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.

References in corpus (2)

• Prime ends for domains in metric spaces
• The Dirichlet problem for p-harmonic functions with respect to the Mazurkiewicz boundary

Cited by in corpus (4)

• Sphericalization and p-harmonic functions on unbounded domains in Ahlfors regular metric spaces
• Resolutivity and invariance for the Perron method for degenerate equations of divergence type
• Boundary regularity for $p$-harmonic functions and solutions of obstacle problems on unbounded sets in metric spaces
• The Perron method associated with finely $p$-harmonic functions on finely open sets