Pinning by holes of multiple vortices in homogenization for Ginzburg-Landau problems

Abstract

We consider a homogenization problem for magnetic Ginzburg-Landau functional in domains with large number of small holes. For sufficiently strong magnetic field, a large number of vortices is formed and they are pinned by the holes. We establish a scaling relation between sizes of holes and the magnitude of the external magnetic field when pinned vortices are multiple 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 $\Gamma$-convergence approach which is 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.