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.
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