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

Journal article

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 threedimensional domains separately and study the correspondence between Landau-de Gennes theory and Ginzburg-Landau theory for superconductors. We treat uniaxial and biaxial cases separately. In the uniaxial case, topological defects correspond to the zero set and we obtain results for the location and dimensionality of the defect set, the solution profile near and away from the defect set. In the three-dimensional case, we establish the C^1,a-convergence of uniaxial small energy solutions to a limiting harmonic map, away from the defect set, for some 0 < a < 1, in the 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.

Authors: Majumdar, A

• Publication date: 2010-01-01