Isoperimetric Upper Bound for the First Eigenvalue of Discrete Steklov Problems

Hélène Perrin

doi:10.1007/s12220-020-00572-2

Abstract

We study upper bounds for the first non-zero eigenvalue of the Steklov problem defined on finite graphs with boundary. For finite graphs with boundary included in a Cayley graph associated to a group of polynomial growth, we give an upper bound for the first non-zero Steklov eigenvalue depending on the number of vertices of the graph and of its boundary. As a corollary, if the graph with boundary also satisfies a discrete isoperimetric inequality, we show that the first non-zero Steklov eigenvalue tends to zero as the number of vertices of the graph tends to infinity. This extends recent results of Han and Hua, who obtained a similar result in the case of mathbb {Z}^n. We obtain the result using metric properties of Cayley graphs associated to groups of polynomial growth.

Highlights

Let M be a compact Riemannian manifold of dimension n ≥ 2 with boundary ∂ M
An important question in studying the spectral geometry of the Steklov problem is to maximize its eigenvalues under a constraint on the volume of the boundary or on the volume of the manifold
We investigate isoperimetric upper bounds for σ1 of the Steklov problem on graphs

Summary

Introduction

Let M be a compact Riemannian manifold of dimension n ≥ 2 with boundary ∂ M. An important question in studying the spectral geometry of the Steklov problem is to maximize its eigenvalues under a constraint on the volume of the boundary or on the volume of the manifold. 2017, a survey of the literature on this question has been given in [10] It was shown in [3] that a Weinstock-type inequality holds in Rn in the class of convex sets, that is, that among all bounded convex sets in Rn with prescribed volume of the boundary, the ball maximizes σ1. We investigate isoperimetric upper bounds for σ1 of the Steklov problem on graphs. The space of real functions defined on the vertices of the boundary, denoted RB, is the Euclidean space of dimension |B|.

Perrin

Groups with Polynomial Growth and Cayley Graphs

Isoperimetric Upper Bound for 1 in Cayley Graphs with Polynomial Growth

Proof of Theorem 1

Application to Subgraphs