Asymptotic shape of isolated magnetic domains

Abstract

We investigate the energy of an isolated magnetized domainΩ⊂Rnforn=2,3. In non-dimensionalized variables, the energy given byE(Ω)=∫Rn|∇χΩ| dx+∫Rn|∇hΩ|2 dxpenalizes the interfacial area of the domain as well as the energy of the corresponding magnetostatic field. Here, the magnetostatic potentialhΩis determined byΔhΩ=∂1χΩ, corresponding to uniform magnetization within the domain. We consider the macroscopic regime|Ω|→∞, in which we derive compactness andΓ-limit which is formulated in terms of the cross-sectional area of the anisotropically rescaled configuration. We then give the solutions for the limit problems.