A priori $L^\infty-$bound for Ginzburg-Landau energy minimizers with divergence penalization

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.