Homogenization of the Landau-Lifshitz-Gilbert equation with natural boundary condition

TL;DR

This paper studies the homogenization of the Landau-Lifshitz-Gilbert equation with natural boundary conditions, establishing convergence results and addressing technical challenges like boundary layers and stray fields.

Contribution

It introduces a high-order Neumann corrector and boundary layer analysis to prove convergence of homogenized solutions in the presence of complex boundary and stray field effects.

Findings

Established convergence between homogenized and original solutions.

Developed high-order correctors to handle boundary layers.

Provided convergence rates in the H^1 norm.

Abstract

The full Landau-Lifshitz-Gilbert equation with periodic material coefficients and natural boundary condition is employed to model the magnetization dynamics in composite ferromagnets. In this work, we establish the convergence between the homogenized solution and the original solution via a Lax equivalence theorem kind of argument. There are a few technical difficulties, including: 1) it is proven the classic choice of corrector to homogenization cannot provide the convergence result in the $H^1$ norm; 2) a boundary layer is induced due to the natural boundary condition; 3) the presence of stray field give rise to a multiscale potential problem. To keep the convergence rates near the boundary, we introduce the Neumann corrector with a high-order modification. Estimates on singular integral for disturbed functions and boundary layer are deduced, to conduct consistency analysis of stray field. Furthermore, inspired by length conservation of magnetization, we choose proper correctors in specific geometric space. These, together with a uniform $W^{1,6}$ estimate on original solution, provide the convergence rates in the $H^1$ sense.