Homogenized description of multiple Ginzburg-Landau vortices pinned by small holes
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1.
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802
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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
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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.
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.
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