The anisotropic $\boldsymbol~p$-capacity and the anisotropic Minkowski inequality

In this paper, we prove a sharp anisotropic $L^p$ Minkowski inequality involving the total$L^p$ anisotropic mean curvature and the anisotropic $p$-capacity for any bounded domains with smooth boundary in $\\rr^n$.As consequences, we obtain an anisotropic Willmore inequality, a sharp anisotropic Minkowski inequality for outward $F$-minimising sets and a sharp volumetric anisotropic Minkowski inequality. For the proof, we utilize a nonlinear potential theoretic approach which has been recently developed by Agostiniani et al. (2019).