Homogenized description of multiple Ginzburg-Landau vortices pinned by small holes

We consider a homogenization problem for the magnetic Ginzburg-Landau functional in domains with a large number of small holes. We establish a scaling relation between sizes of holes and the magnitude of the external magnetic field when the multiple vortices pinned by holes appear in nested subdomains and their homogenized density is described by a hierarchy of variational problems. This stands in sharp contrast with homogeneous superconductors, where all vortices are known to be simple. The proof is based on the G-convergence approach applied to a coupled continuum/discrete variational problem: continuum in the induced magnetic field and discrete in the unknown finite (quantized) values of multiplicity of vortices pinned by holes.