Local minimizers in micromagnetics and related problems

Abstract. 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.