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

TL;DR

This paper investigates the minimizers of the Landau-de Gennes energy in nematic liquid crystals, showing how domain or boundary deformations lead to either torus or split solutions, revealing symmetry-breaking phenomena.

Contribution

It demonstrates the emergence of torus and split minimizers through domain or boundary deformations, extending previous results on nematic liquid crystal configurations.

Findings

Existence of both torus and split minimizers under certain deformations

Deformation of domain or boundary data can induce symmetry breaking

Singular split solutions can be energy minimizers in specific settings

Abstract

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 domains diffeomorphic to a ball (a nematic droplet) and in a restricted class of $\bbS^1$-equivariant configurations. It is known from our previous paper \cite{DMP2} 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 \emph{torus} or of \emph{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, when minimizers among $\bbS^1$-equivariant configurations are singular we derive symmetry breaking result for the minimization among all competitors.