  We consider minimizers $u_\varepsilon$ of the Ginzburg-Landau energy with
quadratic divergence penalization on a simply-connected two-dimensional domain
$\Omega$. On the boundary, strong tangential anchoring is imposed. We prove
that minimizers satisfy a $L^\infty$-bound uniform in $\varepsilon$ when
$\Omega$ has $C^{2,1}-$boundary and that the Lipschitz constant blows up like
$\varepsilon^{-1}$ when $\Omega$ has $C^{3,1}-$boundary. Our theorem extends to
$W^{2,p}-$regularity result for our elliptic system with mixed
Dirichlet-Neumann boundary condition.
