  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.
