Sharp bounds for the anisotropic $p$-capacity of Euclidean compact sets

Abstract

We prove various sharp bounds for the anisotropic p-capacity Cap F,p (K) (1 < p < n) of compact sets K in the Euclidean space R n (n ≥ 3). For example, using the inverse anisotropic mean curvature flow (IAMCF), we get an upper bound of Szegö type (1931) for Cap F,p (K) when ∂K is a smooth, star-shaped and F -mean convex hypersurface in R n (n ≥ 3). Moreover, for such a surface ∂K in R 3 , by introducing the anisotropic Hawking mass and studying its monotonicity property along IAMCF, we obtain an upper bound of Bray-Miao type (2008) for Cap F,p (K).