MiS Preprint Repository

Let $\mathrm{\Omega }\subset R^3$ be a smooth bounded domain and consider the energy functional $${\mathbb{I}}_{ϵ}(m;\mathrm{\Omega }):={\int }_{\mathrm{\Omega }}(\frac{1}{2ϵ}|Dm{|}^2=\psi (m)=\frac{1}{2}|h-m{|}^2)dx+\frac{1}{2}{\int }_{R^3}|h_m{|}^{2}dx$$ Here $ϵ>0$ is a small parameter and the admissible function m lies in the Sobolev space of vector-valued functions $W^{1,2}(\mathrm{\Omega };R^3)$ and satisfies the pointwise constraint |m(x)|=1 for a.e. $x\in \mathrm{\Omega }$. 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 ${\mathbb{I}}_{ϵ}$ arises from the continuum model for ferromagnetic materials known as micromagnetics developed by W.F. Brown.