A fractional De Giorgi isoperimetric type inequality

TL;DR

This paper proves a fractional isoperimetric inequality for level sets in fractional Sobolev spaces, answering a previous open question and linking it to the regularity of fractional De Giorgi classes.

Contribution

It introduces a new fractional isoperimetric inequality and demonstrates its implications for the regularity of functions in fractional De Giorgi classes.

Findings

Established a fractional isoperimetric inequality for level sets.

Connected the inequality to H"older continuity of fractional De Giorgi class functions.

Modified existing estimates for nonlocal interaction functionals.

Abstract

We establish an isoperimetric type inequality for the level sets of functions in fractional Sobolev spaces. This answers a question posed by the first author in a previous paper. To obtain it, we work out a slight modification of some estimates for nonlocal interaction functionals established by Savin and Valdinoci. We also show how said isoperimetric inequality leads to the H\"older continuity of functions in (weak) fractional De Giorgi classes.