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

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