  We investigate minimizers defined on a bounded domain $\Omega$ in
$\mathbb{R}^2$ for singular constrained energy functionals that include Ball
and Majumdar's modification of the Landau-de Gennes Q-tensor model for nematic
liquid crystals. We prove regularity of minimizers with finite energy and show
that their range on compact subdomains of $\Omega$ does not intersect the
boundary of the constraining set.
