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

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 $p$-Poincaré inequality, but they are new also in unweighted ${\mathbf{R}}^n$.

References in corpus (12)

• Obstacle and Dirichlet problems on arbitrary nonopen sets, and fine topology
• The Dirichlet problem for p-harmonic functions with respect to the Mazurkiewicz boundary
• The weak Cartan property for the p-fine topology on metric spaces
• The Cartan, Choquet and Kellogg properties for the fine topology on metric spaces
• The variational capacity with respect to nonopen sets in metric spaces
• Quasiopen and p-path open sets, and characterizations of quasicontinuity
• Sobolev spaces, fine gradients and quasicontinuity on quasiopen sets
• The Dirichlet problem for p-harmonic functions on the topologist's comb
• The Perron method for $p$-harmonic functions in unbounded sets in ${\mathbf{R}}^n$ and metric spaces
• The Dirichlet problem for p-harmonic functions with respect to arbitrary compactifications
• The Dirichlet problem for p-minimizers on finely open sets in metric spaces
• Convergence and local-to-global results for $p$-superminimizers on quasiopen sets