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- W2890971675 abstract "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." @default.
- W2890971675 created "2018-09-27" @default.
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- W2890971675 date "2019-11-01" @default.
- W2890971675 modified "2023-10-03" @default.
- W2890971675 title "A distributional approach to fractional Sobolev spaces and fractional variation: Existence of blow-up" @default.
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- W2890971675 doi "https://doi.org/10.1016/j.jfa.2019.03.011" @default.
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