Title: On approximation of Ginzburg-Landau minimizers by $\mathbb S^1$-valued maps in domains with vanishingly small holes,
Abstract: 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 dsicovered 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 $\mathbb S^1$-valued and $\mathbb C$-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.