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

Abstract

We introduce the new space BVα(Rn) of functions with bounded fractional variation in Rn of order α∈(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 α-perimeter and we define its fractional reduced boundary FαE. We are able to show that Wα,1(Rn)⊂BVα(Rn) continuously and, similarly, that sets with (locally) finite standard fractional α-perimeter have (locally) finite fractional Caccioppoli α-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 α-perimeter, proving existence of blow-ups and giving a first characterisation of these (possibly non-unique) limit sets.