Some topics of Geometric Analysis and Geometric Measure Theory

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

speaker: Giovanni Eugenio Comi (Scuola Normale Superiore di Pisa)

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.

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

Seminario Prof. Comi

<< Go back