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

We consider minimizers u_ε of the Ginzburg-Landau energy with quadratic divergence penalization on a simply-connected two-dimensional domain Ω. On the boundary, strong tangential anchoring is imposed. We prove that minimizers satisfy a L^∞-bound uniform in ε when Ω has C2,1-boundary and that the Lipschitz constant blows up like ε⁻¹ when Ω has C3,1-boundary. Our theorem extends to W2,p-regularity result for our elliptic system with mixed Dirichlet-Neumann boundary condition.