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

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

2000-04-11

arxiv.org/abs/cond-mat/0004172v1

Physics

Condensed Matter

Superconductivity

16 pages. to appear in Com. Contemporary Mathematics

Scientific paper

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 the 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 the domain is a disc of small radius, then any non normal solution is symmetric and unique. Then, in the case of a slab, that is a one dimensional domain, we use the same method to derive that solutions are symmetric. The proofs use a priori estimates and the Poincar\'e inequality.

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