On sets with finite distributional fractional perimeter

TL;DR

This paper advances the understanding of sets with finite distributional fractional perimeter by refining their blow-up characterization, establishing a Leibniz rule, and describing associated non-local boundaries.

Contribution

It introduces new characterizations of blow-ups, proves a Leibniz rule for intersections, and describes non-local boundaries for sets with fractional perimeter.

Findings

Refined the characterization of blow-ups of sets with fractional perimeter.

Proved a Leibniz rule for intersections involving fractional perimeter sets.

Provided a description of non-local boundaries associated with fractional perimeter.

Abstract

We continue the study of the fine properties of sets having locally finite distributional fractional perimeter. We refine the characterization of their blow-ups and prove a Leibniz rule for the intersection of sets with locally finite distributional fractional perimeter with sets with finite fractional perimeter. As a byproduct, we provide a description of non-local boundaries associated with the distributional fractional perimeter.