Torus-like solutions for the Landau-de Gennes model. Part II: Topology of $\mathbb{S}^1$-equivariant minimizers

Federico Dipasquale, Vincent Millot, Adriano Pisante

Published 2020-08-31Version 1

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 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. In case of a nematic droplet, we provide existence of torus solutions, at least when the boundary data are suitably chosen continuous deformations of the radial anchoring. Concerning nonsmooth minimizers (split solutions), we characterize explicitely 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. In the companion paper \cite{DMP2}, we will show how singular solutions or smooth solutions (or even both) for the Euler-Lagrange equations do appear as minimizers under radial anchoring boundary data when the domains are suitable deformation of a spherical droplet.