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

We consider minimizers uε 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 a W2,p−regularity result for our elliptic system with mixed Dirichlet-Neumann boundary condition.