*Preprint*

**Inserted:** 21 sep 2018

**Last Updated:** 24 oct 2018

**Year:** 2018

**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.

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

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