2-d stability of the Neel wall

We are interested in thin-film samples in micromagnetism, where the magnetization m is a 2-d unit-length vector field. More precisely we are interested in transition layers which connect two opposite magnetizations, so called Néel walls. We prove stability of the 1-d transition layer under 2-d perturbations. This amounts to the investigation of the following singularly perturbed energy functional: E2d(m)= ε ∫ |∇m| 2dx + 1/2 ∫ |∇-1/2∇̇m|2dx. The topological structure of this two-dimensional problem allows us to use a duality argument to infer the optimal lower bound. The lower bound relies on an ε-perturbation of the following logarithmically failing interpolation inequality ∫ |∇1/2/φ|2dx ¬ sup|φ| ∫ |∇φ|dx.