Nonexistence of Ginzburg-Landau minimizers with prescribed degree on the boundary of a doubly connected domain

Let ω, Ω be bounded simply connected domains in R², and let over(ω, ̄) ⊂ Ω. In the annular domain A = Ω {set minus} over(ω, ̄) we consider the class J of complex valued maps having modulus 1 and degree 1 on ∂Ω and ∂ω. We prove that, when cap (A) < π, there exists a finite threshold value κ₁ of the Ginzburg-Landau parameter κ such that the minimum of the Ginzburg-Landau energy E_κ not attained in J when κ > κ₁ while it is attained when κ < κ₁. To cite this article: L. Berlyand et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).