Uniaxiality in the Landau-de Gennes theory of nematic liquid crystals

Abstract: We study uniaxial energy minimizers within the Landau-de Gennes theory for nematic liquid crystals, subject to dirichlet boundary conditions. Topological defects in such minimizers correspond to the zeros of the corresponding equilibrium field. We consider two-dimensional and three-dimensional domains separately and study the correspondence between Landau-de Gennes theory and Ginzburg-Landau theory for superconductors. We obtain results for the location and dimensionality of the defect set, the minimizer profile near the defect set and study the qualitative properties of uniaxial energy minimizers away from the defect set, in the physically relevant case of vanishing elastic constant. In the three-dimensional case, we establish the $C^{1,\alpha}$-convergence of uniaxial minimizers to a limiting harmonic map, away from the defect set, for some $0<\alpha<1$. Some generalizations for biaxial minimizers are also discussed. This work is motivated by the study of defects in liquid crystalline systems and their applications.