Pattern Formation for Nematic Liquid Crystals-Modelling, Analysis, and Applications

TL;DR

This paper reviews recent advances in understanding solution landscapes for 2D nematic liquid crystals within the Landau--de Gennes framework, highlighting the effects of geometry, anisotropy, and symmetry on solution multiplicity and regularity.

Contribution

It provides a comprehensive overview of recent analytical, variational, and computational results on solution multiplicity and regularity in 2D nematic liquid crystals, emphasizing the influence of geometry and material properties.

Findings

Multiplicity of solutions depends on domain shape and boundary conditions.

Regularity results vary with asymptotic limits and material anisotropy.

Exotic ordering transitions are possible in 2D nematic frameworks.

Abstract

We summarise some recent results on solution landscapes for two-dimensional (2D) problems in the Landau--de Gennes theory for nematic liquid crystals. We study energy-minimizing and non energy-minimizing solutions of the Euler--Lagrange equations associated with a reduced Landau-de Gennes free energy on 2D domains with Dirichlet tangent boundary conditions. We review results on the multiplicity and regularity of solutions in distinguished asymptotic limits, using variational methods, methods from the theory of nonlinear partial differential equations, combinatorial arguments and scientific computation. The results beautifully canvass the competing effects of geometry (shape, size and symmetry), material anisotropy, and the symmetry of the model itself, illustrating the tremendous possibilities for exotic ordering transitions in 2D frameworks.