Local minimizers in micromagnetics and related problems

Abstract

Let \Omega \subset {\bf R}^3 be a smooth bounded domain and consider the energy functional J_{\varepsilon} (m) := \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 is a small non negative parameter and the space of admissible functions for m is the Sobolev space of vector-valued functions W^{1,2}(\Omega;{\bf R}^3) which satisfy the pointwise constraint |m(x)|^2-1=0 for a.e. x \in \Omega. The integrand \psi:{\bf S}^2 \to {\bf R} is assumed to be a sufficiently smooth non negative density function with a multi-well structure. The function h_m \in L^2 ({\bf R}^3; {\bf R}^3) is related to m via Maxwell´s equations. Finally h \in {\bf R}^3 is a constant vector. The energy functional J_{\varepsilon} arises from the study of continuum models for ferromagnetic materials known as micromagnetics developed by W. Brown \cite{BR}.

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 solutions close to a fixed constant solution. Our main device for doing this is a suitable version of the implicit function theorem. We then show that these solutions are local minimizers of J_{\varepsilon} in appropriate topologies by using 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 of proving our main results we reflect on some related problems.