Asymptotic shape of isolated magnetic domains

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 $|\Omega| \rightarrow \infty$, 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.