# Symmetry and multiplicity of solutions in a two-dimensional Landau-de Gennes model for liquid crystals

## TL;DR

This paper analyzes a 2D Landau-de Gennes model for nematic liquid crystals, proving the existence of exactly two minimizers under certain symmetric boundary conditions with defects, and identifying multiple symmetric critical points.

## Contribution

It establishes the exact number of minimizers and symmetric critical points in a 2D liquid crystal model with topological defects, revealing symmetry-breaking phenomena.

## Key findings

- Exactly two minimizers for large R under symmetric boundary conditions.
- Minimizers do not preserve full symmetry of the boundary data.
- Existence of at least five symmetric critical points.

## Abstract

We consider a variational two-dimensional Landau-de Gennes model in the theory of nematic liquid crystals in a disk of radius $R$. We prove that under a symmetric boundary condition carrying a topological defect of degree $\frac{k}{2}$ for some given {\bf even} non-zero integer $k$, there are exactly two minimizers for all large enough $R$. We show that the minimizers do not inherit the full symmetry structure of the energy functional and the boundary data. We further show that there are at least five symmetric critical points.