  We study global minimizers of a continuum Landau-De Gennes energy functional
for nematic liquid crystals, in three-dimensional domains, subject to uniaxial
boundary conditions. We analyze the physically relevant limit of small elastic
constant and show that global minimizers converge strongly, in $W^{1,2}$, to a
global minimizer predicted by the Oseen-Frank theory for uniaxial nematic
liquid crystals with constant order parameter. Moreover, the convergence is
uniform in the interior of the domain, away from the singularities of the
limiting Oseen-Frank global minimizer. We obtain results on the rate of
convergence of the eigenvalues and the regularity of the eigenvectors of the
Landau-De Gennes global minimizer.
  We also study the interplay between biaxiality and uniaxiality in Landau-De
Gennes global energy minimizers and obtain estimates for various related
quantities such as the biaxiality parameter and the size of admissible strongly
biaxial regions.
