Landau-De Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond

TL;DR

This paper investigates the behavior of Landau-De Gennes energy minimizers for nematic liquid crystals, demonstrating convergence to Oseen-Frank minimizers in the small elastic constant limit and exploring biaxiality effects.

Contribution

It provides a rigorous analysis of the convergence of Landau-De Gennes minimizers to Oseen-Frank solutions and examines biaxiality in three-dimensional domains.

Findings

Global minimizers converge strongly to Oseen-Frank minimizers in the small elastic constant limit.

Convergence is uniform away from singularities of the limiting solution.

Estimates on biaxiality parameters and strongly biaxial regions are obtained.

Abstract

We study global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals, in three-dimensional domains, subject to uniaxial boundary conditions. We analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly, in $W^{1,2}$, to a global minimizer predicted by the Oseen-Frank theory for uniaxial nematic liquid crystals with constant order parameter. Moreover, the convergence is uniform in the interior of the domain, away from the singularities of the limiting Oseen-Frank global minimizer. We obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the Landau-De Gennes global minimizer. We also study the interplay between biaxiality and uniaxiality in Landau-De Gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions.