Revista Matemática Iberoamericana


Full-Text PDF (513 KB) | Metadata | Table of Contents | RMI summary
Volume 34, Issue 3, 2018, pp. 1323–1360
DOI: 10.4171/RMI/1025

Published online: 2018-08-27

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

Anders Björn[1], Jana Björn[2] and Tomas Sjödin[3]

(1) Linköping University, Sweden
(2) Linköping University, Sweden
(3) Linköping University, Sweden

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.

Keywords: Dirichlet problem, harmonizable, invariance, metric space, nonlinear potential theory, Perron solution, $p$-harmonic function, $Q$-compactification, quasi-continuous, resolutive, Wiener solution

Björn Anders, Björn Jana, Sjödin Tomas: The Dirichlet problem for $p$-harmonic functions with respect to arbitrary compactifications. Rev. Mat. Iberoamericana 34 (2018), 1323-1360. doi: 10.4171/RMI/1025