### A compactness result in thin-film micromagnetics and the optimality of the Néel wall

In this paper, we study a model for the magnetization in thin ferromagnetic films. It comes as a variational problem for ${S}^{1}$-valued maps ${m}^{\text{'}}$ (the magnetization) of two variables ${x}^{\text{'}}$: ${E}_{\epsilon}\left({m}^{\text{'}}\right)=\epsilon \int {|{\nabla}^{\text{'}}\xb7{m}^{\text{'}}|}^{2}d{x}^{\text{'}}+\frac{1}{2}\int {\left||{\nabla}^{\text{'}}{|}^{-1/2}{\nabla}^{\text{'}}\xb7{m}^{\text{'}}\right|}^{2}d{x}^{\text{'}}$. We are interested in the behavior of minimizers as $\epsilon \to 0$. They are expected to be ${S}^{1}$-valued maps ${m}^{\text{'}}$ of vanishing distributional divergence ${\nabla}^{\text{'}}\xb7{m}^{\text{'}}=0$, so that appropriate boundary conditions enforce line discontinuities. For finite $\epsilon >0$, these line discontinuities are approximated by smooth transition layers, the so-called Néel walls. Néel...