2-d stability of the Néel wall

Abstract

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:

$$ E_{2d}(m)= \epsilon \int |\nabla m|^2 \,{\rm d}x + \frac{1}{2} \int |\nabla^{-1/2}\nabla \cdot m|^2\,{\rm d}x. $$

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

$$ \int |\nabla^{1/2}\phi|^2 \, {\rm d}x \, \not\lesssim \, {\rm sup} |\phi| \, \int |\nabla \phi| \, {\rm d}x. $$