The Landau-de Gennes theory of nematic liquid crystals: Uniaxiality versus Biaxiality

Abstract

We study small energy solutions within the Landau-de Gennestheory for nematic liquid crystals, subject to Dirichlet boundaryconditions. We consider two-dimensional and three-dimensionaldomains separately. In the two-dimensional case, we establish theequivalence of the Landau-de Gennes and Ginzburg-Landau theory. Inthe three-dimensional case, we give a new definition of thedefect set based on the normalized energy. In thethree-dimensional uniaxial case, we demonstrate the equivalencebetween the defect set and the isotropic set and prove the$C^{1,\alpha}$-convergence of uniaxial small energy solutions to alimiting harmonic map, away from the defect set, for some$0 vanishing core limit. Generalizationsfor biaxial small energy solutions are also discussed, whichinclude physically relevant estimates for the solution and itsscalar order parameters. This work is motivated by the study ofdefects in liquid crystalline systems and their applications.