Ginzburg-Landau minimizers with prescribed degrees. Emergence of vortices and existence/nonexistence of the minimizers

Let Ω be a 2D domain with a hole ω. In the domain A = Ω \ ω consider a class J of complex valued maps having degrees 1 and 1 on ∂Ω, ∂ω respectively. In a joint work with P. Mironescu we show that if cap(A) ≥ π (subcritical domain), minimizers of the Ginzburg-Landau energy E κ exist for each κ. They are vortexless and converge in H(A) to a minimizing S-valued harmonic map as the coherency length κ tends to 0. When cap(A) < π (supercritical domain), for large κ, we establish that the minimizing sequences/minimizers develop exactly two vortices—a vortex of degree 1 near ∂Ω and a vortex of degree −1 near ∂Ω which rapidly converge to ∂A. It was conjectured that the global minimizers do not exist for large κ. In a subsequent joint work with D. Golovaty and V. Rybalko this conjecture was proved. It was shown that, when cap(A) < π , there exists a finite threshold value κ1 of the Ginzburg-Landau parameter such that the minimum of E κ is not attained in J when κ > κ1, while it is attained when κ > κ1. No standard elliptic estimates worked here and our proof is based on an introduction of an auxiliary linear problem which allows for an explicit energy estimate which is sufficiently tight.