Calculus of Variations and Partial Differential Equations

, Volume 27, Issue 2, pp 233-253

First online:

2-d stability of the Néel wall

  • Antonio DeSimoneAffiliated withSISSA, International School for Advanced Mathematics
  • , Hans KnüpferAffiliated withInstitute of Applied Mathematics, University of Bonn
  • , Felix OttoAffiliated withInstitute of Applied Mathematics, University of Bonn

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


Micromagnetics Thin films Néel wall