The Dirichlet problem for p-harmonic functions with respect to the Mazurkiewicz boundary

TL;DR

This paper extends the Perron method to solve the Dirichlet problem for p-harmonic functions in metric spaces using the Mazurkiewicz boundary, improving resolutivity and invariance results.

Contribution

It develops a new approach for p-harmonic functions with Mazurkiewicz boundary values in metric spaces, enhancing existing boundary value problem solutions.

Findings

Established resolutivity for Sobolev and continuous functions.

Proved invariance under small set perturbations.

Improved resolutivity results for standard metric boundaries.

Abstract

In this paper we develop the Perron method for solving the Dirichlet problem for the analog of the p-Laplacian, i.e. for p-harmonic functions, with Mazurkiewicz boundary values. The setting considered here is that of metric spaces, where the boundary of the domain in question is replaced with the Mazurkiewicz boundary. Resolutivity for Sobolev and continuous functions, as well as invariance results for perturbations on small sets, are obtained. We use these results to improve the known resolutivity and invariance results for functions on the standard (metric) boundary. We also illustrate the results of this paper by discussing several examples.