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

摘要

We study p-harmonic functions in complete metric spaces equipped with a doubling Borel measure supporting a weak (1,p)-Poincare inequality, 1 < p < ∞. 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 onesided 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.