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首页> 外文期刊>Communications on pure and applied analysis >The landau-de gennes theory of nematic liquid crystals: Uniaxiality versus biaxiality
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The landau-de gennes theory of nematic liquid crystals: Uniaxiality versus biaxiality

机译：向列液晶的Landau-de gennes理论：单轴性与双轴性

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摘要

We study small energy solutions within the Landau-de Gennes theory for nematic liquid crystals, subject to Dirichlet boundary conditions. We consider two-dimensional and three-dimensional domains separately. In the two-dimensional case, we establish the equivalence of the Landau-de Gennes and Ginzburg-Landau theory. In the three-dimensional case, we give a new definition of the defect set based on the normalized energy. In the threedimensional uniaxial case, we demonstrate the equivalence between the defect set and the isotropic set and prove the C ~(1,α)-convergence of uniaxial small energy solutions to a limiting harmonic map, away from the defect set, for some 0 < a < 1, in the vanishing core limit. Generalizations for biaxial small energy solutions are also discussed, which include physically relevant estimates for the solution and its scalar order parameters. This work is motivated by the study of defects in liquid crystalline systems and their applications.
机译：在Dirichlet边界条件的约束下，我们研究了向列型液晶的Landau-de Gennes理论中的小能量解。我们分别考虑二维域和三维域。在二维情况下，我们建立了Landau-de Gennes和Ginzburg-Landau理论的等价性。在三维情况下，我们基于归一化的能量给出了缺陷集的新定义。在三维单轴情况下，我们证明了缺陷集和各向同性集之间的等价关系，并证明了单轴小能量解的C〜（1，α）-收敛于远离缺陷集的极限谐波图上的约0 <1，在消失的核心限制中。还讨论了双轴小能量解的推广，其中包括该解及其标量阶数参数的物理相关估计。这项工作的动机是研究液晶系统中的缺陷及其应用。

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