薄膜微磁学中边界涡的有效模型
An effective model for boundary vortices in thin-film micromagnetics
摘要：
铁磁材料是由一个非局部、非凸、多尺度的变分原理控制的。主要对象是由一个单位长度的三维向量场，磁化，对应于微磁能的稳定状态。我们的目的是分析一个薄膜制度，捕获的渐近行为所产生的磁化和它们的相互作用能的边界涡。本研究是基于“全球雅可比矩阵”的概念检测的拓扑缺陷，先验可以位于内部和边界的电影。一个主要的困难在于估计微磁能的非局部部分，以隔离对应的拓扑缺陷的确切条款。我们证明了通过$Γ$收敛展开在二阶边界涡周围的能量集中。二阶项是表示边界涡之间的相互作用并控制其最佳位置的重整化能量。我们计算的重整化能量的表达式，我们证明了存在的极小有两个边界涡的多重$1$。紧性的结果也显示了磁化强度和相应的全球雅可比矩阵。
英文摘要：
Ferromagnetic materials are governed by a variational principle which is nonlocal, nonconvex and multiscale. The main object is given by a unit-length three-dimensional vector field, the magnetization, that corresponds to the stable states of the micromagnetic energy. Our aim is to analyze a thin film regime that captures the asymptotic behavior of boundary vortices generated by the magnetization and their interaction energy. This study is based on the notion of "global Jacobian" detecting the topological defects that a priori could be located in the interior and at the boundary of the film. A major difficulty consists in estimating the nonlocal part of the micromagnetic energy in order to isolate the exact terms corresponding to the topological defects. We prove the concentration of the energy around boundary vortices via a $Γ$-convergence expansion at the second order. The second order term is the renormalized energy that represents the interaction between the boundary vortices and governs their optimal position. We compute the expression of the renormalized energy for which we prove the existence of minimizers having two boundary vortices of multiplicity $1$. Compactness results are also shown for the magnetization and the corresponding global Jacobian.
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