Sharp Riesz capacity inequalities and the Pólya--Szegő conjecture

Abstract

For $n\ge 2$ and $p\in (n-2,n)$, we prove that closed balls maximize the ratio of Riesz $p$-capacity to Newtonian capacity among all compact subsets of ${\mathbb{R}}^n$. This resolves the classical Pólya--Szegő capacity ratio conjecture and its higher-dimensional extensions. For $n\ge 4$, a sharp inequality for the ratio of Newtonian capacity to logarithmic capacity is established. All equality cases are characterized.

arXiv commentsecond version; 22 pages