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

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