Asymptotic analysis for micromagnetics of thin films governed by indefinite material coefficients

Rejeb Hadiji,Ken Shirakawa

doi:10.3934/cpaa.2010.9.1345

Abstract

In this paper, a class of minimization problems, associated with the micromagnetics of thin films,is dealt with. Each minimization problem is distinguished by the thickness of the thin film,denoted by $ 0 < h < 1 $, and it is considered under spatial indefinite and degenerativesetting of the material coefficients.On the basis of the fundamental studies of the governing energy functionals,the existence of minimizers, for every $ 0 < h < 1 $, and the 3D-2D asymptotic analysisfor the observing minimization problems, as $ h \to 0 $, will be demonstrated inthe main theorem of this paper.

Highlights

Let S ⊂ R2 be a two-dimensional bounded domain with a smooth boundary, and let Ω ⊂ R3 be a three-dimensional cylindrical domain, given by Ω := S × (0, 1)
The main theme of this study is to verify whether some analogous conclusion can be obtained even under degenerative situations of α, or not
The first theorem is concerned with a Hilbert space, associated with the effective domains of convex parts of energy functionals

Summary

Introduction

Let S ⊂ R2 be a two-dimensional bounded domain with a smooth boundary, and let Ω ⊂ R3 be a three-dimensional cylindrical domain, given by Ω := S × (0, 1). (II) Under the conditions (a1)-(a2), there exist a sequence {hi | i = 1, 2, 3, · · · } ⊂ (0, 1) and a limiting function m◦ ∈ L2(S; R3) of two variables, such that: (i) hi ց 0, m(hi) → m◦ in L2(Ω; R3), E (hi)(m(hi)) → E ◦(m◦), and

Results

Conclusion