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

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.