Condenser capacities and capacitary potentials for unbounded sets, and global $p$-harmonic Green functions on metric spaces

Abstract

We study the condenser capacity ${\mathrm{cap}}_p(E,\hat{I}\text{c◯})$ on \emph{unbounded} open sets $\hat{I}\text{c◯}$ in a proper connected metric space $X$ equipped with a locally doubling measure supporting a local $p$-Poincaré inequality, where $1<p<\infty$. Using a new definition of capacitary potentials, we show that ${\mathrm{cap}}_p$ is countably subadditive and that it is a Choquet capacity. We next obtain formulas for the capacity of superlevel sets for the capacitary potential. These are then used to show that $p$-harmonic Green functions exist in an unbounded domain $\hat{I}\text{c◯}$ if and only if either $X$ is $p$-hyperbolic or the Sobolev capacity $C_p(X\setminus \hat{I}\text{c◯})>0$. As an application, we deduce new results for Perron solutions and boundary regularity for the Dirichlet boundary value problem for $p$-harmonic functions in unbounded open sets.