向列相壳中的缺陷：离散到连续的$Γ$收敛方法
Defects in Nematic Shells: a $Γ$-convergence discrete-to-continuum approach
摘要：
在本文中，我们严格调查的出现缺陷的向列壳的亏格不同。这种现象与壳层的拓扑结构和指向矢场的排列之间的非平凡相互作用有关。为此，我们考虑一个离散的$XY$系统上的壳$M$，由一个切向量字段与单位范数坐在一个三角形的壳的顶点。当我们让三角剖分的网格步长为零时，即在离散到连续极限时，就会出现缺陷。在本文中，我们研究了离散到连续极限的$Γ$ -收敛在两个不同的渐近制度。第一个缩放促进了有限数量的缺陷的出现，其电荷是根据壳$M$的拓扑结构，通过庞加莱-霍普夫定理。第二个标度产生了所谓的重整化能量，它控制着有缺陷的构型的平衡。
英文摘要：
In this paper we rigorously investigate the emergence of defects on Nematic Shells with genus different from one. This phenomenon is related to a non trivial interplay between the topology of the shell and the alignment of the director field. To this end, we consider a discrete $XY$ system on the shell $M$, described by a tangent vector field with unit norm sitting at the vertices of a triangulation of the shell. Defects emerge when we let the mesh size of the triangulation go to zero, namely in the discrete-to-continuum limit. In this paper we investigate the discrete-to-continuum limit in terms of $Γ$-convergence in two different asymptotic regimes. The first scaling promotes the appearance of a finite number of defects whose charges are in accordance with the topology of shell $M$, via the Poincaré-Hopf Theorem. The second scaling produces the so called Renormalized Energy that governs the equilibrium of the configurations with defects.
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