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

Léonard Tschanz

Abstract

We study the Steklov problem on a subgraph with boundary $(Ω,B)$ of a polynomial growth Cayley graph $Γ$. We prove that for each $k \in \mathbb{N}$, the $k^{\mbox{th}}$ eigenvalue tends to $0$ proportionally to $1/|B|^{\frac{1}{d-1}}$, where $d$ represents the growth rate of $Γ$. The method consists in associating a manifold $M$ to $Γ$ and a bounded domain $N \subset M$ to a subgraph $(Ω, B)$ of $Γ$. We find upper bounds for the Steklov spectrum of $N$ and transfer these bounds to $(Ω, B)$ by discretizing $N$ and using comparison Theorems.

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

Abstract

We study the Steklov problem on a subgraph with boundary of a polynomial growth Cayley graph . We prove that for each , the eigenvalue tends to proportionally to , where represents the growth rate of . The method consists in associating a manifold to and a bounded domain to a subgraph of . We find upper bounds for the Steklov spectrum of and transfer these bounds to by discretizing and using comparison Theorems.

Paper Structure

This paper contains 7 sections, 14 theorems, 47 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Cayley graphs and isoperimetric inequality
  3. Manifold modeled on graphs
  4. Proof of the main Theorem
  5. Domain associated to a subgraph
  6. Isoperimetric control of a domain associated to a subgraph
  7. Discretization of a Riemannian manifold

Key Result

Theorem 4

Let $\mathbb{Z}^d$ be the integer lattice of dimension $d$. Let $(\Omega, B)$ be a subgraph of $\mathbb{Z}^d$. Then we have where $\bar{C}= (64 d^3 \omega_d^{\frac{1}{d}})^{-1}, C'=\frac{1}{32d}$ and $\omega_d$ is the volume of the unit ball in $\mathbb{R}^d$.

Figures (5)

  • Figure 1: A fundamental piece associated with the lattice $\mathbb{Z}^2$.
  • Figure 2: Example of a manifold modeled on the lattice $\mathbb{Z}^2$.
  • Figure 3: Example of a fundamental piece associated with the lattice $\mathbb{Z}^2$. Removing the ball $B(z, 1)$ leads to a fifth hole like the other four ones. The picture on the right is a view from the side.
  • Figure 4: Example of a subgraph of $\mathbb{Z}^2$ and a domain associated. On the left, the big dots represent the boundary $B$ while the small ones represent the interior $\Omega$. On the right, the grey balls are removed from the domain.
  • Figure 5: Without the trick of the annulus, the boundary structure of the subgraph might not be reproduced on the associated domain : there is no boundary component near the central piece.

Theorems & Definitions (41)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 4: Han, Hua, 2019
  • Theorem 5: Perrin, 2020
  • Theorem 6
  • Corollary 7
  • Remark 8
  • Example 9
  • Proposition 10
  • ...and 31 more