Abstract

In this talk, I will present a recent joint work with Rossella Giorgio and Hidde Sch\”onberger, where we conduct the joint homogenization and localization of a nonlocal version of the micromagnetic functional including symmetric and antisymmetric exchange energies.  The common local symmetric exchange term, which resembles the well-known Dirichlet energy, naturally admits a nonlocal representation in the spirit of Bourgain, Brezis and Mironescu. It was proven first by Ponce that this nonlocal formulation indeed \(\Gamma\)-converges to its local counterpart under sufficient conditions on the localizing kernels. Recently, this result has been extended by Davoli, Di Fratta and Giorgio to the case where an additional antisymmetric exchange term representing the chiral Dzyaloshinskii-Moriya interaction (DMI) is considered. We take this as a motivation to study the variational multiscale behavior of this nonlocal functional in the highly heterogeneous framework and undertake its homogenization in the regime where the scale of the inhomogeneities and the horizon of the nonlocal interactions are comparable. We develop a version of nonlocal two-scale convergence to perform the homogenization procedure. By adapting to the \(\mathbb{S}^2\)-constraint—a hallmark of micromagnetics—, we extend the relatively young research direction of nonlocal homogenization to the setting of manifold constraints.