土星环缺陷周围的球形粒子沉浸在液晶
Saturn ring defect around a spherical particle immersed in nematic liquid crystal
摘要：
我们考虑一种在球形胶体颗粒外部占据三维区域的液晶。该粒子受到狄利克雷边界条件的约束，该边界条件强制粒子分子与粒子表面正交附着。我们的主要兴趣是理解Landau-de Gennes $Q$ -张量模型在零关联长度极限下的能量临界组态的行为。我们证明存在的配置与一个单一的土星环缺陷接近赤道的粒子，没有其他线或点缺陷。我们通过分析两个对称约束下的能量最小化的渐近性：围绕垂直轴的旋转等方差和在水平面上的反射。在环缺陷处的能量爆炸是构造消除点缺陷的可能性所需的良好行为的比较图的重大障碍。我们开发的边界估计，以解决这个问题是新的，应该适用于更广泛的一类问题。
英文摘要：
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.
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