THE OBSTACLE AND DIRICHLET PROBLEMS ASSOCIATED WITH p-HARMONIC FUNCTIONS IN UNBOUNDED SETS IN Rn AND METRIC SPACES

The obstacle problem associated with p-harmonic functions is extended to unbounded open sets, whose complement has positive capacity, in the setting of a proper metric measure space supporting a (p,p)-Poincare inequality, 1 < p < infinity, and the existence of a unique solution is proved. Furthermore, if the measure is doubling, then it is shown that a continuous obstacle implies that the solution is continuous, and moreover p-harmonic in the set where it does not touch the obstacle. This includes, as a special case, the solution of the Dirichlet problem for p-harmonic functions with Sobolev type boundary data.