Homogenization results for a Landau–Lifshitz–Gilbert equation in composite materials with transmission defects

Abstract

We study the homogenization of Landau–Lifshitz–Gilbert equation in a ϵ-periodic composite material formed by two constituents, separated by an imperfect interface ${\mathrm{\Gamma }}^{ϵ}$, on which we prescribe the continuity of the conormal derivatives and a jump of the solution proportional to the conormal derivative, by means of a coefficient of order ${ϵ}^{\gamma }$. We use the periodic unfolding method together with extension operators for handling the nonlinearities to identify the limit problem when tuning up the parameter γ in $\mathbb{R}$.

Keywords:

• Homogenization
• Landau–Lifshitz–Gilbert equation
• ferromagnetic materials
• periodic unfolding method
• composite materials
• imperfect interface

2010 Mathematics Subject Classifications:

• 35B27
• 78A25
• 35Q60
• 35B40
• 82D40
• 74A40

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Notice that the surface integrals make sense since ${\mathrm{\Gamma }}^{ϵ}$ is a Lipschitz surface.

2 The existence of ${\phi }_0^{ϵ}\in W_{bp}({\mathbb{R}}^3)$ follows from the Lax–Milgram theorem, see e.g. [25]. See also [22] for further results.