Boundary regularity for p-harmonic functions and solutions of the obstacle problem on metric spaces

Abstract

We study $p$-harmonic functions in complete metric spaces equipped with a doubling Borel measure supporting a weak $(1,p)$-Poincaré inequality, . We establish the barrier classification of regular boundary points from which it also follows that regularity is a local property of the boundary. We also prove boundary regularity at the fixed (given) boundary for solutions of the one-sided obstacle problem on bounded open sets. Regularity is further characterized in several other ways.

Our results apply also to Cheeger $p$-harmonic functions and in the Euclidean setting to $A$-harmonic functions, with the usual assumptions on $A$.