The Dirichlet Problem for p -minimizers on Finely Open Sets in Metric Spaces

We initiate the study of fine p -(super)minimizers, associated with p -harmonic functions, on finely open sets in metric spaces, where 1 < p < ∞ . 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 R n . We build this theory in a complete metric space equipped with a doubling measure supporting a p -Poincaré inequality.