论文标题
土星环形在一个浸入列明液晶的球形粒子周围
Saturn ring defect around a spherical particle immersed in nematic liquid crystal
论文作者
论文摘要
我们认为列出了一个列在球形胶体颗粒外部的三维结构域的列液晶体。列表受dirichlet边界条件的约束，该条件会强制列内分子在粒子表面上的正交附着。我们的主要兴趣是了解Landau-de Gennes $ Q $ tensor模型的能源关键构型的行为，该模型在消失的相关长度的极限下。我们证明了具有单个土星的缺陷的配置，即接近粒子的赤道，而没有其他线或点缺陷。我们通过在两个对称约束下分析能量最小化器的渐近造型来证明这一点：围绕垂直轴的旋转模棱两可和整个水平平面的反射。环形缺陷处的能量爆炸是构建行为良好的比较图所需的重要障碍，以消除点缺陷的可能性。我们为解决此问题而开发的边界估计是新的，应该适用于更广泛的问题。
We consider a nematic liquid crystal occupying the three-dimensional domain in the exterior of a spherical colloid particle. The nematic is subject to Dirichlet boundary conditions that enforce orthogonal attachment of nematic molecules to the surface of the particle. Our main interest is to understand the behavior of energy-critical configurations of the Landau-de Gennes $Q$-tensor model in the limit of vanishing correlation length. We demonstrate existence of configurations with a single Saturn-ring defect approaching the equator of the particle and no other line or point defects. We show this by analyzing asymptotics of energy minimizers under two symmetry constraints: rotational equivariance around the vertical axis and reflection across the horizontal plane. Energy blow-up at the ring defect is a significant obstacle to constructing well-behaved comparison maps needed to eliminate the possibility of point defects. The boundary estimates we develop to address this issue are new and should be applicable to a wider class of problems.
