Obstacle and Dirichlet problems on arbitrary nonopen sets in metric spaces, and fine topology

Abstract

We study the double obstacle problem for p-harmonic functions on arbitrary bounded nonopen sets E in quite general metric spaces. The Dirichlet and single obstacle problems are included as special cases. We obtain the Adams criterion for the solubility of the single obstacle problem and establish connections with fine potential theory. We also study when the minimal $p$-weak upper gradient of a function remains minimal when restricted to a nonopen subset. Many of the results are new even for open $E$ (apart from those which are trivial in this case) and also on ${\mathbb{R}}^n$.