论文标题
Landau-de Gennes模型的圆环样解决方案。第二部分：$ \ mathbb {s}^1 $ - equivariant minimizers的拓扑
Torus-like solutions for the Landau-de Gennes model. Part II: Topology of $\mathbb{S}^1$-equivariant minimizers
论文作者
论文摘要
我们研究了在三维轴对称结构域以及受限制类别的$ \ mathbb {s}^1 $ -Equivariant（即轴向对称）配置中，在三维轴对称结构域以及在受限类的$ \ mathbb {s}^1 $ \ mathbb {s}^1 $ \ mathbb中，能量最小化。我们假设光滑且非趋势的$ \ Mathbb {s}^1 $ -Equivariant（例如同型）dirichlet边界条件和内部具有物理相关的规范约束（Lyuksyutov约束）。依赖于\ cite {dmp1}的结果，在非对称设置中，我们证明了最小化器的部分规律性，而不是在对称轴上的一组有限的内部奇异点上。对于适当的域和边界数据，我们表明，对于平滑的最小化器（圆环解决方案），签名的双轴性的水平集是革命的摩托车的一般有限结合。 Concerning nonsmooth minimizers (split solutions), we characterize their asymptotic behavior around any singular point in terms of explicit $\mathbb{S}^1$-equivariant harmonic maps into $\mathbb{S}^4$, whence the generic level sets of the signed biaxiality contains invariant topological spheres.最后，在列液滴的模型情况下，我们至少在边界数据是径向锚固的单轴单轴变形时，并存在用于边界数据的拆分溶液的存在，而边界数据是合适的线性充分谐波球。
We study energy minimization of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in three-dimensional axisymmetric domains and in a restricted class of $\mathbb{S}^1$-equivariant (i.e., axially symmetric) configurations. We assume smooth and nonvanishing $\mathbb{S}^1$-equivariant (e.g. homeotropic) Dirichlet boundary conditions and a physically relevant norm constraint (Lyuksyutov constraint) in the interior. Relying on results in \cite{DMP1} in the nonsymmetric setting, we prove partial regularity of minimizers away from a possible finite set of interior singularities lying on the symmetry axis. For a suitable class of domains and boundary data we show that for smooth minimizers (torus solutions) the level sets of the signed biaxiality are generically finite union of tori of revolution. Concerning nonsmooth minimizers (split solutions), we characterize their asymptotic behavior around any singular point in terms of explicit $\mathbb{S}^1$-equivariant harmonic maps into $\mathbb{S}^4$, whence the generic level sets of the signed biaxiality contains invariant topological spheres. Finally, in the model case of a nematic droplet, we provide existence of torus solutions, at least when the boundary data are suitable uniaxial deformations of the radial anchoring, and existence of split solutions for boundary data which are suitable linearly full harmonic spheres.
