铁磁薄膜中非局域相互作用畴壁的分离及其重正化能的$Γ$ -收敛
Separation of domain walls with nonlocal interaction and their renormalised energy by $Γ$-convergence in thin ferromagnetic films
摘要：
我们分析了两个变种的非凸变分模型从微磁学与非局部能量泛函，取决于一个小参数$ε> 0$。该模型产生的过渡层，称为Néel墙，我们研究他们的行为限制$ε\to 0$。该分析与Ginzburg-Landau涡旋理论有相似之处。特别是，它产生了一个重新归一化的能量，决定了Néel壁之间的相互作用（吸引或排斥），以领先秩序。但是，尽管金兹伯格-朗道涡旋对相同符号的程度表现出吸引力，对相反符号的程度表现出排斥力，但在这个模型中，这种模式是相反的。在以前的论文中，我们在假设Néel壁彼此分离的情况下确定了这里研究的一个模型的重整化能量。在本文中，我们提出了一个更深入的分析，特别是消除这一假设。该理论产生了一个有效的变分问题的墙壁的位置，封装在一个$Γ$收敛的结果。在本文的第二部分，我们将注意力转向另一个更物理的模型，包括各向异性项。我们表明，它允许一个类似的理论，但各向异性改变了重整化能量在意想不到的方式，需要不同的方法来找到它。
英文摘要：
We analyse two variants of a nonconvex variational model from micromagnetics with a nonlocal energy functional, depending on a small parameter $ε> 0$. The model gives rise to transition layers, called Néel walls, and we study their behaviour in the limit $ε\to 0$. The analysis has some similarity to the theory of Ginzburg-Landau vortices. In particular, it gives rise to a renormalised energy that determines the interaction (attraction or repulsion) between Néel walls to leading order. But while Ginzburg-Landau vortices show attraction for degrees of the same sign and repulsion for degrees of opposite signs, the pattern is reversed in this model. In a previous paper, we determined the renormalised energy for one of the models studied here under the assumption that the Néel walls stay separated from each other. In this paper, we present a deeper analysis that in particular removes this assumption. The theory gives rise to an effective variational problem for the positions of the walls, encapsulated in a $Γ$-convergence result. In the second part of the paper, we turn our attention to another, more physical model, including an anisotropy term. We show that it permits a similar theory, but the anisotropy changes the renormalised energy in unexpected ways and requires different methods to find it.
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