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

created by comi on 21 Sep 2018
modified on 24 Oct 2018

[BibTeX]

Preprint

Inserted: 21 sep 2018
Last Updated: 24 oct 2018

Year: 2018

ArXiv: 1809.08575 PDF

Abstract:

We introduce the new space $BV^{\alpha}(\mathbb{R}^n)$ of functions with fractional bounded 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 characterization of these (possibly non-unique) limit sets.