# Symmetry Breaking in Annular Domains for a Ginzburg-Landau Superconductivity Model

## Abstract

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

*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*

**R>Rc***, 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*

**δ***R*and

*. 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*

**δ***R*. This minimizer is rotationally invariant and converges uniformly to the solution of the corresponding harmonic map problem as

*.*

**δ→0**## Keywords

Critical Thickness Blaschke Product Unique Minimizer Separate Form Regular Perturbation## Preview

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