On some isoperimetric inequalities for the Newtonian capacity

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 $t$.