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arXiv · 1801.04477

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

Michael R. Novack
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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.

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Michael R. Novack. 2018-01-13. Dimension Reduction for the Landau-de Gennes Model: The Vanishing Nematic Correlation Length Limit. https://arxiv.org/abs/1801.04477

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