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

2018/09/30 by Giovanni E. Comi, Giorgio Stefani · 1 citation
Mathematics · #math.FA #msc:26A33 #msc:35B44 #msc:47B38 #msc:26B30

paper · pdf · doi:10.1016/j.jfa.2019.03.011

published as J. Funct. Anal. 277 (2019), no. 10, 3373-3435 · 46 pages

arxiv created 2019/10/29 · arxiv updated 2019/10/30

Abstract

We introduce the new space BV^α(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 \mathscrF^α E. We are able to show that W^α,1(ℝⁿ)⊂ BV^α(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.

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