**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. Thanks to the continuous inclusion $W^{{\alpha,} 1}(\mathbb{R}^{{n})} \subset BV^{{\alpha}}(\mathbb{R}^{{n})$,} our theory provides a natural extension of the known fractional framework. In analogy with the classical $BV$ theory, we define sets with (locally) finite fractional Caccioppoli $\alpha$-perimeter and we partially extend De Giorgi's Blow-up Theorem to such sets, proving existence of blow-ups on points of the naturally defined fractional reduced boundary.

Wed 17 Apr, 11:50 - 12:20, Aula Dini

Seminario Prof. Comi

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