Landau–De Gennes Theory of Nematic Liquid Crystals: the Oseen–Frank Limit and Beyond
KAUST Grant NumberKUK-C1-013-04
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AbstractWe study global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals, in three-dimensional domains, subject to uniaxial boundary conditions. We analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly, in W1,2, to a global minimizer predicted by the Oseen-Frank theory for uniaxial nematic liquid crystals with constant order parameter. Moreover, the convergence is uniform in the interior of the domain, away from the singularities of the limiting Oseen-Frank global minimizer. We obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the Landau-De Gennes global minimizer. We also study the interplay between biaxiality and uniaxiality in Landau-De Gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions. © Springer-Verlag (2009).
CitationMajumdar A, Zarnescu A (2009) Landau–De Gennes Theory of Nematic Liquid Crystals: the Oseen–Frank Limit and Beyond. Archive for Rational Mechanics and Analysis 196: 227–280. Available: http://dx.doi.org/10.1007/s00205-009-0249-2.
SponsorsA. Majumdar was supported by a Royal Commission for the Exhibition of 1851 Research Fellowship till October 2008. She is now supported by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST), to the Oxford Centre for Collaborative Applied Mathematics. A. Zarnescu is supported by the EPSRC Grant EP/E010288/1-Equilibrium Liquid Crystal Configurations: Energetics, Singularities and Applications. We thank John M. Ball and Christ of Melcher for stimulating discussions.