\(p\)-harmonic functions on graphs and manifolds
type: Item
instance of: scholarly article
title: \(p\)-harmonic functions on graphs and manifolds
zbMATH Open document ID: 0898.31007
DOI: 10.1007/BF02677841
author: Ilkka Holopainen
author: Paolo M. Soardi
published in: Manuscripta Mathematica
publication date: 1998-10-13T00:00:00Z
full work available at URL: https://eudml.org/doc/156326
review text: Let \(M^n\) be a noncompact, connected, and oriented Riemannian \(C^\infty\) manifold of dimension \(n\geq 2\) equipped with a Riemannian metric. Assume its Ricci curvature to be uniformly bounded from below and its injectivity radius to be positive (bounded geometry). On an open set \(G\subset M^n\) a function \(u\in C(G)\cap W^{1,p}_{\text{loc}}(G)\), \(1<p<\infty\), is called \(p\)-harmonic if it is a weak solution of \(-\text{div}(| \nabla u| ^{p-2} \nabla u) =0\). Let \(\Gamma\) be a connected infinite graph (no self-loops) of uniformly bounded degree endowed with the shortest path metric. A function \(u\) is said to be \(p\)-harmonic in a vertex \(x\), \(1<p<\infty\), if \[ \Delta_p u(x) = \sum_{x\sim y} \text{sign}(u(y)-u(x))| u(y) -u(x)| ^{p-1} =0, \] where ``\(\sim\)'' denotes the neighborhood relation. A manifold or graph has the \(D_p\)-Liouville property if every \(p\)-Dirichlet finite \(p\)-harmonic function on it is constant. It is known that this property is invariant under rough isometries between either graphs of bounded degree or Riemannian manifolds of bounded geometry. The authors prove a mixed version: Let \(M^n\) be a Riemannian manifold of bounded geometry and let \(\Gamma\) be a graph of bounded degree which is roughly isometric to \(M^n\). Then \(M^n\) and \(\Gamma\) have the Liouville \(D_p\)-property simultaneously. This result is new even in the classical case (\(p=2\)) and nonlinear for \(p\neq 2\). The proof first relates \(M^n\) to its \(\kappa\)-graph and then the \(\kappa\)-graph to \(\Gamma\).
Mathematics Subject Classification ID: 31C12
Mathematics Subject Classification ID: 31C20
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J05
zbMATH DE Number: 1138676
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: Liouville property
zbMATH Keywords: \(p\)-harmonic function
zbMATH Keywords: connected infinite graph
zbMATH Keywords: path metric
reviewed by: Volker Metz
MaRDI profile type: Publication
cites work: Instability of the Liouville property for quasi-isometric graphs and manifolds of polynomial volume growth
cites work: Heat Kernel Lower Bounds on Riemannian Manifolds Using the Old Ideas of Nash
cites work: Noyau de la chaleur et discrétisation d'une variété riemannienne. (Heat kernel and discretization of a Riemannian manifold)
cites work: Isometric Riemannian manifolds at infinity
cites work: Difference Equations, Isoperimetric Inequality and Transience of Certain Random Walks
cites work: Q3789868
cites work: Q4693180
cites work: Q3996137
cites work: Positive Solutions of Quasilinear Elliptic Equations on Riemannian Manifolds
cites work: Rough isometries and \(p\)-harmonic functions with finite Dirichlet integral
cites work: Solutions of elliptic equations on manifolds with roughly Euclidean ends
cites work: Rough isometries, and combinatorial approximations of geometries of non- compact Riemannian manifolds
cites work: Rough isometries and the parabolicity of Riemannian manifolds
cites work: The Wiener test and potential estimates for quasilinear elliptic equations
cites work: Instability of Liouville property for quasi-isometric Riemannian manifolds and reversible Markov chains
cites work: Function theory, random paths and covering spaces
cites work: Transient Random Walks on Graphs and Metric Spaces with Applications to Hyperbolic Surfaces
cites work: Q4033071
cites work: Rough Isometries and Dirichlet Finite Harmonic Functions on Graphs
cites work: Potential theory on infinite networks
cites work: Parabolic index and rough isometries
cites work: Classification of infinite networks and its application
cites work: Brownian motion and random walks on manifolds
cites work: Parabolic and hyperbolic infinite networks
cites work: Q3205541
description: scientific article; zbMATH DE number 1138676