Size of planar domains and existence of minimizers of the Ginzburg–Landau energy with semistiff boundary conditions

The Ginzburg–Landau energy with semistiff boundary conditions is an intermediate model between the full Ginzburg–Landau equations, which leads to the appearance of both a condensate wave function and a magnetic potential, and the simplified Ginzburg–Landau model, coupling the condensate wave function to a Dirichlet boundary condition. In the semistiff model, there is no magnetic potential. The boundary data are not fixed, but circulation is prescribed on the boundary. Mathematically, this leads to prescribing the degrees on the components of the boundary. The corresponding problem is variational, but noncompact: in general, energy minimizers do not exist. Existence of minimizers is governed by the topology and the size of the underlying domain. We propose here various notions of domain size related to existence of minimizers, and discuss existence of minimizers or critical points, as well as their uniqueness and asymptotic behavior. We also present the state of the art in the study of this model, accounting for results obtained during the last decade by Berlyand, Dos Santos, Farina, Golovaty, Rybalko, Sandier, and the author.