Dimension Reduction for the Landau-de Gennes Model: The Vanishing Nematic Correlation Length Limit

TL;DR

This paper analyzes the behavior of nematic liquid crystalline films in the limit of vanishing thickness and correlation length, establishing a rigorous mathematical framework for their energy minimization and structure.

Contribution

It proves $ ext{Gamma}$-convergence for a sequence of singularly perturbed Landau-de Gennes functionals, revealing the limiting energy structure and existence of local minimizers.

Findings

Limiting energy includes perimeter and vortex-related terms.

Existence of local minimizers despite high-dimensional wells.

Connection to Allen-Cahn and Ginzburg-Landau models.

Abstract

We study nematic liquid crystalline films within the framework of the Landau-de Gennes theory in the limit when both the thickness of the film and the nematic correlation length are vanishingly small compared to the lateral extent of the film. We prove $\Gamma$-convergence for a sequence of singularly perturbed functionals with a potential vanishing on a high-dimensional set and a Dirichlet condition imposed on admissible functions. This then allows us to prove the existence of local minimizers of the Landau-de Gennes energy in the spirit of a theorem due to Kohn and Sternberg despite the lack of compactness arising from the high-dimensional structure of the wells. The limiting energy consists of leading order perimeter terms, similar to Allen-Cahn models, and lower order terms arising from vortex structures reminiscent of Ginzburg-Landau models.