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

TL;DR

This paper establishes sharp bounds for the anisotropic p-capacity of compact sets in Euclidean space using inverse anisotropic mean curvature flow, extending classical results to anisotropic settings with new geometric tools.

Contribution

It introduces new sharp bounds for anisotropic p-capacity employing IAMCF and anisotropic Hawking mass, extending classical bounds to anisotropic geometric contexts.

Findings

Derived Szeg"o-type upper bounds for smooth, star-shaped, F-mean convex hypersurfaces.

Established Bray--Miao-type upper bounds in three dimensions using anisotropic Hawking mass.

Applied inverse anisotropic mean curvature flow to obtain monotonicity properties and bounds.

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 $\mathrm{Cap}_{F,p}(K)$.