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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

  • 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.

    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

    Related Papers:

  • 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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  • This article has been cited by:

    1. Roberto Alicandro, Andrea Braides, Marco Cicalese, Lucia De Luca, Andrey Piatnitski, Topological Singularities in Periodic Media: Ginzburg–Landau and Core-Radius Approaches, 2022, 243, 0003-9527, 559, 10.1007/s00205-021-01731-7
    2. Leonid Berlyand, Dmitry Golovaty, Oleksandr Iaroshenko, Volodymyr Rybalko, On approximation of Ginzburg–Landau minimizers by S1-valued maps in domains with vanishingly small holes, 2018, 264, 00220396, 1317, 10.1016/j.jde.2017.09.037
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