A quantitative stability inequality for fractional capacities

TL;DR

This paper establishes a quantitative stability inequality for fractional capacities, linking the isocapacitary deficit to Fraenkel asymmetry, and explores the behavior as the fractional parameter approaches 1.

Contribution

It provides a non-sharp quantitative stability version of the fractional isocapacitary inequality and analyzes the asymptotic behavior as the fractional parameter varies.

Findings

Lower bound for the isocapacitary deficit in terms of Fraenkel asymmetry

Asymptotic behavior of fractional capacity as s approaches 1

Stability of the estimate with respect to the parameter s

Abstract

The aim of this work is to show a non-sharp quantitative stability version of the fractional isocapacitary inequality. In particular, we provide a lower bound for the isocapacitary deficit in terms of the Fraenkel asymmetry. In addition, we provide the asymptotic behaviour of the $s$-fractional capacity when $s$ goes to $1$ and the stability of our estimate with respect to the parameter $s$.