Homogenized description of multiple Ginzburg-Landau vortices pinned by small holes
Leonid V. Berlyand, Volodymyr Rybalko · Networks and Heterogeneous Media · 2013
We consider a homogenization problem for the magnetic Ginzburg-Landaufunctional in domains with a large number of small holes. Weestablish a scaling relation between sizes of holes and themagnitude of the external magnetic field when the multiple vorticespinned by holes appear in nested subdomains and their homogenizeddensity is described by a hierarchy of variational problems. Thisstands in sharp contrast with homogeneous superconductors, whereall vortices are known to be simple. The proof is based on the$\Gamma$-convergence approach applied to a coupledcontinuum/discrete variational problem: continuum in the inducedmagnetic field and discrete in the unknown finite (quantized)values of multiplicity of vortices pinned by holes.