二维多边形外简化Nematic Liquid Crystal模型的多稳定性
Multistability for a Reduced Nematic Liquid Crystal Model in the Exterior of 2D Polygons
摘要：
在一个简化的Landau-de Gennes框架下，研究了无界区域中具有带$K$边的二维正多边形孔洞的平衡问题.这补充了我们之前关于限制在正多边形内的非线性均衡的“内部问题”的工作（SIAM Journal on Applied Mathematics，80（4）：1678-1703，2020）。两个基本的无量纲模型参数是$λ$ --多边形孔的边长与微扰相关长度的比值，以及一个附加的自由度，$γ^*$ --无穷远处的微扰指向矢。在$λ\to 0$极限中，除了三角形和正方形之外，极限轮廓在通用多边形孔外部具有两个内部点缺陷。对于正方形孔，极限轮廓要么没有内部缺陷，要么有两个线缺陷，这取决于$γ^*$，而对于三角形孔，在孔外存在唯一的内部点缺陷。在$λ\to\infty$极限中，至少存在由两个弯曲顶点的位置区分的$\binom{K}{2}$稳定状态%，并且与内部问题相比，多稳定性通过$γ^*$增强。我们的工作为如何调整缺陷的存在、位置和维度提供了新的见解。
英文摘要：
We study nematic equilibria in an unbounded domain, with a two-dimensional regular polygonal hole with $K$ edges, in a reduced Landau-de Gennes framework. This complements our previous work on the "interior problem" for nematic equilibria confined inside regular polygons (SIAM Journal on Applied Mathematics, 80(4):1678-1703, 2020). The two essential dimensionless model parameters are $λ$--the ratio of the edge length of polygon hole to the nematic correlation length, and an additional degree of freedom, $γ^*$--the nematic director at infinity. In the $λ\to 0$ limit, the limiting profile has two interior point defects outside a generic polygon hole, except for a triangle and a square. For a square hole, the limiting profile has either no interior defects or two line defects depending on $γ^*$, and for a triangular hole, there is a unique interior point defect outside the hole. In the $λ\to\infty$ limit, there are at least $\binom{K}{2}$ stable states %differentiated by the location of two bend vertices and the multistability is enhanced by $γ^*$, compared to the interior problem. Our work offers new insights into how to tune the existence, location, and dimensionality of defects.
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