The Dirichlet problem for p-harmonic functions with respect to arbitrary compactifications

TL;DR

This paper investigates the Dirichlet problem for p-harmonic functions on metric spaces using arbitrary compactifications, introducing Sobolev-Perron solutions and establishing new resolutivity and invariance results.

Contribution

It introduces Sobolev-Perron solutions for the Dirichlet problem, advancing the understanding of resolutivity and invariance in nonlinear potential theory on metric spaces.

Findings

Most previously resolutive functions are Sobolev-resolutive.

Established invariance results for Sobolev-Perron solutions.

Connected (Sobolev)-Wiener solutions to Perron solutions.

Abstract

We study the Dirichlet problem for p-harmonic functions on metric spaces with respect to arbitrary compactifications. A particular focus is on the Perron method, and as a new approach to the invariance problem we introduce Sobolev-Perron solutions. We obtain various resolutivity and invariance results, and also show that most functions that have earlier been proved to be resolutive are in fact Sobolev-resolutive. We also introduce (Sobolev)-Wiener solutions and harmonizability in this nonlinear context, and study their connections to (Sobolev)-Perron solutions, partly using Q-compactifications.