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Asymptotic shape of isolated magnetic domains

We investigate the energy of an isolated magnetized domain $Ω\subset \mathbb{R}^n$ for $n=2,3$. In non-dimensionalized variables, the energy given by $$ \mathcal{E}(Ω) \ = \ \int_{\mathbb{R}^n} |\nabla χ_Ω| \ dx + \int_{\mathbb{R}^n} |\nabla h_Ω|^2 \ dx $$ penalizes the interfacial area of the domain as well as the energy of the corresponding magnetostatic field. Here, the magnetostatic potential $h_Ω$ is determined by $Δh_Ω= \partial_1 χ_Ω$, corresponding to uniform magnetization within the domain. We consider the macroscopic regime $|Ω| \rightarrow \infty$, in which we derive compactness and $Γ$-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.