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The theory of boundary regularity for  p -harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a  p -Poincaré inequality, 1 &lt;  p  &lt; ∞. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.

Boundary Regularity for p-Harmonic Functions and Solutions of Obstacle Problems on Unbounded Sets in Metric Spaces