Convergence of Meissner minimisers of the Ginzburg-Landau energy as kappa ->+infinity

We consider Meissner solutions, critical points of the Ginzbug-Landau energ y epsilon (kappa) for an infinite cylinder of smooth bounded section Ohm su bset of R-2. These Meissner solutions do not contain vortices. satisfy a co nvexity condition, and are local minimisers. For an exterior magnetic field H-0 less than a critical field H-0*. we prove the existence and uniqueness of such a Meissner solution u(kappa)(H0) for kappa large enough. Moreover, we prove the convergence of u(kappa)(H0) towards u(infinity)(H0) as kappa --> +infinity, where u(infinity)(H0) is a local minimiser of some energy ep silon (infinity), and can be understood as a Meissner solution for kappa = +infinity, On another hand we prove that epsilon (infinity) has no global m inimisers. (C) 2000 Academie des sciences/Editions scientifiques et medical es Elsevier SAS.