On some isoperimetric inequalities for the Newtonian capacity

TL;DR

This paper establishes new isoperimetric inequalities for Newtonian capacity in higher dimensions, relating it to geometric measures like perimeter and mean curvature, with applications to Wiener sausage capacity.

Contribution

It introduces novel upper bounds for Newtonian capacity based on geometric properties and provides a quantitative refinement involving Fraenkel asymmetry.

Findings

Upper bounds for capacity in terms of perimeter and mean curvature.

Equality cases characterized by balls.

Refinement involving Fraenkel asymmetry.

Abstract

Upper bounds are obtained for the Newtonian capacity of compact sets in $\R^d,\,d\ge 3$ in terms of the perimeter of the $r$-parallel neighbourhood of $K$. For compact, convex sets in $\R^d,\,d\ge 3$ with a $C^2$ boundary the Newtonian capacity is bounded from above by $(d-2)M(K)$, where $M(K)>0$ is the integral of the mean curvature over the boundary of $K$ with equality if $K$ is a ball. For compact, convex sets in $\R^d,\,d\ge 3$ with non-empty interior the Newtonian capacity is bounded from above by $\frac{(d-2)P(K)^2}{d|K|}$ with equality if $K$ is a ball. Here $P(K)$ is the perimeter of $K$ and $|K|$ is its measure. A quantitative refinement of the latter inequality in terms of the Fraenkel asymmetry is also obtained. An upper bound is obtained for expected Newtonian capacity of the Wiener sausage in $\R^d,\,d\ge 5$ with radius $\varepsilon$ and time length $t$.