Asymptotic shape of isolated magnetic domains

Abstract

We investigate the energy of an isolated magnetized domain $\Omega \subset {\mathbb{R}}^n$ for $n=2,3$. In non-dimensionalized variables, the energy given by $$\mathcal{E}(\Omega )={\int }_{{\mathbb{R}}^n}|\nabla {\chi }_{\Omega }|dx+{\int }_{{\mathbb{R}}^n}|\nabla h_{\Omega }{|}^{2}dx$$ penalizes the interfacial area of the domain as well as the energy of the corresponding magnetostatic field. Here, the magnetostatic potential $h_{\Omega }$ is determined by $\Delta h_{\Omega }={\partial }_1{\chi }_{\Omega }$, corresponding to uniform magnetization within the domain. We consider the macroscopic regime $|\Omega |\to \infty$, in which we derive compactness and $\Gamma$-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.

arXiv comment27 pages, 2 figures, final version accepted for publication in Proc. R. Soc. A

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Cited by in corpus (1)

• Minimal Energy for Geometrically Nonlinear Elastic Inclusions in Two Dimensions