Abrikosov lattices in finite domains

In 1957 Abrikosov published his work on periodic solutions to the linearized Ginzburg-Landau equations. Abrikosov's analysis assumes periodic boundary conditions, which are very different from the natural boundary conditions the minimizer of the Ginzburg-Landau energy functional should satisfy. In the present work we prove that the global minimizer of the fully non-linear functional can be approximated, in every rectangular subset of the domain, by one of the periodic solution to the linearized Ginzburg-Landau equations in the plane. Furthermore, we prove that the energy of this solution is close to the minimum of the energy over all Abrikosov's solutions in that rectangle.