On approximation of Ginzburg–Landau minimizers by S¹-valued maps in domains with vanishingly small holes

We consider a two-dimensional Ginzburg–Landau problem on an arbitrary domain with a finite number of vanishingly small circular holes. A special choice of scaling relation between the material and geometric parameters (Ginzburg–Landau parameter vs. hole radius) is motivated by a recently discovered phenomenon of vortex phase separation in superconducting composites. We show that, for each hole, the degrees of minimizers of the Ginzburg–Landau problems in the classes of S¹-valued and C-valued maps, respectively, are the same. The presence of two parameters that are widely separated on a logarithmic scale constitutes the principal difficulty of the analysis that is based on energy decomposition techniques.