Liouville type results for local minimizers of the micromagnetic energy
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- 作者：Fran?ois Alouges ; Giovanni Di Fratta…
- 关键词：35B35 ; 35B53 ; 49K20 ; 49K40 ; 49S05 ; 74G65 ; 82D40
- 刊名：Calculus of Variations and Partial Differential Equations
- 出版年：2015
- 出版时间：July 2015
- 年：2015
- 卷：53
- 期：3-4
- 页码：525-560
- 全文大小：780 KB
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Giovanni Di Fratta (2)
Benoit Merlet (1)
1. CMAP, école Polytechnique, route de Saclay, 91128?, Palaiseau Cedex, France
2. School of Mathematics, University of Bristol, Bristol, UK - 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Analysis
Systems Theory and Control
Calculus of Variations and Optimal Control
Mathematical and Computational Physics - 出版者：Springer Berlin / Heidelberg
- ISSN：1432-0835
文摘
We study local minimizers of the micromagnetic energy in small ferromagnetic 3d convex particles for which we justify the Stoner–Wohlfarth approximation: given a uniformly convex shape \(\Omega \subset \varvec{\mathbf {R}}^3\), there exist \(\delta _c\)>0 and \(C > 0\) such that for \(0 < \delta \le \delta _c\) any local minimizer \(\mathbf {m}\) of the micromagnetic energy in the particle \(\delta \Omega \) satisfies \(\Vert \nabla \mathbf {m} \Vert _{L^2} \leqslant C \delta ^2\). In the case of ellipsoidal particles we strengthen this result by proving that, for \(\delta \) small enough, local minimizers are exactly spatially uniform. This last result extends W.F. Brown’s fundamental theorem for fine 3d ferromagnetic particles Brown (J Appl Phys 39:463-88, 1968), Di Fratta et al. (Physica B 407(9):1368-371, 2011) which states the same result but only for global minimizers. As a by-product of the method that we use, we establish a new Liouville type result for locally minimizing \(p\)-harmonic maps with values into a closed subset of a Hilbert space. Namely, we establish that in a smooth uniformly convex domain of \(\mathbf {R}^d\) any local minimizer of the \(p\)-Dirichlet energy (\(p > 1\), \(p \ne d\)) is constant.