Vol. 312, No. 1, 2021

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The limit of first eigenfunctions of the $p$-Laplacian on graphs

Huabin Ge, Bobo Hua and Wenfeng Jiang

Vol. 312 (2021), No. 1, 103–112

We study the limit of first eigenfunctions of (discrete) p-Laplacian on a finite subset of a graph with Dirichlet boundary condition, as p 1. We prove that up to a subsequence, they converge to a summation of characteristic functions of Cheeger cuts of the graph. We give an example to show that the limit may not be a characteristic function of a single Cheeger cut.

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$p$-Laplacian, graph, isoperimetric inequalities
Mathematical Subject Classification 2010
Primary: 35R02
Secondary: 35P30
Received: 21 December 2018
Revised: 12 December 2019
Accepted: 29 June 2020
Published: 4 August 2021
Huabin Ge
School of Mathematics
Renmin University of China
Beijing 100872
Bobo Hua
School of Mathematical Sciences, LMNS
Fudan University
Shanghai 200433
Wenfeng Jiang
School of Mathematics (Zhuhai)
Sun Yat-Sen University
Zhuhai 519082