ. For (Ω l , B l ) l≥1 a sequence of subraph of Γ such that |Ω l | −→ ∞, we prove that for each k ∈ N, the k th eigenvalue tends to 0 proportionally to 1/|B l |. The idea of this proof consists in finding a bounded domain (N, Σ) of the hyperbolic plane which is roughly isometric to (Ω, B), giving an upper bound for the Steklov eigenvalues of (N, Σ) and transferring this bound to (Ω, B) via a process called discretization.