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Local minimizers in micromagnetics and related problems
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Let \(\Omega \subset{\bf R}^3\) be a smooth bounded domain and consider the energy functional
\({\mathcal J}_{\varepsilon} (m; \Omega) := \int_{\Omega} \left ( \frac{1}{2 \varepsilon} Dm^2 + \psi(m) + \frac{1}{2} hm^2 \right) dx + \frac{1}{2} \int_{{\bf R}^3} h_m^2 dx. \)
Here \(\varepsilon>0\) is a small parameter and the admissible function m lies in the Sobolev space of vectorvalued functions \(W^{1,2}(\Omega;{\bf R}^3)\) and satisfies the pointwise constraint \(m(x)=1\) for a.e. \(x \in \Omega\). The induced magnetic field \(h_m \in L^2({\bf R}^3;{\bf R}^3)\) is related to m via Maxwell's equations and the function \(\psi:{\bf S}^2 \to{\bf R}\) is assumed to be a sufficiently smooth, nonnegative energy density with a multiwell structure. Finally \(h \in{\bf R}^3\) is a constant vector. The energy functional \({\mathcal J}_{\varepsilon}\) arises from the continuum model for ferromagnetic materials known as micromagnetics developed by W.F. Brown [9].
In this paper we aim to construct local energy minimizers for this functional. Our approach is based on studying the corresponding EulerLagrange equation and proving a local existence result for this equation around a fixed constant solution. Our main device for doing so is a suitable version of the implicit function theorem. We then show that these solutions are local minimizers of \({\mathcal J}_{\varepsilon}\) in appropriate topologies by use of certain sufficiency theorems for local minimizers.
Our analysis is applicable to a much broader class of functionals than the ones introduced above and on the way to proving our main results we reflect on some related problems.
Received: 20 November 2000 / Accepted: 4 December 2000 / Published online: 4 May 2001
 Title
 Local minimizers in micromagnetics and related problems
 Journal

Calculus of Variations and Partial Differential Equations
Volume 14, Issue 1 , pp 127
 Cover Date
 20020101
 DOI
 10.1007/s005260100085
 Print ISSN
 09442669
 Online ISSN
 14320835
 Publisher
 SpringerVerlag
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