On approximation of Ginzburg-Landau minimizers by $\mathbb S^1$-valued maps in domains with vanishingly small holes

TL;DR

This paper studies how Ginzburg-Landau minimizers in 2D domains with tiny holes can be approximated by circle-valued maps, revealing vortex phase separation phenomena in superconducting composites.

Contribution

It demonstrates the equivalence of degrees of minimizers for S^1-valued and C-valued maps under specific scaling, advancing understanding of vortex behavior in perforated domains.

Findings

Degrees of minimizers are the same for S^1 and C-valued maps.

Energy decomposition techniques handle widely separated parameters.

Vortex phase separation occurs in superconducting composites.

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.