The Perron method for $p$-harmonic functions in unbounded sets in $\mathbf{R}^n$ and metric spaces

Abstract

The Perron method for solving the Dirichlet problem for $p$-harmonic functions is extended to unbounded open sets in the setting of a complete metric space with a doubling measure supporting a $p$-Poincar\'e inequality, $1<p<\infty$. The upper and lower ($p$-harmonic) Perron solutions are studied for $p$-parabolic open sets. It is shown that continuous functions and quasicontinuous Dirichlet functions are resolutive (i.e., that their upper and lower Perron solutions coincide) and that the Perron solution coincides with the $p$-harmonic extension. It is also shown that Perron solutions are invariant under perturbation of the function on a set of capacity zero.