凸多面体中液晶的弹性能
Elastic energy of liquid crystals in convex polyhedra
摘要：
我们考虑在一个有界的凸多面体中的液晶液晶，该多面体由指向矢场n（r）描述，且满足切向边界条件。我们推导出一个常数的弹性能量的拓扑不变量的下界。对于一个直角棱镜和一个大类的拓扑结构，我们推导出上界引入测试配置构造的局部共形解的欧拉-拉格朗日方程。上下边界的比值仅取决于棱镜的纵横比。随着纵横比的变化，最小能量共形态经历了一个急剧的过渡，从光滑的边缘上有奇点。
英文摘要：
We consider nematic liquid crystals in a bounded, convex polyhedron described by a director field n(r) subject to tangent boundary conditions. We derive lower bounds for the one-constant elastic energy in terms of topological invariants. For a right rectangular prism and a large class of topologies, we derive upper bounds by introducing test configurations constructed from local conformal solutions of the Euler-Lagrange equation. The ratio of the upper and lower bounds depends only on the aspect ratios of the prism. As the aspect ratio is varied, the minimum-energy conformal state undergoes a sharp transition from being smooth to having singularities on the edges.
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