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# Communications in Analysis and Geometry

## Volume 31 (2023)

### Number 2

### Steklov eigenvalue problem on subgraphs of integer lattices

Pages: 343 – 366

DOI: https://dx.doi.org/10.4310/CAG.2023.v31.n2.a4

#### Authors

#### Abstract

We study the eigenvalues of the Dirichlet-to-Neumann operator on a finite subgraph of the integer lattice $\mathbb{Z}^n$. We estimate the first $n + 1$ eigenvalues using the number of vertices of the subgraph. As a corollary, we prove that the first non-trivial eigenvalue of the Dirichlet-to-Neumann operator tends to zero as the number of vertices of the subgraph tends to infinity.

Received 23 June 2019

Accepted 16 September 2020

Published 6 December 2023