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

Leonid Berlyand; Volodymyr Rybalko; Leonid Berlyand; Volodymyr Rybalko

doi:10.3934/nhm.2013.8.115

Networks and Heterogeneous Media

2013, Volume 8, Issue 1: 115-130. doi: 10.3934/nhm.2013.8.115

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

Leonid Berlyand ^{1
,},
Volodymyr Rybalko ^{2
,}

1.
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802
2.
Mathematical Division, B.Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Lenin Ave., 61103 Kharkiv

Received: 01 January 2012 Revised: 01 May 2012
Primary: 49K20, 35J66, 35J50; Secondary: 47H11.

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 $\Gamma$-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.
- Homogenization,
- Ginzburg-Landau functional,
- pinning,
- vortices
Citation: Leonid Berlyand, Volodymyr Rybalko. Homogenized description of multiple Ginzburg-Landau vortices pinned by small holes[J]. Networks and Heterogeneous Media, 2013, 8(1): 115-130. doi: 10.3934/nhm.2013.8.115

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Abstract

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 $\Gamma$-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.

References

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