Obstacle and Dirichlet problems on arbitrary nonopen sets, and fine topology

TL;DR

This paper investigates obstacle and Dirichlet problems for p-harmonic functions on arbitrary nonopen sets within metric spaces, establishing new criteria for solvability and exploring connections with fine potential theory.

Contribution

It introduces Adams' criterion for obstacle problem solvability on nonopen sets and links these problems to fine potential theory in general metric spaces.

Findings

Adams' criterion for single obstacle problem solvability

Connections established between obstacle problems and fine potential theory

Most results are new for nonopen sets and in Euclidean spaces

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 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. Most of the results are new for open E (apart from those which are trivial in this case) and also on R^n.