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

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We study the condenser capacity cap p ⁡(E, Ω) on unbounded open sets Ω in a proper connected metric space X equipped with a locally doubling measure supporting a local p-Poincaré inequality, where 1 < p < ∞ . Using a new definition of capacitary potentials, we show that 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 Ω if and only if either X is p-hyperbolic or the Sobolev capacity C p ( X ∖ Ω ) > 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.