Characterization of the subdifferential and minimizers for the anisotropic p-capacity
Esther Cabezas-Rivas, Salvador Moll, Marcos Solera · arXiv (Cornell University) · 2023
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 $\mathbb R^N \setminus \overlineΩ$ 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.