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

Abstract

We introduce the new space $B{V}^{\alpha }({\mathbb{R}}^n)$ 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 ${\mathcal{F}}^{\alpha }E$. We are able to show that $W^{\alpha ,1}({\mathbb{R}}^n)\subset B{V}^{\alpha }({\mathbb{R}}^n)$ 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.

46 pages