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

摘要： 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.