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

  • Leonid Berlyand
  • Karl Voss
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 91)


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.


Critical Thickness Blaschke Product Unique Minimizer Separate Form Regular Perturbation 
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Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Leonid Berlyand
    • 1
  • Karl Voss
    • 1
  1. 1.Department of MathematicsThe Pennsylvania State UniversityUniversity Park

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