# American Institute of Mathematical Sciences

May  2012, 11(3): 1303-1337. doi: 10.3934/cpaa.2012.11.1303

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

 1 Mathematical Institute, University of Oxford, 24-29 St.Giles, Oxford, United Kingdom

Received  November 2010 Revised  November 2011 Published  December 2011

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 $0 < \alpha < 1$, 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.
Citation: Apala Majumdar. The Landau-de Gennes theory of nematic liquid crystals: Uniaxiality versus Biaxiality. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1303-1337. doi: 10.3934/cpaa.2012.11.1303
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