A compactness result for Landau state in thin-film micromagnetics

We deal with a nonconvex and nonlocal variational problem coming from thin-film micromagnetics. It consists in a free-energy functional depending on two small parameters $\eps$ and $\eta$ and defined over $S^2-$vector fields $m$ that are tangent at the boundary of a two-dimensional domain $\Omega$. We are interested in the behavior of minimizers as $\eps, \eta \to 0$. The minimizers tend to be in-plane away from a region of length scale $\eps$ (generically, an interior vortex ball or two boundary vortex balls) and of vanishing divergence, so that $S^1-$transition layers of length scale $\eta$ (N\'eel walls) are enforced by the boundary condition. We first prove an upper bound for the minimal energy that corresponds to the cost of a vortex and the configuration of N\'eel walls associated to the viscosity solution, so-called Landau state. Our main result concerns the compactness of vector fields $m_{\eps, \eta}$ of energies close to the Landau state in the regime where a vortex is energetically more expensive than a N\'eel wall. Our method uses techniques developed for the Ginzburg-Landau type problems for the concentration of energy on vortex balls, together with an approximation argument of $S^2-$vector fields by $S^1-$vector fields away from the vortex balls.