Minimizers of a Landau-de Gennes Energy with a Subquadratic Elastic Energy

TL;DR

This paper investigates a modified Landau-de Gennes model with subquadratic elastic energy growth, analyzing minimizers' behavior, convergence, and biaxial core formation in nematic liquid crystals in low-temperature regimes.

Contribution

It introduces a subquadratic elastic energy term in the Landau-de Gennes model and studies the resulting minimizers' properties and singularity structures.

Findings

Uniform convergence of minimizers and gradients away from singularities

Presence of maximally biaxial cores at low temperatures in 2D

Analysis in small defect core regimes

Abstract

We study a modified Landau-de Gennes model for nematic liquid crystals, where the elastic term is assumed to be of subquadratic growth in the gradient. We analyze the behaviour of global minimizers in two- and three-dimensional domains, subject to uniaxial boundary conditions, in the asymptotic regime where the length scale of the defect cores is small compared to the length scale of the domain. We obtain uniform convergence of the minimizers and of their gradients, away from the singularities of the limiting uniaxial map. We also demonstrate the presence of maximally biaxial cores in minimizers on two-dimensional domains, when the temperature is sufficiently low.