Dimension reduction for the Landau-de Gennes model on curved nematic thin films

TL;DR

This paper employs $mma$-convergence to analyze the asymptotic behavior of the Landau-de Gennes model for nematic liquid crystals on curved thin films, revealing the influence of curvature and anchoring conditions on minimizers.

Contribution

It generalizes previous planar surface analysis to curved surfaces, incorporating curvature effects and weak anchoring conditions in the limiting energy framework.

Findings

Convergence to a surface energy involving the normal component of the tensor gradient.

Curvature impacts the structure of minimizers, as shown in the frustrum example.

Anchoring energy dominates in the thin film limit, shaping admissible configurations.

Abstract

We use the method of $\Gamma$-convergence to study the behavior of the Landau-de Gennes model for a nematic liquid crystalline film attached to a general fixed surface in the limit of vanishing thickness. This paper generalizes the approach that we used previously to study a similar problem for a planar surface. Since the anchoring energy dominates when the thickness of the film is small, it is essential to understand its influence on the structure of the minimizers of the limiting energy. In particular, the anchoring energy dictates the class of admissible competitors and the structure of the limiting problem. We assume general weak anchoring conditions on the top and the bottom surfaces of the film and strong Dirichlet boundary conditions on the lateral boundary of the film when the surface is not closed. We establish a general convergence result to an energy defined on the surface that involves a somewhat surprising remnant of the normal component of the tensor gradient. Then we exhibit one effect of curvature through an analysis of the behavior of minimizers to the limiting problem when the substrate is a frustrum.