Homogenization of nonlocal exchange energies in micromagnetics

TL;DR

This paper investigates the asymptotic behavior of nonlocal micromagnetic energies with heterogeneities, deriving an effective local functional through a specialized two-scale convergence approach that respects the magnetization's manifold constraint.

Contribution

It introduces a novel homogenization framework for nonlocal micromagnetic functionals, incorporating a tailored two-scale convergence method that accounts for the sphere constraint on magnetization.

Findings

Derived the $ ext{Γ}$-limit of nonlocal energies as an effective local functional.

Developed a two-scale convergence technique adapted to nonlocal differences and manifold constraints.

Established that microscopic oscillations are confined to the tangent space of the sphere.

Abstract

We study the homogenization of nonlocal micromagnetic functionals incorporating both symmetric and antisymmetric exchange contributions under the physical constraint that the magnetization field takes values in the unit sphere. Assuming that the nonlocal interaction range and the scale of heterogeneities vanish simultaneously, we capture the asymptotic behavior of the nonlocal energies by identifying their $\Gamma$-limit, leading to an effective local functional expressed through a tangentially constrained nonlocal cell problem. Our proof builds upon a tailored notion of two-scale convergence, which takes into account oscillations only in specific directions. It enables us to describe the two-scale limit of suitable nonlocal difference quotients, yielding a nonlocal analog of the classical limit decomposition result for gradient fields. To deal with the manifold constraint of the magnetization, we additionally prove that the microscopic oscillations in the two-scale limit are constrained to lie in the tangent space of the sphere.