液晶与多面体畴中的调和映射
Liquid crystals and harmonic maps in polyhedral domains
摘要：
凸多面体$P$上的单位矢量场$\nvec$服从切线边界条件，提供了原型LCD显示器中的LCD液晶的简单模型。平衡和亚稳构型对应于Dirichlet能量的极小值和局部极小值，可以看作是$P$上的$S^2$值调和映射.我们考虑从$P$的顶点开始连续的单位向量场。对于给定的同伦类的下确界Dirichlet能量的下界得到的分数缺陷之间的最小连接的总和在$P$的顶点。在某些情况下，这个下界可以通过引入某些非阿贝尔同伦不变量来改进。对于矩形棱柱，从Euler-Lagrange方程的局部共形解得到了下确界Dirichlet能量的上界，上界和下界的比值与同伦类型无关.然而，由于同伦类不是弱闭的，下确界可能无法实现;给定同伦类型的连续局部极小的存在性和正则性是一个悬而未决的问题。数值结果表明，一些同伦类总是包含光滑极小，而其他人可能会或可能不会取决于$P$的几何形状。数值模拟结果表明，所观察到的双折射配置可以区分拓扑。
英文摘要：
Unit-vector fields $\nvec$ on a convex polyhedron $P$ subject to tangent boundary conditions provide a simple model of nematic liquid crystals in prototype bistable displays. The equilibrium and metastable configurations correspond to minimisers and local minimisers of the Dirichlet energy, and may be regarded as $S^2$-valued harmonic maps on $P$. We consider unit-vector fields which are continuous away from the vertices of $P$. A lower bound for the infimum Dirichlet energy for a given homotopy class is obtained as a sum of minimal connections between fractional defects at the vertices of $P$. In certain cases, this lower bound can be improved by incorporating certain nonabelian homotopy invariants. For a rectangular prism, upper bounds for the infimum Dirichlet energy are obtained from locally conformal solutions of the Euler-Lagrange equations, with the ratio of the upper and lower bounds bounded independently of homotopy type. However, since the homotopy classes are not weakly closed, the infimum may not be realised; the existence and regularity properties of continuous local minimisers of given homotopy type are open questions. Numerical results suggest that some homotopy classes always contain smooth minimisers, while others may or may not depending on the geometry of $P$. Numerical results modelling a bistable device suggest that the observed nematic configurations may be distinguished topologically.
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