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

Wen Han (School of Mathematical Sciences, Fudan University, Shanghai, China)

Bobo Hua (School of Mathematical Sciences, LMNS, and Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China)

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.

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Received 23 June 2019

Accepted 16 September 2020

Published 6 December 2023