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

2018; Society for Industrial and Applied Mathematics; Volume: 50; Issue: 6 Linguagem: Inglês

10.1137/18m1165189

1095-7154

Michael Novack,

Stochastic processes and statistical mechanics

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 [R. V. Kohn and P. Sternberg, 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.