Symmetry Breaking in Annular Domains for a Ginzburg-Landau Superconductivity Model
Leonid V. Berlyand, Karl F. Voss · Kluwer Academic Publishers eBooks · 2006
A minimization problem for the Ginzburg-Landau functional is considered on an annulus with thickness R in the class of functions $$ \mathcal{F}^d$$ where only the degree (the winding number), d, on the boundary is prescribed. The existence of a critical thickness R c which depends on the Ginzburg-Landau parameter δ and d is established. It is shown that if R>Rc then any function in separated form in polar coordinates can not be a global minimizer in the class $$ \mathcal{F}^d$$ and therefore the functions in a minimizing sequence can not be of this form. Hence for a sufficiently thick annulus, the variational Ginzburg-Landau problem is not a regular perturbation, in δ, of the corresponding problem for harmonic maps. The proof is based on an explicit construction of a sequence of admissible functions with vortices approaching the boundary. The key point of the proof is that the Ginzburg-Landau energy of the functions in this sequence approaches a constant independent of R and δ. By contrast, in the class of functions with separated form in polar coordinates and the same boundary conditions, there exists a unique minimizer whose energy depends on R. This minimizer is rotationally invariant and converges uniformly to the solution of the corresponding harmonic map problem as δ→0.