Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A distributional approach to fractional Sobolev spaces and fractional
  variation: existence of blow-up   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
characterisation of these (possibly non-unique) limit sets.

16
The categories of this paper are: stat
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
18
The categories of this paper are: econ
37
The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
19
The categories of this paper are: eess
9
The categories of this paper are: nlin
29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
30
The categories of this paper are: q-alg
14
The categories of this paper are: math-ph
1
The categories of this paper are: cs
26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
22
The categories of this paper are: chao-dyn