Ginzburg-Landau minimizers with high topological degrees in an annulus

Motivated by recent experiments on fermionic rings, we study the asymptotic behaviour of minimizers of the Ginzburg-Landau (GL) energy in an annulus with a Dirichlet data which depends on the GL parameter on the outer boundary. We show that there is a critical degree of order $\mathrm{\mid }\ln \epsilon \mathrm{\mid }$ under which the ground state displays a giant vortex and above which minimizers exhibit a combination of a giant vortex and vortices which tend to the outer boundary as the GL parameter tends to zero. Our analysis relies on the construction of suitable upper and lower bounds, on the extension to a slightly bigger annulus and on the minimization of the mean-field energy appearing in the lower bound. In order to be able to derive the minimum of this energy we use the symmetry of the domain and criticality with respect to inner variations.