Convergence of Meissner Minimizers of the Ginzburg--Landau Energy of Superconductivity as $\kappa\to +\infty$

Abstract

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 ${\cal E}_\kappa$. 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 ${\cal E}_\infty$ in the limit as $\kappa \rightarrow \infty$. Finally, it is proved that for $\kappa$ large enough the Meissner solution is not a global minimizer of ${\cal E}_\kappa$.