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

Abstract

We study nematic liquid crystalline films within the framework of the Landaude 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 Γ-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 [R. V. Kohn and P. Sternberg, Local minimisers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), pp. 69-84] 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.

A Model Problem for Nematic-Isotropic Transitions with Highly Disparate Elastic Constants

“…We will deal with u ε on {x : −m 2/3 ε < d(x) 0} and on R ε at the end. It can be shown, by calculations similar to those preceding [31,Equation 3.33] that lim sup…”

“…As observed in [31], this rather weak form of compactness is nonetheless sufficient to imply the existence of local minimizers of F ε given a local minimizer of F 0 which is isolated in this weaker topology, by modifying an argument of [22]. For example, on a "dumbbell"-type domain, there always exist local minimizers of F ε for ε sufficiently small, cf.…”

“…For example, on a "dumbbell"-type domain, there always exist local minimizers of F ε for ε sufficiently small, cf. [31,Theorems 4.2,5.1].…”

Active nematodynamics on curved surfaces -- the influence of geometric forces on motion patterns of topological defects

“…Given the discussion of [25] we highlight the impact of the choice of β. With β = 0 we yield a surface model for uniaxial nematics [19,16], which essentially has the same properties as a two-dimensional model, while β = 0 results in thin film models of nematics [24,14,15,35,28,31] with several properties shared with three-dimensional models. The differences result from the consideration of extrinsic curvature contributions.…”

Variational Analysis of Nematic Shells

“…For other dimension reduction results which apply to liquid crystals, see e.g. [24,36]. (In these papers, the authors adopt a different modelling framework for the liquid crystals, i.e.…”