Torus-like solutions for the Landau-de Gennes model. Part III: torus vs split minimizers

We study the behaviour of global minimizers of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in three-dimensional axially symmetric domains diffeomorphic to a ball (a nematic droplet) and in a restricted class of S1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}<^>1$$\end{document}-equivariant configurations. It is known from our previous paper (Dipasquale et al. in J Funct Anal 286:110314, 2024) that, assuming smooth and uniaxial (e.g. homeotropic) boundary conditions and a physically relevant norm constraint in the interior (Lyuksyutov constraint), minimizing configurations are either of torus or of split type. Here, starting from a nematic droplet with the homeotropic boundary condition, we show how singular (split) solutions or smooth (torus) solutions (or even both) for the Euler-Lagrange equations do appear as energy minimizers by suitably deforming either the domain or the boundary data. As a consequence, we derive symmetry breaking results for the minimization among all competitors.