The Perron method associated with finely $p$-harmonic functions on finely open sets

TL;DR

This paper develops a Perron method for solving the nonlinear Dirichlet problem for p-harmonic functions on finely open sets, establishing conditions for resolutivity and fine continuity in a metric space setting.

Contribution

It introduces four types of Perron solutions for the p-energy minimization problem on finely open sets and proves their equivalence and properties under natural assumptions.

Findings

Four Perron solutions are equal quasieverywhere.

Perron solutions coincide with boundary data that are Sobolev or uniformly continuous.

Perron solutions are finely continuous and p-harmonic for uniformly continuous data.

Abstract

Given a bounded finely open set $V$ and a function $f$ on the fine boundary of $V$, we introduce four types of upper Perron solutions to the nonlinear Dirichlet problem for $p$-energy minimizers, $1<p<\infty$, with $f$ as boundary data. These solutions are given as pointwise infima of suitable families of fine $p$-superminimizers in $V$. We show (under natural assumptions) that the four upper Perron solutions are equal quasieverywhere and that they are fine $p$-minimizers of the $p$-energy integral. We moreover show that the upper and lower Perron solutions coincide quasieverywhere for Sobolev and for uniformly continuous boundary data, i.e.\ that such boundary data are resolutive. For the uniformly continuous boundary data, the Perron solutions are also shown to be finely continuous and thus finely $p$-harmonic. We prove our results in a complete metric space $X$ equipped with a doubling measure supporting a $p$-Poincar\'e inequality, but they are new also in unweighted $\mathbf{R}^n$.