The sharp quantitative isocapacitary inequality (the case of $p$-capacity)

TL;DR

This paper establishes a precise quantitative version of the isocapacitary inequality for general p-capacity, extending previous work that focused on the case p=2, thereby broadening the understanding of capacity inequalities.

Contribution

It generalizes the sharp quantitative isocapacitary inequality from the specific case p=2 to all p, providing a more comprehensive mathematical framework.

Findings

Proved a sharp quantitative inequality for general p-capacity.

Extended previous results from p=2 to arbitrary p.

Enhanced understanding of capacity inequalities in geometric analysis.

Abstract

We prove a sharp quantitative form of isocapacitary inequality in the case of a general $p$. This work is a generalization of the author's paper with Guido De Philippis and Michele Marini, where we treated the case of $2$-capacity.