A strong quantitative form of the fractional isoperimetric inequality

TL;DR

This paper establishes a strong fractional quantitative isoperimetric inequality linking the isoperimetric deficit to boundary oscillation and asymmetry, extending previous local results and deriving stability estimates for fractional Cheeger inequalities.

Contribution

It introduces a novel strong version of the fractional isoperimetric inequality that controls boundary oscillation and asymmetry, generalizing prior local results with a different proof approach.

Findings

The fractional isoperimetric deficit bounds boundary oscillation.

The inequality extends Fusco and Julin's local result to a stronger form.

Stability estimates for fractional Cheeger inequalities are derived.

Abstract

We show a strong version of the fractional quantitative isoperimetric inequality, in which the isoperimetric deficit controls not only the Fraenkel asymmetry but also a sort of oscillation of the boundary. This generalizes the local result by Fusco and Julin in \cite{FJ}. The proof follows a regularization process as in \cite{FJ} but it is quite different in its spirit. Then, as a consequence of the quantitative inequality, we prove some stability estimates for a fractional Cheeger inequality.