# American Institute of Mathematical Sciences

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

## Ginzburg-Landau model with small pinning domains

 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.
Citation: Mickaël Dos Santos, Oleksandr Misiats. Ginzburg-Landau model with small pinning domains. Networks and 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), 339-372. 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), 095102, 39 pp. [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), 705-729. [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), 45-73, 75-98. doi: 10.1007/s002050050083. [5] H. Aydi and A. Kachmar, Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint. II, Commun. Pure Appl. Anal., 8 (2009), 977-998. doi: 10.3934/cpaa.2009.8.977. [6] J. Bardeen and M. Stephen, Theory of the motion of vortices in superconductors, Phys. Rev, 140 (1965), 1197-1207. doi: 10.1103/PhysRev.140.A1197. [7] P. Bauman, N. Carlson and D. Phillips, On the zeros of solutions to Ginzburg-Landau type systems, SIAM J. Math. Anal., 24 (1993), 1283-1293. doi: 10.1137/0524073. [8] L. Berlyand and P. Mironescu, Ginzburg-Landau minimizers in perforated domains with prescribed degrees, preprint, 2006. Available from: http://math.univ-lyon1.fr/~mironescu/prepublications.htm. [9] L. Berlyand and P. Mironescu, Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain, Netw. Heterog. Media, 3 (2008), 461-487. doi: 10.3934/nhm.2008.3.461. [10] F. Bethuel, H. Brezis and F. Hélein, Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. Partial Differential Equations, 1 (1993), 123-148. [11] F. Bethuel, H. Brezis and F. Hélein, "Ginzburg-Landau Vortices," Progress in Nonlinear Differential Equations and their Applications, 13, Birkhäuser Boston, Inc., Boston, MA, 1994. [12] H. Brezis, Équations de Ginzburg-Landau et singularités, Notes de cours rédigées par Vicentiu Radulescu, 2001. Available from: http://inf.ucv.ro/~radulescu/articles/coursHB.pdf. [13] H. Brezis, New questions related to the topological degree, in "The Unity of Mathematics," 137-154, Progr. Math., 244, Birkhäuser Boston, Boston, MA, 2006. [14] M. del Pino and P. Felmer, On the basic concentration estimate for the Ginzburg-Landau equation, Differ Integr Equat., 11 (1998), 771-779. [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., to appear. [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), 771-779. [17] D. Larbalestier, A. Gurevich, M. Feldmann and A. Polyanskii, High-Tc superconducting material for electric power applications, Nature, 414 (2001), 368-377. doi: 10.1038/35104654. [18] L. Lassoued and P. Mironescu, Ginzburg-Landau type energy with discontinuous constraint, J. Anal. Math., 77 (1999), 1-26. doi: 10.1007/BF02791255. [19] C. Lefter and V. Radulescu, Minimization problems and corresponding renormalized energies, Differential Integral Equations, 9 (1996), 903-917. [20] C. Lefter and V. Radulescu, On the Ginzburg-Landau energy with weight, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 171-184. [21] F. Lin and Q. Du, Ginzburg-Landau vortices, dynamics, pinning, and hysteresis, SIAM J. Math. Anal., 28 (1997), 1265-1293. doi: 10.1137/S0036141096298060. [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), 189-206. [23] P. Mironescu, Explicit bounds for solutions to a Ginzburg-Landau type equation, Rev. Roumaine Math. Pures Appl., 41 (1996), 263-271. [24] C. Morrey, Jr., "Multiple Integrals in the Calculus of Variations," Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. [25] P. Newton and G. Chamoun, Vortex lattice theory: A particle interaction perspective, SIAM Rev., 51 (2009), 501-542. doi: 10.1137/07068597X. [26] J. Rubinstein, On the equilibrium position of Ginzburg Landau vortices, Z. Angew. Math. Phys., 46 (1995), 739-751. doi: 10.1007/BF00949077. [27] E. Sandier and S. Serfaty, "Vortices in the Magnetic Ginzburg-Landau Model,'' Progress in Nonlinear Differential Equations and their Applications, 70, Birkhäuser Boston, Inc., Boston, MA, 2007. [28] I. Sigal and F. Ting, Pinning of magnetic vortices by an external potential, St. Petersburg Math. J., 16 (2005), 211-236. doi: 10.1090/S1061-0022-04-00848-9.

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##### 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), 095102, 39 pp. [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), 705-729. [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), 45-73, 75-98. doi: 10.1007/s002050050083. [5] H. Aydi and A. Kachmar, Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint. II, Commun. Pure Appl. Anal., 8 (2009), 977-998. doi: 10.3934/cpaa.2009.8.977. [6] J. Bardeen and M. Stephen, Theory of the motion of vortices in superconductors, Phys. Rev, 140 (1965), 1197-1207. doi: 10.1103/PhysRev.140.A1197. [7] P. Bauman, N. Carlson and D. Phillips, On the zeros of solutions to Ginzburg-Landau type systems, SIAM J. Math. Anal., 24 (1993), 1283-1293. doi: 10.1137/0524073. [8] L. Berlyand and P. Mironescu, Ginzburg-Landau minimizers in perforated domains with prescribed degrees, preprint, 2006. Available from: http://math.univ-lyon1.fr/~mironescu/prepublications.htm. [9] L. Berlyand and P. Mironescu, Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain, Netw. Heterog. Media, 3 (2008), 461-487. doi: 10.3934/nhm.2008.3.461. [10] F. Bethuel, H. Brezis and F. Hélein, Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. Partial Differential Equations, 1 (1993), 123-148. [11] F. Bethuel, H. Brezis and F. Hélein, "Ginzburg-Landau Vortices," Progress in Nonlinear Differential Equations and their Applications, 13, Birkhäuser Boston, Inc., Boston, MA, 1994. [12] H. Brezis, Équations de Ginzburg-Landau et singularités, Notes de cours rédigées par Vicentiu Radulescu, 2001. Available from: http://inf.ucv.ro/~radulescu/articles/coursHB.pdf. [13] H. Brezis, New questions related to the topological degree, in "The Unity of Mathematics," 137-154, Progr. Math., 244, Birkhäuser Boston, Boston, MA, 2006. [14] M. del Pino and P. Felmer, On the basic concentration estimate for the Ginzburg-Landau equation, Differ Integr Equat., 11 (1998), 771-779. [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., to appear. [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), 771-779. [17] D. Larbalestier, A. Gurevich, M. Feldmann and A. Polyanskii, High-Tc superconducting material for electric power applications, Nature, 414 (2001), 368-377. doi: 10.1038/35104654. [18] L. Lassoued and P. Mironescu, Ginzburg-Landau type energy with discontinuous constraint, J. Anal. Math., 77 (1999), 1-26. doi: 10.1007/BF02791255. [19] C. Lefter and V. Radulescu, Minimization problems and corresponding renormalized energies, Differential Integral Equations, 9 (1996), 903-917. [20] C. Lefter and V. Radulescu, On the Ginzburg-Landau energy with weight, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 171-184. [21] F. Lin and Q. Du, Ginzburg-Landau vortices, dynamics, pinning, and hysteresis, SIAM J. Math. Anal., 28 (1997), 1265-1293. doi: 10.1137/S0036141096298060. [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), 189-206. [23] P. Mironescu, Explicit bounds for solutions to a Ginzburg-Landau type equation, Rev. Roumaine Math. Pures Appl., 41 (1996), 263-271. [24] C. Morrey, Jr., "Multiple Integrals in the Calculus of Variations," Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. [25] P. Newton and G. Chamoun, Vortex lattice theory: A particle interaction perspective, SIAM Rev., 51 (2009), 501-542. doi: 10.1137/07068597X. [26] J. Rubinstein, On the equilibrium position of Ginzburg Landau vortices, Z. Angew. Math. Phys., 46 (1995), 739-751. doi: 10.1007/BF00949077. [27] E. Sandier and S. Serfaty, "Vortices in the Magnetic Ginzburg-Landau Model,'' Progress in Nonlinear Differential Equations and their Applications, 70, Birkhäuser Boston, Inc., Boston, MA, 2007. [28] I. Sigal and F. Ting, Pinning of magnetic vortices by an external potential, St. Petersburg Math. J., 16 (2005), 211-236. doi: 10.1090/S1061-0022-04-00848-9.
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