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

We introduce the new space BV alpha(R-n) of functions with bounded fractional variation in R-n of order a is an element of (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 (FE)-E-alpha. We are able to show that W-alpha,W-1 (R-n) subset of BV alpha (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. (C) 2019 Elsevier Inc. All rights reserved.