Torus-like Solutions for the Landau-de Gennes Model. Part I: The Lyuksyutov Regime

We study global minimizers of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in three-dimensional domains, under a Dirichlet boundary condition. In a relevant range of parameters (which we call the Lyuksyutov regime), the main result establishes the nontrivial topology of the biaxiality sets of minimizers for a large class of boundary conditions including the homeotropic boundary data. To achieve this result, we first study minimizers subject to a physically relevant norm constraint (the Lyuksyutov constraint), and show their regularity up to the boundary. From this regularity, we rigorously derive the norm constraint from the asymptotic Lyuksyutov regime. As a consequence, isotropic melting is avoided by unconstrained minimizers in this regime, which then allows us to analyse their biaxiality sets. In the case of a nematic droplet, this also implies that the radial hedgehog is an unstable equilibrium in the same regime of parameters. Technical results of this paper will be largely employed in Dipasquale et al. (Torus-like solutions for the Landau- de Gennes model. Part II: topology of S-1-equivariant minimizers. ; Torus-like solutions for the Landau- de Gennes model. Part III: torus solutions vs split solutions (In preparation)), where we prove that biaxiality level sets are generically finite unions of tori for smooth configurations minimizing the energy in restricted classes of axially symmetric maps satisfying a topologically nontrivial boundary condition.

Automated summary

In this study, the global minimizers of a continuum Landau-de Gennes energy functional for nematic liquid crystals in three-dimensional domains under Dirichlet boundary conditions are investigated. The nontrivial topology of the biaxiality sets of minimizers is established in a specific parameter range, known as the Lyuksyutov regime. The results provide insights into the regularity and behavior of minimizers, particularly in relation to isotropic melting and the stability of equilibrium configurations.