Abstract

In this paper, we prove a sharp anisotropic Lᵖ Minkowski inequality involving the total Lᵖ anisotropic mean curvature and the anisotropic p-capacity for any bounded domains with smooth boundary in Rⁿ. 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).