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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

E. J. Balder, "Lectures on Young Measures," Cah. de Ceremade, 1995. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

M. Dos Santos, The Ginzburg-Landau functional with a discontinuous and rapidly oscillating pinning term. Part II: The non-zero degree case,, preprint., ().   Google Scholar

[10]

I. Ekeland and R. Temam, "Analyse Convexe et Problemes Variationnels," (French) Collection Etudes Mathematiques. Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974.  Google Scholar

[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.  Google Scholar

[12]

L. Lassoued and P. Mironescu, Ginzburg-Landau type energy with discontinuous constraint, J. Anal. Math., 77 (1999), 1-26. doi: 10.1007/BF02791255.  Google Scholar

[13]

P. Pedregal, "Parametrized Measures and Variational Principles," Birkhauser, 1997. doi: 10.1007/978-3-0348-8886-8.  Google Scholar

[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.  Google Scholar

[15]

M. Valadier, Young measures, Methods of Nonconvex Analysis, Lecture Notes Math., Springer, (1990), 152-188. doi: 10.1007/BFb0084935.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

E. J. Balder, "Lectures on Young Measures," Cah. de Ceremade, 1995. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

M. Dos Santos, The Ginzburg-Landau functional with a discontinuous and rapidly oscillating pinning term. Part II: The non-zero degree case,, preprint., ().   Google Scholar

[10]

I. Ekeland and R. Temam, "Analyse Convexe et Problemes Variationnels," (French) Collection Etudes Mathematiques. Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974.  Google Scholar

[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.  Google Scholar

[12]

L. Lassoued and P. Mironescu, Ginzburg-Landau type energy with discontinuous constraint, J. Anal. Math., 77 (1999), 1-26. doi: 10.1007/BF02791255.  Google Scholar

[13]

P. Pedregal, "Parametrized Measures and Variational Principles," Birkhauser, 1997. doi: 10.1007/978-3-0348-8886-8.  Google Scholar

[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.  Google Scholar

[15]

M. Valadier, Young measures, Methods of Nonconvex Analysis, Lecture Notes Math., Springer, (1990), 152-188. doi: 10.1007/BFb0084935.  Google Scholar

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