We study the Steklov problem on a subgraph with boundary $$(\Omega ,B)$$ ( Ω , B ) of a polynomial growth Cayley graph $$\Gamma$$ Γ . For $$(\Omega _l, B_l)_{l=1}^\infty$$… Click to show full abstract
We study the Steklov problem on a subgraph with boundary $$(\Omega ,B)$$(Ω,B) of a polynomial growth Cayley graph $$\Gamma$$Γ. For $$(\Omega _l, B_l)_{l=1}^\infty$$(Ωl,Bl)l=1∞ a sequence of subgraphs of $$\Gamma$$Γ such that $$|\Omega _l| \longrightarrow \infty$$|Ωl|⟶∞, we prove that for each $$k \in {\mathbb {N}}$$k∈N, the kth eigenvalue tends to 0 proportionally to $$1/|B|^{\frac{1}{d-1}}$$1/|B|1d-1, where d represents the growth rate of $$\Gamma$$Γ. The method consists in associating a manifold M to $$\Gamma$$Γ and a bounded domain $$N \subset M$$N⊂M to a subgraph $$(\Omega , B)$$(Ω,B) of $$\Gamma$$Γ. We find upper bounds for the Steklov spectrum of N and transfer these bounds to $$(\Omega , B)$$(Ω,B) by discretizing N and using comparison theorems.
