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

Anders Björn, Jana Björn, Visa Latvala

Published 2022-09-02Version 1

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 $\mathbf{R}^n$.