Liquid crystal defects in the Landau-de Gennes theory in two dimensions-beyond the one-constant approximation

TL;DR

This paper analyzes two-dimensional liquid crystal defects within the Landau-de Gennes framework, exploring how elastic constants influence the existence, stability, and structure of symmetric critical points beyond the one-constant approximation.

Contribution

It extends the analysis of liquid crystal defects by considering multiple elastic constants and characterizing the existence and stability of symmetric solutions in two dimensions.

Findings

Critical points exist only for k=2 with symmetric boundary conditions.

Identified three types of radial profiles with different component configurations.

Numerical results show stability depends on elastic constants and physical regimes.

Abstract

We consider the two-dimensional Landau-de Gennes energy with several elastic constants, subject to general $k$-radially symmetric boundary conditions. We show that for generic elastic constants the critical points consistent with the symmetry of the boundary conditions exist only in the case $k=2$. In this case we identify three types of radial profiles: with two, three of full five components and numerically investigate their minimality and stability depending on suitable parameters. We also numerically study the stability properties of the critical points of the Landau-de Gennes energy and capture the intricate dependence of various qualitative features of these solutions on the elastic constants and the physical regimes of the liquid crystal system.