On the minimizers of the Ginzburg-Landau energy for high kappa: the axially symmetric case

The Ginzburg-Landau theory of superconductivity is examined in the case of a special geometry of the sample, the infinite cylinder. We restrict to axi ally symmetric solutions and consider models with and without vortices. Fir st putting the Ginzburg-Landau parameter kappa formally equal to infinity, the existence of a minimizer of this reduced Ginzburg-Landau energy is prov ed. Then asymptotic behaviour for large kappa of minimizers of the full Gin zburg-Landau energy is analyzed and different convergence results are obtai ned. Our main result states that, when kappa is large, the minimum of the e nergy is reached when there are about kappa vortices at the center of the c ylinder. Numerical computations illustrate the various behaviours. (C) Else vier, Paris.