The Perron method for \(p\)-harmonic functions in unbounded sets in \(\mathbf {R}^n\) and metric spaces
type: Item
instance of: scholarly article
title: The Perron method for \(p\)-harmonic functions in unbounded sets in \(\mathbf {R}^n\) and metric spaces
zbMATH Open document ID: 1397.31004
author: Daniel Hansevi
published in: Mathematische Zeitschrift
publication date: 2018-03-21T00:00:00Z
review text: The authors extend the classical notions of \(p\)-subharmonic and \(p\)-superharmonic function, that is a function satisfying, respectively, the inequalities \(-\Delta_p u\leq 0\) and \(-\Delta_p u\geq 0\) in the Euclidean space \(\mathbb{R}^n\), to the setting of a metric space \((X,\mathcal{M},\mu,d)\), equipped with a metric \(d\) and with a complete measure \(\mu\) defined on the completion \(\mathcal{M}\) of the Borel \(\sigma\)-algebra. To this end, the authors introduce the notion of \(p\)-weak upper subgradient and the related notions of Newtonian and Dirichlet spaces on \(X\). Then, given an open set \(\Omega\) in \(X\) which satisfies a \(p\)-parabolic condition if unbounded, the authors apply the Perron method in this setting to find \(p\)-harmonic functions on \(\Omega\) satisfying, in the sense specified below, an assigned boundary condition on \(\partial \Omega\). Going into details, given a real-extended function \(f\) defined on \(\partial \Omega\), the authors consider the set \(\mathcal{U}_f(\Omega)\) of all bounded below superharmonic functions \(u\) in \(\Omega\) such that \(\liminf_{y\rightarrow x}u(y)\geq f(x)\), for all \(x\in \partial \Omega\), and the set \(\mathcal{L}_f(\Omega)\) of all bounded above subharmonic functions \(u\) in \(\Omega\) such that \(\limsup_{y\rightarrow x}u(y)\leq f(x)\), for all \(x\in \partial \Omega\). Then the authors find sufficient conditions on \(f\) in order to \[ P_\Omega(x):=\inf_{u\in \mathcal{U}_f(\Omega)}u(x)=\sup_{u\in \mathcal{L}_f(\Omega)}u(x)\in \mathbb{R}, \] for all \(x\in \Omega\). When the above equality holds, the function \(f\) is said to be resolutive with respect to \(\Omega\), and \(P_\Omega\) is called the Perron solution of \(f\). After introducing a new suitable concept of \(p\)-capacity for the subsets of \(\overline{\Omega}\), the authors prove that a function \(f:\overline{\Omega}\rightarrow \overline{\mathbb{R}}\) is resolutive with respect to \(\Omega\) if the following conditions hold…
Mathematics Subject Classification ID: 31E05
Mathematics Subject Classification ID: 31C45
zbMATH DE Number: 6851617
zbMATH Keywords: metric measure space
zbMATH Keywords: \(p\)-harmonic functions
zbMATH Keywords: \( p\)-subharmonic functions
zbMATH Keywords: \(p\)-superharmonic functios
zbMATH Keywords: Dirichlet problem
reviewed by: Giovanni Anello
MaRDI profile type: Publication
full work available at URL: s00209-017-1877-0
OpenAlex ID: W2964112729
Wikidata QID: Q59611747
cites work: A weak Kellogg property for quasiminimizers
cites work: The Dirichlet problem for \(p\)-harmonic functions on the topologist's comb
cites work: Nonlinear potential theory on metric spaces
cites work: Obstacle and Dirichlet problems on arbitrary nonopen sets in metric spaces, and fine topology
cites work: Lebesgue points and the fundamental convergence theorem for superharmonic functions on metric spaces
cites work: The Dirichlet problem for p-harmonic functions on metric spaces
cites work: The Perron method for \(p\)-harmonic functions in metric spaces.
cites work: Q5302131
cites work: The Dirichlet problem for \(p\)-harmonic functions with respect to the Mazurkiewicz boundary, and new capacities
cites work: Geometry of prime end boundary and the Dirichlet problem for bounded domains in metric measure spaces
cites work: Continuous dependence on obstacles for the double obstacle problem on metric spaces
cites work: Note on the PWB-method in the nonlinear case
cites work: Sobolev met Poincaré
cites work: The obstacle and Dirichlet problems associated with p-harmonic functions in unbounded sets in R^n and metric spaces
cites work: Lectures on analysis on metric spaces
cites work: Q5294569
cites work: From local to global in quasiconformal structures.
cites work: Quasiconformal maps in metric spaces with controlled geometry
cites work: Sobolev Spaces on Metric Measure Spaces
cites work: Singular functions on metric measure spaces.
cites work: Q4207003
cites work: Q4896353
cites work: Nonlinear potential theory of metric spaces
cites work: Polar sets on metric spaces
cites work: Q4238869
cites work: Q4354419
cites work: Q4340096
cites work: Newtonian spaces: An extension of Sobolev spaces to metric measure spaces
cites work: Harmonic functions on metric spaces
cites work: Some Convergence Results for p-Harmonic Functions on Metric Measure Spaces
DOI: 10.1007/S00209-017-1877-0
description: scientific article; zbMATH DE number 6851617