Toruslike solutions for the Landaude Gennes model. Part II: Topology of $\mathbb{S}^1$equivariant minimizers
Abstract
We study energy minimization of a continuum Landaude Gennes energy functional for nematic liquid crystals, in threedimensional 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2020
 arXiv:
 arXiv:2008.13676
 Bibcode:
 2020arXiv200813676D
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 35J50;
 35B40 (Primary) 82D30;
 76A15 (Secondary)
 EPrint:
 Proposition 6.1 added