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

We study small energy solutions within the Landau-de Gennes theory for nematic liquid crystals, subject to Dirichlet boundary conditions. We consider two-dimensional and three-dimensional domains separately. In the two-dimensional case, we establish the equivalence of the Landau-de Gennes and Ginzburg-Landau theory. In the three-dimensional case, we give a new definition of the defect set based on the normalized energy. In the three-dimensional uniaxial case, we demonstrate the equivalence between the defect set and the isotropic set and prove the $C^{1,\alpha}$-convergence of uniaxial small energy solutions to a limiting harmonic map, away from the defect set, for some $0 vanishing core limit . Generalizations for biaxial small energy solutions are also discussed, which include physically relevant estimates for the solution and its scalar order parameters. This work is motivated by the study of defects in liquid crystalline systems and their applications.