Upper bounds for Steklov eigenvalues of subgraphs of polynomial growth Cayley graphs

Léonard Tschanz

doi:10.1007/s10455-021-09799-w

Abstract

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

Highlights

Given a smooth compact orientable Riemannian manifold M of dimension d ≥ 2 with a smooth boundary M, the Steklov problem on M is to find all ∈ R such that there exists a non trivial function u satisfyingΔu = 0 on M u = u on M, where Δ is the Laplace-Beltrami operator acting on functions on M, and is the outward normal derivative along M
In [2] upper bounds for all the eigenvalues of domains of the Euclidean space were given. As it is already done for the Laplace operator, one can define a discrete Steklov problem, which is a problem defined on graphs with boundary and similar to the Steklov problem defined above
One can observe that the bound depends on the cardinal of the boundary in the same way as Perrin’s one in Theorem 4. This leads to a consequence that extends the one of Theorems 3 and 4: Corollary 6 Let Γ be a polynomial growth Cayley graph of order d ≥ 2 and (Ωl, Bl)∞l=1 be a sequence of subgraphs of such that |Ωl|

Summary

Introduction

Given a smooth compact orientable Riemannian manifold M of dimension d ≥ 2 with a smooth boundary M , the Steklov problem on M is to find all ∈ R such that there exists a non trivial function u satisfying. In [2] upper bounds for all the eigenvalues of domains of the Euclidean space were given As it is already done for the Laplace operator, one can define a discrete Steklov problem, which is a problem defined on graphs with boundary and similar to the Steklov problem defined above. One can observe that the bound depends on the cardinal of the boundary in the same way as Perrin’s one in Theorem 4 This leads to a consequence that extends the one of Theorems 3 and 4: Corollary 6 Let Γ be a polynomial growth Cayley graph of order d ≥ 2 and (Ωl, Bl)∞l=1 be a sequence of subgraphs of such that |Ωl|.

Cayley graphs and isoperimetric inequality

Manifold modeled on graphs

Proof of the main Theorem

Isoperimetric control of a domain associated to a subgraph

Findings

Discretization of a Riemannian manifold