  We study global minimizers of a continuum Landau-De Gennes energy functional
for nematic liquid crystals, in three-dimensional domains. Assuming smooth and
uniaxial (e.g. homeotropic) boundary conditions and a corresponding physically
relevant norm constraint (Lyuksyutov constraint) in the interior, we prove full
regularity up to the boundary for the minimizers. As a consequence, in a
relevant range (which we call the Lyuksyutov regime) of parameters of the model
we show that even without the norm constraint isotropic melting is anyway
avoided in the energy minimizing configurations. Finally, we describe a class
of boundary data including radial anchoring which yield in both the previous
situations as minimizers smooth configurations whose level sets of the
biaxiality carry nontrivial topology. Results in this paper will be largely
employed and refined in the next papers of our series. In particular, in
[DMP2], we will prove that for smooth minimizers in a restricted class of
axially symmetric configurations, the level sets of the biaxiality are
generically finite union of tori of revolution.
