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

Abstract

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