Characterization of the subdifferential and minimizers for the anisotropic p-capacity

We obtain existence of minimizers for the p-capacity functional defined with respect to a centrally symmetric anisotropy for 1 < p < ∞ {1<p<\infty} , including the case of a crystalline norm in ℝ N {\mathbb{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 ¯ {\mathbb{R}^{N}\setminus\overline{\Omega}} 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.