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December  2011, 6(4): 715-753. doi: 10.3934/nhm.2011.6.715

Ginzburg-Landau model with small pinning domains

Mickaël Dos Santos 1, and  Oleksandr Misiats 2,

1. 

Université de Lyon, Université Lyon 1, Institut Camille Jordan CNRS UMR 5208, 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne, France
2. 

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

Received  March 2011 Revised  October 2011 Published  December 2011

We consider a Ginzburg-Landau type energy with a piecewise constant pinning term $a$ in the potential $(a^2 - |u|^2)^2$. The function $a$ is different from 1 only on finitely many disjoint domains, called the pinning domains. These pinning domains model small impurities in a homogeneous superconductor and shrink to single points in the limit $\epsilon\to0$; here, $\epsilon$ is the inverse of the Ginzburg-Landau parameter. We study the energy minimization in a smooth simply connected domain $\Omega \subset \mathbb{C}$ with Dirichlet boundary condition $g$ on $\partial \Omega$, with topological degree ${\rm deg}_{\partial \Omega} (g) = d >0$. Our main result is that, for small $\epsilon$, minimizers have $d$ distinct zeros (vortices) which are inside the pinning domains and they have a degree equal to $1$. The question of finding the locations of the pinning domains with vortices is reduced to a discrete minimization problem for a finite-dimensional functional of renormalized energy. We also find the position of the vortices inside the pinning domains and show that, asymptotically, this position is determined by local renormalized energy which does not depend on the external boundary conditions.
Keywords: Ginzburg-Landau Functional, vortices, pinning domain, degree..
Mathematics Subject Classification: Primary: 49K20, 35J66, 35J50; Secondary: 47H1.
Citation: Mickaël Dos Santos, Oleksandr Misiats. Ginzburg-Landau model with small pinning domains. Networks & Heterogeneous Media, 2011, 6 (4) : 715-753. doi: 10.3934/nhm.2011.6.715
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.  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).   Google Scholar

[3]

N. André, P. Bauman and D. Phillips, Vortex pinning with bounded fields for the Ginzburg-Landau equation,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 705.   Google Scholar

[4]

N. André and I. Shafrir, Asymptotic behavior of minimizers for the Ginzburg-Landau functional with weight. I, II,, Arch. Rational Mech. Anal., 142 (1998), 45.  doi: 10.1007/s002050050083.  Google Scholar

[5]

H. Aydi and A. Kachmar, Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint. II,, Commun. Pure Appl. Anal., 8 (2009), 977.  doi: 10.3934/cpaa.2009.8.977.  Google Scholar

[6]

J. Bardeen and M. Stephen, Theory of the motion of vortices in superconductors,, Phys. Rev, 140 (1965), 1197.  doi: 10.1103/PhysRev.140.A1197.  Google Scholar

[7]

P. Bauman, N. Carlson and D. Phillips, On the zeros of solutions to Ginzburg-Landau type systems,, SIAM J. Math. Anal., 24 (1993), 1283.  doi: 10.1137/0524073.  Google Scholar

[8]

L. Berlyand and P. Mironescu, Ginzburg-Landau minimizers in perforated domains with prescribed degrees,, preprint, (2006).   Google Scholar

[9]

L. Berlyand and P. Mironescu, Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain,, Netw. Heterog. Media, 3 (2008), 461.  doi: 10.3934/nhm.2008.3.461.  Google Scholar

[10]

F. Bethuel, H. Brezis and F. Hélein, Asymptotics for the minimization of a Ginzburg-Landau functional,, Calc. Var. Partial Differential Equations, 1 (1993), 123.   Google Scholar

[11]

F. Bethuel, H. Brezis and F. Hélein, "Ginzburg-Landau Vortices,", Progress in Nonlinear Differential Equations and their Applications, 13 (1994).   Google Scholar

[12]

H. Brezis, Équations de Ginzburg-Landau et singularités,, Notes de cours rédigées par Vicentiu Radulescu, (2001).   Google Scholar

[13]

H. Brezis, New questions related to the topological degree,, in, 244 (2006), 137.   Google Scholar

[14]

M. del Pino and P. Felmer, On the basic concentration estimate for the Ginzburg-Landau equation,, Differ Integr Equat., 11 (1998), 771.   Google Scholar

[15]

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,, Commun. Contemp. Math., ().   Google Scholar

[16]

B. A. Glowacki and M. Majoros, Superconducting-magnetic heterostructures: A method of decreasing AC losses and improving critical current density in multifilamentary conductors,, J. Phys.: Condens. Matter, 21 (2009), 771.   Google Scholar

[17]

D. Larbalestier, A. Gurevich, M. Feldmann and A. Polyanskii, High-Tc superconducting material for electric power applications,, Nature, 414 (2001), 368.  doi: 10.1038/35104654.  Google Scholar

[18]

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

[19]

C. Lefter and V. Radulescu, Minimization problems and corresponding renormalized energies,, Differential Integral Equations, 9 (1996), 903.   Google Scholar

[20]

C. Lefter and V. Radulescu, On the Ginzburg-Landau energy with weight,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 171.   Google Scholar

[21]

F. Lin and Q. Du, Ginzburg-Landau vortices, dynamics, pinning, and hysteresis,, SIAM J. Math. Anal., 28 (1997), 1265.  doi: 10.1137/S0036141096298060.  Google Scholar

[22]

N. G. Meyers, An L$^p$-estimate for the gradient of solutions of second order elliptic divergence equations,, Ann. Scuola Norm. Sup. Pisa (3), 17 (1963), 189.   Google Scholar

[23]

P. Mironescu, Explicit bounds for solutions to a Ginzburg-Landau type equation,, Rev. Roumaine Math. Pures Appl., 41 (1996), 263.   Google Scholar

[24]

C. Morrey, Jr., "Multiple Integrals in the Calculus of Variations,", Die Grundlehren der mathematischen Wissenschaften, (1966).   Google Scholar

[25]

P. Newton and G. Chamoun, Vortex lattice theory: A particle interaction perspective,, SIAM Rev., 51 (2009), 501.  doi: 10.1137/07068597X.  Google Scholar

[26]

J. Rubinstein, On the equilibrium position of Ginzburg Landau vortices,, Z. Angew. Math. Phys., 46 (1995), 739.  doi: 10.1007/BF00949077.  Google Scholar

[27]

E. Sandier and S. Serfaty, "Vortices in the Magnetic Ginzburg-Landau Model,'', Progress in Nonlinear Differential Equations and their Applications, 70 (2007).   Google Scholar

[28]

I. Sigal and F. Ting, Pinning of magnetic vortices by an external potential,, St. Petersburg Math. J., 16 (2005), 211.  doi: 10.1090/S1061-0022-04-00848-9.  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.  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).   Google Scholar

[3]

N. André, P. Bauman and D. Phillips, Vortex pinning with bounded fields for the Ginzburg-Landau equation,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 705.   Google Scholar

[4]

N. André and I. Shafrir, Asymptotic behavior of minimizers for the Ginzburg-Landau functional with weight. I, II,, Arch. Rational Mech. Anal., 142 (1998), 45.  doi: 10.1007/s002050050083.  Google Scholar

[5]

H. Aydi and A. Kachmar, Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint. II,, Commun. Pure Appl. Anal., 8 (2009), 977.  doi: 10.3934/cpaa.2009.8.977.  Google Scholar

[6]

J. Bardeen and M. Stephen, Theory of the motion of vortices in superconductors,, Phys. Rev, 140 (1965), 1197.  doi: 10.1103/PhysRev.140.A1197.  Google Scholar

[7]

P. Bauman, N. Carlson and D. Phillips, On the zeros of solutions to Ginzburg-Landau type systems,, SIAM J. Math. Anal., 24 (1993), 1283.  doi: 10.1137/0524073.  Google Scholar

[8]

L. Berlyand and P. Mironescu, Ginzburg-Landau minimizers in perforated domains with prescribed degrees,, preprint, (2006).   Google Scholar

[9]

L. Berlyand and P. Mironescu, Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain,, Netw. Heterog. Media, 3 (2008), 461.  doi: 10.3934/nhm.2008.3.461.  Google Scholar

[10]

F. Bethuel, H. Brezis and F. Hélein, Asymptotics for the minimization of a Ginzburg-Landau functional,, Calc. Var. Partial Differential Equations, 1 (1993), 123.   Google Scholar

[11]

F. Bethuel, H. Brezis and F. Hélein, "Ginzburg-Landau Vortices,", Progress in Nonlinear Differential Equations and their Applications, 13 (1994).   Google Scholar

[12]

H. Brezis, Équations de Ginzburg-Landau et singularités,, Notes de cours rédigées par Vicentiu Radulescu, (2001).   Google Scholar

[13]

H. Brezis, New questions related to the topological degree,, in, 244 (2006), 137.   Google Scholar

[14]

M. del Pino and P. Felmer, On the basic concentration estimate for the Ginzburg-Landau equation,, Differ Integr Equat., 11 (1998), 771.   Google Scholar

[15]

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,, Commun. Contemp. Math., ().   Google Scholar

[16]

B. A. Glowacki and M. Majoros, Superconducting-magnetic heterostructures: A method of decreasing AC losses and improving critical current density in multifilamentary conductors,, J. Phys.: Condens. Matter, 21 (2009), 771.   Google Scholar

[17]

D. Larbalestier, A. Gurevich, M. Feldmann and A. Polyanskii, High-Tc superconducting material for electric power applications,, Nature, 414 (2001), 368.  doi: 10.1038/35104654.  Google Scholar

[18]

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

[19]

C. Lefter and V. Radulescu, Minimization problems and corresponding renormalized energies,, Differential Integral Equations, 9 (1996), 903.   Google Scholar

[20]

C. Lefter and V. Radulescu, On the Ginzburg-Landau energy with weight,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 171.   Google Scholar

[21]

F. Lin and Q. Du, Ginzburg-Landau vortices, dynamics, pinning, and hysteresis,, SIAM J. Math. Anal., 28 (1997), 1265.  doi: 10.1137/S0036141096298060.  Google Scholar

[22]

N. G. Meyers, An L$^p$-estimate for the gradient of solutions of second order elliptic divergence equations,, Ann. Scuola Norm. Sup. Pisa (3), 17 (1963), 189.   Google Scholar

[23]

P. Mironescu, Explicit bounds for solutions to a Ginzburg-Landau type equation,, Rev. Roumaine Math. Pures Appl., 41 (1996), 263.   Google Scholar

[24]

C. Morrey, Jr., "Multiple Integrals in the Calculus of Variations,", Die Grundlehren der mathematischen Wissenschaften, (1966).   Google Scholar

[25]

P. Newton and G. Chamoun, Vortex lattice theory: A particle interaction perspective,, SIAM Rev., 51 (2009), 501.  doi: 10.1137/07068597X.  Google Scholar

[26]

J. Rubinstein, On the equilibrium position of Ginzburg Landau vortices,, Z. Angew. Math. Phys., 46 (1995), 739.  doi: 10.1007/BF00949077.  Google Scholar

[27]

E. Sandier and S. Serfaty, "Vortices in the Magnetic Ginzburg-Landau Model,'', Progress in Nonlinear Differential Equations and their Applications, 70 (2007).   Google Scholar

[28]

I. Sigal and F. Ting, Pinning of magnetic vortices by an external potential,, St. Petersburg Math. J., 16 (2005), 211.  doi: 10.1090/S1061-0022-04-00848-9.  Google Scholar

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