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

TL;DR

This paper investigates the properties of condenser capacities and capacitary potentials in unbounded sets within metric spaces, establishing their mathematical characteristics and linking them to the existence of $p$-harmonic Green functions and boundary regularity.

Contribution

It introduces a new definition of capacitary potentials, proves that capacity is a Choquet capacity, and characterizes the existence of $p$-harmonic Green functions in unbounded domains.

Findings

Capacity is countably subadditive and a Choquet capacity.

Formulas for capacities of superlevel sets of capacitary potentials.

Existence of $p$-harmonic Green functions characterized by hyperbolicity or capacity positivity.

Abstract

We study the condenser capacity $\mathrm{cap}_p(E,\Omega)$ on \emph{unbounded} open sets $\Omega$ in a proper connected metric space $X$ equipped with a locally doubling measure supporting a local $p$-Poincar\'e 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 $\Omega$ if and only if either $X$ is $p$-hyperbolic or the Sobolev capacity $C_p(X\setminus \Omega)>0$. 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.