Original article

Calculus of Variations and Partial Differential Equations

, Volume 14, Issue 1, pp 1-27

First online:

Local minimizers in micromagnetics and related problems

  • J.M. BallAffiliated withMathematical Institute, University of Oxford, Oxford, UK
  • , A. TaheriAffiliated withMathematical Institute, University of Oxford, Oxford, UK
  • , M. WinterAffiliated withMathematisches Institut A, Universität Stuttgart, Stuttgart, Germany

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Let \(\Omega \subset{\bf R}^3\) be a smooth bounded domain and consider the energy functional

\({\mathcal J}_{\varepsilon} (m; \Omega) := \int_{\Omega} \left ( \frac{1}{2 \varepsilon} |Dm|^2 + \psi(m) + \frac{1}{2} |h-m|^2 \right) dx + \frac{1}{2} \int_{{\bf R}^3} |h_m|^2 dx. \)

Here \(\varepsilon>0\) is a small parameter and the admissible function m lies in the Sobolev space of vector-valued 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, non-negative energy density with a multi-well 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 Euler-Lagrange 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.