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

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March 2013, 8(1): 115-130. doi: 10.3934/nhm.2013.8.115

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

Leonid Berlyand ^1, and Volodymyr Rybalko ^2,

1.	Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States
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, Ukraine

Received January 2012 Revised May 2012 Published April 2013

Abstract
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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.

Keywords: vortices., pinning, Ginzburg-Landau functional, Homogenization.

Mathematics Subject Classification: Primary: 49K20, 35J66, 35J50; Secondary: 47H1.

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

References:

[1]	A. Aftalion, E. Sandier and S. Serfaty, Pinning phenomena in the Ginzburg-Landau model of superconductivity, J. Math. Pures Appl. (9), 80 (2001), 339-372. doi: 10.1016/S0021-7824(00)01180-6.
[2]	S. Alama and L. Bronsard, Pinning effects and their breakdown for a Ginzburg-Landau model with normal inclusions, J. Math. Phys., 46 (2005), 39 pp. doi: 10.1063/1.2010354.
[3]	S. Alama and L. Bronsard, Vortices and pinning effects for the Ginzburg-Landau model in multiply connected domains, Comm. Pure Appl. Math., 59 (2006), 36-70. doi: 10.1002/cpa.20086.
[4]	H. Aydi and A. Kachmar, Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint. II, Commun. Pure Appl. Anal., 8 (2009), 977-998. doi: 10.3934/cpaa.2009.8.977.
[5]	E. J. Balder, "Lectures on Young Measures," Cah. de Ceremade, 1995.
[6]	G. R. Berdiyorov, M. V. Milosević and F. M. Peeters, Novel commensurability effects in superconducting films with antidot arrays, Phys. Rev. Lett., 96 (2006). doi: 10.1103/PhysRevLett.96.207001.
[7]	M. Dos Santos and O. Misiats, Ginzburg-Landau model with small pinning domains, Netw. Heterog. Media, 6 (2011), 715-753. doi: 10.3934/nhm.2011.6.715.
[8]	M. Dos Santos, P. Mironescu and O. Misiats, The Ginzburg-Landau functional with a discontinuous and rapidly oscillating pinning term. Part I : The zero degree case, Comm. Contemp. Math., 13 (2011), 885-914. doi: 10.1142/S021919971100449X.
[9]	M. Dos Santos, The Ginzburg-Landau functional with a discontinuous and rapidly oscillating pinning term. Part II: The non-zero degree case, preprint.
[10]	I. Ekeland and R. Temam, "Analyse Convexe et Problemes Variationnels," (French) Collection Etudes Mathematiques. Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974.
[11]	A. Kachmar, Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint, ESAIM Control Optim. Calc. Var., 16 (2010), 545-580. doi: 10.1051/cocv/2009009.
[12]	L. Lassoued and P. Mironescu, Ginzburg-Landau type energy with discontinuous constraint, J. Anal. Math., 77 (1999), 1-26. doi: 10.1007/BF02791255.
[13]	P. Pedregal, "Parametrized Measures and Variational Principles," Birkhauser, 1997. doi: 10.1007/978-3-0348-8886-8.
[14]	E. Sandier and S. Serfaty, Global minimizers for the Ginzburg-Landau functional below the first critical magnetic field, Annales IHP, Analyse Non Linéaire, 17 (2000), 119-145. doi: 10.1016/S0294-1449(99)00106-7.
[15]	M. Valadier, Young measures, Methods of Nonconvex Analysis, Lecture Notes Math., Springer, (1990), 152-188. doi: 10.1007/BFb0084935.

show all references

References:

[1]	A. Aftalion, E. Sandier and S. Serfaty, Pinning phenomena in the Ginzburg-Landau model of superconductivity, J. Math. Pures Appl. (9), 80 (2001), 339-372. doi: 10.1016/S0021-7824(00)01180-6.
[2]	S. Alama and L. Bronsard, Pinning effects and their breakdown for a Ginzburg-Landau model with normal inclusions, J. Math. Phys., 46 (2005), 39 pp. doi: 10.1063/1.2010354.
[3]	S. Alama and L. Bronsard, Vortices and pinning effects for the Ginzburg-Landau model in multiply connected domains, Comm. Pure Appl. Math., 59 (2006), 36-70. doi: 10.1002/cpa.20086.
[4]	H. Aydi and A. Kachmar, Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint. II, Commun. Pure Appl. Anal., 8 (2009), 977-998. doi: 10.3934/cpaa.2009.8.977.
[5]	E. J. Balder, "Lectures on Young Measures," Cah. de Ceremade, 1995.
[6]	G. R. Berdiyorov, M. V. Milosević and F. M. Peeters, Novel commensurability effects in superconducting films with antidot arrays, Phys. Rev. Lett., 96 (2006). doi: 10.1103/PhysRevLett.96.207001.
[7]	M. Dos Santos and O. Misiats, Ginzburg-Landau model with small pinning domains, Netw. Heterog. Media, 6 (2011), 715-753. doi: 10.3934/nhm.2011.6.715.
[8]	M. Dos Santos, P. Mironescu and O. Misiats, The Ginzburg-Landau functional with a discontinuous and rapidly oscillating pinning term. Part I : The zero degree case, Comm. Contemp. Math., 13 (2011), 885-914. doi: 10.1142/S021919971100449X.
[9]	M. Dos Santos, The Ginzburg-Landau functional with a discontinuous and rapidly oscillating pinning term. Part II: The non-zero degree case, preprint.
[10]	I. Ekeland and R. Temam, "Analyse Convexe et Problemes Variationnels," (French) Collection Etudes Mathematiques. Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974.
[11]	A. Kachmar, Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint, ESAIM Control Optim. Calc. Var., 16 (2010), 545-580. doi: 10.1051/cocv/2009009.
[12]	L. Lassoued and P. Mironescu, Ginzburg-Landau type energy with discontinuous constraint, J. Anal. Math., 77 (1999), 1-26. doi: 10.1007/BF02791255.
[13]	P. Pedregal, "Parametrized Measures and Variational Principles," Birkhauser, 1997. doi: 10.1007/978-3-0348-8886-8.
[14]	E. Sandier and S. Serfaty, Global minimizers for the Ginzburg-Landau functional below the first critical magnetic field, Annales IHP, Analyse Non Linéaire, 17 (2000), 119-145. doi: 10.1016/S0294-1449(99)00106-7.
[15]	M. Valadier, Young measures, Methods of Nonconvex Analysis, Lecture Notes Math., Springer, (1990), 152-188. doi: 10.1007/BFb0084935.

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