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MiS Preprint
83/2000
Local minimizers in micromagnetics and related problems
John M. Ball, Ali Taheri and M. Winter
Abstract
Let $\Omega \subset R^3 $ be a smooth bounded domain and consider the energy functional $$\mathbb{I}_\epsilon (m;\Omega):= \int_\Omega \left( \frac{1}{2\epsilon} |Dm|^2 =\psi (m )= \frac{1}{2} |h-m|^2 \right) dx + \frac{1}{2} \int_{R^3} |h_m|^2 dx$$ Here $\epsilon > 0$ is a small parameter and the admissible function m lies in the Sobolev space of vector-valued functions $W^{1,2} (\Omega ; R^3)$ and satisfies the pointwise constraint |m(x)|=1 for a.e. $x \in \Omega$. The induced magnetic field $h_m \in L^2 (R^3;R^3)$ is related to m via Maxwell\'s equations and the function $\psi : S^2 \to R$ is assumed to be a sufficiently smooth, non-negative energy density with a multi-well structure. Finally $h \in R^3$ is a constant vector. The energy functional $\mathbb{I}_\epsilon$ arises from the continuum model for ferromagnetic materials known as micromagnetics developed by W.F. Brown.
