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

Abstract

We obtain existence of minimizers for the $p$-capacity functional defined with respect to a centrally symmetric anisotropy for $1<p<\infty$, including the case of a crystalline norm in ${\mathbb{R}}^N$. The result is obtained by a characterization of the corresponding subdifferential and it applies for unbounded domains of the form ${\mathbb{R}}^N\setminus \overline{\Omega }$ under mild regularity assumptions (Lipschitz-continuous boundary) and no convexity requirements on the bounded domain $\Omega$. 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.