The Dirichlet problem for p-minimizers on finely open sets in metric spaces

Abstract

We initiate the study of fine $p$-(super)minimizers, associated with $p$-harmonic functions, on finely open sets in metric spaces, where $1<p<\infty$. After having developed their basic theory, we obtain the $p$-fine continuity of the solution of the Dirichlet problem on a finely open set with continuous Sobolev boundary values, as a by-product of similar pointwise results. These results are new also on unweighted ${\mathbf{R}}^n$. We build this theory in a complete metric space equipped with a doubling measure supporting a $p$-Poincaré inequality.

arXiv comment22 pages

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• The Cartan, Choquet and Kellogg properties for the fine topology on metric spaces
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Cited by in corpus (3)

• Condenser capacities and capacitary potentials for unbounded sets, and global $p$-harmonic Green functions on metric spaces
• Convergence and local-to-global results for $p$-superminimizers on quasiopen sets
• The Perron method associated with finely $p$-harmonic functions on finely open sets