On the symmetry and uniqueness of solutions of the Ginzburg-Landau equations for small domains

In this paper, we study the Ginzburg-Landau equations for a two dimensional domain which has small size. We prove that if the domain is small, then th e solution has no zero, that is no vortex. More precisely, we show that the order parameter Psi is almost constant. Additionnally, we obtain that if t he domain is a disc of small radius, then any non normal solution is symmet ric and unique. Then, in the case of a slab, that is a one dimensional doma in, we use the same method to derive that solutions are symmetric. The proo fs use a priori estimates and the Poincare inequality.