Convergence of Meissner minimizers of the Ginzburg-Landau energy of superconductivity as κ→+∞

The Meissner solution of a smooth cylindrical superconducting domain subject to a uniform applied axial magnetic field is examined. Under an additional convexity condition the uniqueness of the Meissner solution is proved. It is then shown that it is a local minimizer of the Ginzburg-Landau energy epsilon(k), For applied fields less than a critical value, the existence of the Meissner solution is proved for large enough Ginzburg-Landau parameter kappa. Moreover it is proved that the Meissner solution converges to a local minimizer of a certain energy epsilon(infinity) in the limit as kappa --> infinity. Finally, it is proved that for large enough the Meissner solution is not a global minimizer of epsilon(kappa).