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 $1 < p < \infty$\mathrm{.}Afterhavingdevelopedtheirbasictheory,weobtainthe$p$-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 $p$-Poincar'e inequality.