We study the Steklov problem on a subgraph with boundary $$(\Omega ,B)$$ ( Ω , B ) of a polynomial growth Cayley graph $$\Gamma$$ Γ . For $$(\Omega _l, B_l)_{l=1}^\infty$$ ( Ω l , B l ) l = 1 ∞ a sequence of subgraphs of $$\Gamma$$ Γ such that $$|\Omega _l| \longrightarrow \infty$$ | Ω l | ⟶ ∞ , 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 | 1 d - 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.

We introduce a graph $$\Gamma $$ Γ which is roughly isometric to the hyperbolic plane, and we study the Steklov eigenvalues of a subgraph with boundary $$\Omega $$ Ω of $$\Gamma $$ Γ . For $$(\Omega _l)_{l\ge 1}$$ ( Ω l ) l ≥ 1 a sequence of subgraphs of $$\Gamma $$ Γ such that $$|\Omega _l| \longrightarrow \infty $$ | Ω l | ⟶ ∞ , we prove that for each $$k \in \mathbb {N}$$ k ∈ N , the $$k^{\text{ th }}$$ k th eigenvalue tends to 0 proportionally to $$1/|B_l|$$ 1 / | B l | . The idea of the proof consists in finding a bounded domain N of the hyperbolic plane which is roughly isometric to $$\Omega $$ Ω , giving an upper bound for the Steklov eigenvalues of N and transferring this bound to $$\Omega $$ Ω via a process called discretization.

We investigate the question of sharp upper bounds for the Steklov eigenvalues of a hypersurface of revolution of the Euclidean space with two boundary components isometric to two copies of S n−1 . For the case of the first non zero Steklov eigenvalue, we give a sharp upper bound B n (L) (that depends only on the dimension n ≥ 3 and the meridian length L > 0) which is reached by a degenerated metric g * , that we compute explicitly. We also give a sharp upper bound B n which depends only on n. Our method also permits us to prove some stability properties of these upper bounds.

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.