Conference on Calculus of Variations in Lille - 3rd edition - July 4-6 2022

Fractional Allen–Cahn systems with multi-well potential and nonlocal minimal partitions

Jul 5, 2022, 3:00 PM

30m

M2 building, Cité Scientifique - Meeting room, 1st floor (Laboratoire Paul Painlevé)

Speaker

Thomas Gabard (Université Paris-Est Créteil )

Description

The aim of this talk is to present results on the asymptotic analysis of a fractional version of the vectorial Allen–Cahn equation with multiple-well in arbitrary dimension. In contrast to usual Allen–Cahn equations, the Laplace operator is replaced by the fractional Laplacian as defined in Fourier space. Our results concern the singular limit $\epsilon \to 0$ and show that arbitrary solutions with uniformly bounded energy converge both in the energetic and geometric sense to nonlocal minimal partitions in $\mathrm{\Omega }$. The notion of nonlocal minimal partition corresponds to the stationary version of the nonlocal minimizing clusters introduced by M. Colombo & F. Maggi (2017) and A. Cesaroni & M. Novaga (2020), and generalizing the nonlocal minimal surfaces of L. Caffarelli, J.M. Roquejoffre, & O. Savin (2010).