We study -harmonic functions in complete metric spaces equipped with a doubling Borel measure supporting a weak -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 -harmonic functions and in the Euclidean setting to -harmonic functions, with the usual assumptions on .
Anders BJÖRN. Jana BJÖRN. "Boundary regularity for p-harmonic functions and solutions of the obstacle problem on metric spaces." J. Math. Soc. Japan 58 (4) 1211 - 1232, October, 2006. https://doi.org/10.2969/jmsj/1179759546