Calculus of Variations and Geometric Measure Theory

G. E. Comi - G. Stefani

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

created by comi on 21 Sep 2018
modified on 29 Oct 2019


Published Paper

Inserted: 21 sep 2018
Last Updated: 29 oct 2019

Journal: Journal of Functional Analysis
Volume: 277
Number: 10
Pages: 3373-3435
Year: 2019
Doi: 10.1016/j.jfa.2019.03.011

ArXiv: 1809.08575 PDF


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 characterization of these (possibly non-unique) limit sets.

Keywords: fractional Sobolev spaces, blow-up, Fractional Gradient, fractional calculus, fractional perimeter, fractional derivative, fractional divergence, function with bounded fractional variation