SIAM Journal on Applied Mathematics, Volume 80, Issue 4, Page 1678-1703, January 2020.
We study reduced nematic equilibria on regular two-dimensional polygons with Dirichlet tangent boundary conditions in a reduced two-dimensional Landau--de Gennes framework, discussing their relevance in the full three-dimensional framework too. We work at a fixed temperature and study the reduced stable equilibria in terms of the edge length, $\lambda$, of the regular polygon, $E_K$, with $K$ edges. We analytically compute a novel “ring solution” in the $\lambda \to 0$ limit, with a unique point defect at the center of the polygon for $K \neq 4$. The ring solution is unique. For sufficiently large $\lambda$, we deduce the existence of at least $[K/2 ]$ classes of stable equilibria and numerically compute bifurcation diagrams for reduced equilibria on a pentagon and hexagon, as a function of $\lambda^2$, thus illustrating the effects of geometry on the structure, locations, and dimensionality of defects in this framework.

中文翻译：

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二维多边形向列平衡的简化研究

SIAM应用数学杂志，第80卷，第4期，第1678-1703页，2020年1月。
我们在简化的二维Landau-de Gennes框架中研究具有Dirichlet切线边界条件的规则二维多边形的简化向列平衡，并讨论它们在整个三维框架中的相关性。我们在固定温度下工作，并根据具有$ K $边的规则多边形$ E_K $的边长$ \ lambda $来研究降低的稳定平衡。我们以$ \ lambda \至0 $的极限分析计算出一种新颖的“环解”，对于$ K \ neq 4 $，在多边形中心具有唯一的点缺陷。环网解决方案是独一无二的。对于足够大的$ \ lambda $，我们推导出至少$ [K / 2] $类稳定均衡的存在，并根据$ \ lambda ^ 2 $的函数对五边形和六边形上的简化均衡进行数值计算的分叉图。 ，