论文标题
弯曲的手性dirichlet能量的薄膜极限
Curved thin-film limits of chiral Dirichlet energies
论文作者
论文摘要
我们调查了一个在$ h^1 $ sobolev映射的空间中，在$（n-1）$ - dimensional submanifold $ n $ of $ h^1 $ h^1 $ sobolev映射的空间中，弯曲的薄膜极限$ \ mathbb {r}^m $。我们考虑的扰动$ \ mathsf {k} $由$ m $上定义的矩阵值函数表示，并在$ \ mathbb {r}^{m \ times n} $中以值表示。在自然规律性的假设上，$ n $，$ m $和$ \ m rathsf {k} $，我们表明，从$γ$ convergence的意义上讲，这些能量的家族会收敛于意外形式的$ n $，这对磁性天际理论特别感兴趣。作为我们结果的副产品，我们得到的是，在弯曲的薄膜极限下，反对称交换相互作用也以各向异性术语表现出来，其特异性既取决于薄膜的曲率和目标歧管的曲率。微磁性变分理论中各种类型的反对称交换相互作用是我们工作的灵感和动机的来源。
We investigate the curved thin-film limit of a family of perturbed Dirichlet energies in the space of $H^1$ Sobolev maps defined in a tubular neighborhood of an $(n - 1)$-dimensional submanifold $N$ of $\mathbb{R}^n$ and with values in an $(m - 1)$-dimensional submanifold $M$ of $\mathbb{R}^m$. The perturbation $\mathsf{K}$ that we consider is represented by a matrix-valued function defined on $M$ and with values in $\mathbb{R}^{m \times n}$. Under natural regularity hypotheses on $N$, $M$, and $\mathsf{K}$, we show that the family of these energies converges, in the sense of $Γ$-convergence, to an energy functional on $N$ of an unexpected form, which is of particular interest in the theory of magnetic skyrmions. As a byproduct of our results, we get that in the curved thin-film limit, antisymmetric exchange interactions also manifest under an anisotropic term whose specific shape depends both on the curvature of the thin film and the curvature of the target manifold. Various types of antisymmetric exchange interactions in the variational theory of micromagnetism are a source of inspiration and motivation for our work.
