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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> {1<p<∞} , including the case of a crystalline norm in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:math> {RN} . The result is obtained by a characterization of the corresponding subdifferential and it applies to unbounded domains of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> <m:mo>∖</m:mo> <m:mover accent="true"> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>¯</m:mo> </m:mover> </m:mrow> </m:math> {RNΩ̄} 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.