On some isoperimetric inequalities for the Newtonian capacity

Abstract

Upper bounds are obtained for the Newtonian capacity of compact sets in [Formula: see text] in terms of the perimeter of the [Formula: see text]-parallel neighborhood of [Formula: see text]. For compact, convex sets in [Formula: see text] with a [Formula: see text] boundary the Newtonian capacity is bounded from above by [Formula: see text], where [Formula: see text] is the integral of the mean curvature over the boundary of [Formula: see text] with equality if [Formula: see text] is a ball. For compact, convex sets in [Formula: see text] with non-empty interior the Newtonian capacity is bounded from above by [Formula: see text] with equality if [Formula: see text] is a ball. Here, [Formula: see text] is the perimeter of [Formula: see text] and [Formula: see text] 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 [Formula: see text] with radius [Formula: see text] and time length [Formula: see text].