We obtain existence of minimizers for the p-capacity functional defined with respect to a centrally symmetric anisotropy for 1 < p < ∞, including the case of a crystalline norm in ℝ^N. The result is obtained by a characterization of the corresponding subdifferential and it applies to unbounded domains of the form ℝ^N \ Ω under mild regularity assumptions (Lipschitz-continuous boundary) and no convexity requirements on the bounded domain Ω. If we further assume an interior ball condition (where the Wulff shape plays the role of a ball), then any minimizer is shown to be Lipschitz continuous.
