Boundary regularity for $p$-harmonic functions and solutions of obstacle problems on unbounded sets in metric spaces

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 $p$-Poincar'e inequality, $1<p<\mathrm{\infty }$. 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.