Duke Mathematical Journal

Limiting vorticities for the Ginzburg-Landau equations

Etienne Sandier and Sylvia Serfaty

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study the asymptotic limit of solutions of the Ginzburg-Landau equations in two dimensions with or without magnetic field. We first study the Ginzburg-Landau system with magnetic field describing a superconductor in an applied magnetic field, in the "London limit" of a Ginzburg-Landau parameter $\kappa$ tending to $\infty$. We examine the asymptotic behavior of the "vorticity measures" associated to the vortices of the solution, and we prove that passing to the limit in the equations (via the "stress-energy tensor") yields a criticality condition on the limiting measures. This condition allows us to describe the possible locations and densities of the vortices. We establish analogous results for the Ginzburg-Landau equation without magnetic field.

Article information

Source
Duke Math. J., Volume 117, Number 3 (2003), 403-446.

Dates
First available in Project Euclid: 26 May 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1085598401

Digital Object Identifier
doi:10.1215/S0012-7094-03-11732-9

Mathematical Reviews number (MathSciNet)
MR1979050

Zentralblatt MATH identifier
1035.82045

Subjects
Primary: 82D55: Superconductors
Secondary: 35B25: Singular perturbations 35J20: Variational methods for second-order elliptic equations 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 58E50: Applications

Citation

Sandier, Etienne; Serfaty, Sylvia. Limiting vorticities for the Ginzburg-Landau equations. Duke Math. J. 117 (2003), no. 3, 403--446. doi:10.1215/S0012-7094-03-11732-9. https://projecteuclid.org/euclid.dmj/1085598401


Export citation

References

  • 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.
  • G. Alberti, S. Baldo, and G. Orlandi, Variational convergence for functionals of Ginzburg-Landau type, preprint, 2002, http://cvgmt.sns.it/papers/albbalorl02a/
  • J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of superconductivity, Phys. Rev. (2) 108 (1957), 1175--1204.
  • F. Bethuel, H. Brezis, and F. Hélein, Ginzburg-Landau Vortices, Progr. Nonlinear Differential Equations Appl. 13, Birkhäuser, Boston, 1994.
  • F. Bethuel and T. Rivière, Vortices for a variational problem related to superconductivity, Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995), 243--303.
  • A. Bonnet and R. Monneau, Distribution of vortices in a type-II superconductor as a free boundary problem: Existence and regularity via Nash-Moser theory, Interfaces Free Bound. 2 (2000), 181--200.
  • H. Brezis, remark in L'injection du cône positif de $H^-1$ dans $W^-1,q$ est compacte pour tout $q < 2$, by F. Murat, J. Math. Pures Appl. (9) 60 (1981), 321--322.
  • H. Brezis and S. Serfaty, A variational formulation for the two-sided obstacle problem with measure data, Commun. Contemp. Math. 4 (2001), 357--374. \CMP1 901 150
  • J.-Y. Chemin, Fluides parfaits incompressibles, Astérisque 230, Soc. Math. France, Montrouge, 1995.
  • S. J. Chapman, J. Rubinstein, and M. Schatzman, A mean-field model of superconducting vortices, European J. Appl. Math. 7 (1996), 97--111.
  • P.-G. de Gennes, Superconductivity of Metals and Alloys, Frontiers in Phys., Benjamin, New York, 1966.
  • J.-M. Delort, Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc. 4 (1991), 553--586.
  • L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC Press, Boca Raton, Fla., 1992.
  • P. C. Fife and L. A. Peletier, On the location of defects in stationary solutions of the Ginzburg-Landau equation in $\mathbbR^2$, Quart. Appl. Math. 54 (1996), 85--104.
  • J. Frehse, Capacity methods in the theory of partial differential equations, Jahresber. Deutsch. Math.-Verein. 84 (1982), 1--44.
  • T. Giorgi and D. Phillips, The breakdown of superconductivity due to strong fields for the Ginzburg-Landau model, SIAM J. Math. Anal. 30 (1999), 341--359.
  • E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monogr. Math. 80, Birkhäuser, Basel, 1984.
  • F. Hélein, Symétries dans les problèmes variationnels et applications harmoniques, preprint, 1998, no. 9821, Centre de Mathématiques et de Leurs Applications, Cachan, France, http://cmla.ens-cachan.fr/Cmla/Publications/1998/
  • A. Jaffe and C. Taubes, Vortices and Monopoles: Structure of Static Gauge Theories, Progr. Phys. 2, Birkhaüser, Boston, 1980.
  • R. L. Jerrard, Lower bounds for generalized Ginzburg-Landau functionals, SIAM J. Math. Anal. 30 (1999), 721--746.
  • R. L. Jerrard and H. M. Soner, The Jacobian and the Ginzburg-Landau energy, Calc. Var. Partial Differential Equations 14 (2002), 151--191.
  • --. --. --. --., Limiting behavior of the Ginzburg-Landau functional, J. Funct. Anal. 192 (2002), 524--561. \CMP1 923 413
  • L. D. Landau, Collected Papers of L. D. Landau, ed. D. ter Haar, Gordon and Breach, New York, 1967.
  • F. Murat, L'injection du cône positif de $H^-1$ dans $W^-1,q$ est compacte pour tout $q<2$, J. Math. Pures Appl. (9) 60 (1981), 309--322.
  • E. Sandier, Lower bounds for the energy of unit vector fields and applications, J. Funct. Anal. 152 (1998), 379--403.
  • E. Sandier and S. Serfaty, Global minimizers for the Ginzburg-Landau functional below the first critical magnetic field, Ann. Inst. H. Poincaré Anal. Non Linéaire. 17 (2000), 119--145.
  • --. --. --. --., On the energy of type-II superconductors in the mixed phase, Rev. Math. Phys. 12 (2000), 1219--1257.
  • --. --. --. --., A rigorous derivation of a free-boundary problem arising in superconductivity, Ann. Sci. École Norm. Sup. (4) 33 (2000), 561--592.
  • --------, Ginzburg-Landau minimizers near the first critical field have bounded vorticity, to appear in Calc. Var. Partial Differential Equations.
  • E. Sandier and M. Soret, $S^1$-valued harmonic maps with high topological degree: asymptotic behavior of the singular set, Potential Anal. 13 (2000), 169--184.
  • S. Serfaty, Local minimizers for the Ginzburg-Landau energy near critical magnetic field, I, Commun. Contemp. Math. 1 (1999), 213--254.
  • --. --. --. --., Local minimizers for the Ginzburg-Landau energy near critical magnetic field, II, Commun. Contemp. Math. 1 (1999), 295--333.
  • --. --. --. --., Stable configurations in superconductivity: Uniqueness, multiplicity, and vortex-nucleation, Arch. Ration. Mech. Anal. 149 (1999), 329--365.
  • M. Tinkham, Introduction to Superconductivity, 2d ed., Internat. Ser. Pure Appl. Phys. McGraw-Hill, New York, 1996.
  • W. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, Grad. Texts in Math. 120, Springer, New York, 1989.