“A distributional approach to fractional Sobolev spaces and fractional variation: asymptotic analysis”

Mercoledì 11 Dicembre 2019, ore 14:30 - Aula 2BC30 - Giorgio Stefani (Scuola Normale Superiore di Pisa)

Abstract

In a recent paper in collaboration with G. E. Comi, we have introduced a new notion of fractional $\alpha$-variation for all $\alpha\in(0,1)$ via a distributional approach exploiting suitable notions of fractional gradient and fractional divergence already existing in the literature. In this talk, after a brief summary on the relevant features of our fractional BV space, I will present some recent results in collaboration with E. Brué, M. Calzi and G. E. Comi about the asymptotic behaviour of the fractional $\alpha$-variation as the parameter $\alpha$ tends to 0 and 1. We are able to prove natural analogues of the asymptotic results by Davila-Maz’ya-Shaposhnikova via new interpolation inequalities. In addition, when $\alpha$ tends to 1, we are also able to prove the $\Gamma$-convergence of the fractional variation to the De Giorgi’s variation, in perfect analogy with the well-known $\Gamma$-convergence result by Ambrosio-De Philippis-Martinazzi.