#### 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
##### Abstract

We study the limit of first eigenfunctions of (discrete) $p$-Laplacian on a finite subset of a graph with Dirichlet boundary condition, as $p\to 1$. 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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##### Keywords
$p$-Laplacian, graph, isoperimetric inequalities
Primary: 35R02
Secondary: 35P30
##### Milestones
Received: 21 December 2018
Revised: 12 December 2019
Accepted: 29 June 2020
Published: 4 August 2021
##### Authors
 Huabin Ge School of Mathematics Renmin University of China Beijing 100872 China Bobo Hua School of Mathematical Sciences, LMNS Fudan University Shanghai 200433 China Wenfeng Jiang School of Mathematics (Zhuhai) Sun Yat-Sen University Zhuhai 519082 China