  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.
