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.

2-d stability of the Neel wall

DeSimone A.;
2006

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: 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.
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Desimone, A.; Knupfer, H.; Otto, F.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/16233
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