A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up

TL;DR

This paper develops a new fractional variation space using a distributional approach, extending classical BV theory to fractional contexts and proving a blow-up theorem for sets with finite fractional perimeter.

Contribution

Introduces the space $BV^{eta}( ^n)$ with a novel distributional approach, extending classical BV theory to fractional variation and perimeter, and proves a fractional blow-up theorem.

Findings

Established continuous embedding of $W^{eta,1}$ into $BV^{eta}$.

Connected fractional perimeter with fractional Caccioppoli sets.

Proved existence and characterization of blow-ups for fractional perimeter sets.

Abstract

We introduce the new space $BV^{\alpha}(\mathbb{R}^n)$ of functions with bounded fractional variation in $\mathbb{R}^n$ of order $\alpha \in (0, 1)$ via a new distributional approach exploiting suitable notions of fractional gradient and fractional divergence already existing in the literature. In analogy with the classical $BV$ theory, we give a new notion of set $E$ of (locally) finite fractional Caccioppoli $\alpha$-perimeter and we define its fractional reduced boundary $\mathscr{F}^{\alpha} E$. We are able to show that $W^{\alpha,1}(\mathbb{R}^n)\subset BV^\alpha(\mathbb{R}^n)$ continuously and, similarly, that sets with (locally) finite standard fractional $\alpha$-perimeter have (locally) finite fractional Caccioppoli $\alpha$-perimeter, so that our theory provides a natural extension of the known fractional framework. Our main result partially extends De Giorgi's Blow-up Theorem to sets of locally finite fractional Caccioppoli $\alpha$-perimeter, proving existence of blow-ups and giving a first characterisation of these (possibly non-unique) limit sets.