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 -harmonic functions, on finely open sets in metric spaces, where $1 < \infty $ 1 < p ∞ . After having developed their basic theory, we obtain -fine continuity solution Dirichlet problem a set continuous Sobolev boundary values, as by-product similar pointwise results. These results are new also unweighted R n build this theory complete space equipped doubling measure supporting -Poincaré inequality.