Ginzburg-Landau vortices : the static model

Tristan Rivière

Séminaire Bourbaki (1999-2000)

  • Volume: 42, page 73-103
  • ISSN: 0303-1179

How to cite

top

Rivière, Tristan. "Ginzburg-Landau vortices : the static model." Séminaire Bourbaki 42 (1999-2000): 73-103. <http://eudml.org/doc/110284>.

@article{Rivière1999-2000,
author = {Rivière, Tristan},
journal = {Séminaire Bourbaki},
keywords = {energy functional; London limit; gauge invariance; superconductivity},
language = {eng},
pages = {73-103},
publisher = {Société Mathématique de France},
title = {Ginzburg-Landau vortices : the static model},
url = {http://eudml.org/doc/110284},
volume = {42},
year = {1999-2000},
}

TY - JOUR
AU - Rivière, Tristan
TI - Ginzburg-Landau vortices : the static model
JO - Séminaire Bourbaki
PY - 1999-2000
PB - Société Mathématique de France
VL - 42
SP - 73
EP - 103
LA - eng
KW - energy functional; London limit; gauge invariance; superconductivity
UR - http://eudml.org/doc/110284
ER -

References

top
  1. [Ab] A. Abrikosov - On the magnetic properties of superconductors of the second type, Soviet Phys. JETP5, 1174-1182 (1957). 
  2. [Af] A. Aftalion - On the minimizers of the Ginzburg-Landau energy for high kappa: the one-dimensional case, European J. Appl. Math.8, 331-345 (1997). Zbl0886.34019MR1471596
  3. [AT] A. Aftalion and C. Troy - On the solutions of the one dimensional Ginzburg-Landau Equations, Preprint LMENS (1998). 
  4. [AB1] L. Almeida and F. Bethuel - Multiplicity results for the Ginzburg-Landau equation in presence of symmetries, Houston J. Math.23, 733-764 (1997). Zbl0901.35029MR1687389
  5. [AB2] L. Almeida and F. Bethuel - Topological methods for the Ginzburg-Landau equations, J. Math. Pures Appl.77, 1-49 (1998). Zbl0904.35023MR1617594
  6. [BBC] H. Berestycki, A. Bonnet and J. Chapman - A semi-elliptic system arising in the theory of type-II superconductivity, Comm. Appl. Non-linear Anal.1, 1-21 (1994). Zbl0866.35030
  7. [BC] M.S. Berger and Y.Y. Chen - Symmetric vortices for the Ginzburg-Landau equations of superconductivity and the nonlinear desingularization, phenomenon, Journal of Funct. Anal.82, 259-295 (1989). Zbl0685.46051MR987294
  8. [BBH0] F. Bethuel, H. Brezis, and F. Hélein - Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. Partial Differ. Equ.1, 2, 123-148 (1993). Zbl0834.35014MR1261720
  9. [BBH] F. Bethuel, H. Brezis and F. Hélein - Ginzburg-Landau vortices, Birkhäuser (1994). Zbl0802.35142MR1269538
  10. [BR1] F. Bethuel and T. Rivière - Vortices for a variational problem related to supraconductivity, Ann. Inst. Henri Poincaré, Anal. Non Linéaire12, 3, 243-303 (1995). Zbl0842.35119MR1340265
  11. [BR2] F. Bethuel and T. Rivière - Vorticité dans les modèles de Ginzburg-Landau pour la supraconductivité, Seminaire EDP de l'École Polytechnique, exposé XVI (1994). Zbl0876.35112MR1300912
  12. [Bog] B. Bogomol'nyi — Soviet J. Nucl. Phys.24, 449 (1976). MR443719
  13. [BH1] C. Bolley and B. Helffer - Rigorous results on Ginzburg-Landau models in a film submitted to an exterior parallel magnetic field I and II, Non Linear Stud.3, 1-29 and 121-152 (1996). Zbl0869.34016MR1396033
  14. [BH2] C. Bolley and B. Helffer - The ginzburg-Landau equations in a semi-infinite superconducting film in the large κ limit, European J. Appl. Math.8, 347-367 (1997). Zbl0891.35144
  15. [BCM] A. Bonnet, J. Chapman and R. Monneau - Convergence of Meissner minimizers of the Ginzburg-Landau energy of superconductivity as κ → +∞, preprint (1999). 
  16. [BM] A. Bonnet and R. Monneau - Existence of a smooth free-boundary in a superconductor with a Nash-Moser inverse function theorem argument, preprint (1999). 
  17. [BBM] J. Bourgain, H. Brezis and P. Mironescu - Lifting properties of Sobolev maps, in preparation. 
  18. [BMR] H. Brezis, F. Merle and T. Rivière - Quantization effects for -Δu = u(1 - |u|2) in R2, Arch. Rat. Mech. Analysis. 126, 123-145 (1994). Zbl0809.35019
  19. [DL] Q. Du and H. Lin - Ginzburg-Landau vortices: dynamics, pinning and hysteresis, Siam J. Math. Anal.28, 1265-1293 (1997). Zbl0888.35054MR1474214
  20. [D] M. Dutour - Bifurcation vers l'état d'Abrikosov et diagramme de phase, thèse de l'Université Paris-Sud, Orsay (1999). 
  21. [dG] G. De Gennes - Superconductivity of Metal and Alloys, Benjamin, New-York and Amsterdam (1966). Zbl0138.22801
  22. [GS] S. Gueron and I. Shafrir - On a discrete variational problem involving interacting particles, to appear in SIAM J. Appl. Math. (1999). Zbl0962.49025MR1740832
  23. [JT] A. Jaffe and C. Taubes - Vortices and Monopoles, Birkhäuser (1980). Zbl0457.53034MR614447
  24. [J] R. Jerrard - Lower bounds for generalized Ginzburg-Landau functionals, SIAM J. Math. Anal.30, 721-746 (1999). Zbl0928.35045MR1684723
  25. [LL] H. Lin and C. Lin - Minimax solutions of the Ginzburg-Landau equations, Selecta Mathematica, New Series 3, 99-113 (1997). Zbl0876.49006MR1454087
  26. [LR] H. Lin and T. Rivière - Complex Ginzburg-Landau equations and codimension 2 minimal surfaces, J.E.M.S.1, 3 (1999). 
  27. [LR2] H. Lin and T. Rivière - A quantization property for static Ginzburg-Landau vortices, Comm. Pure and App. Math. (2000). Zbl1033.58013
  28. [Mi] P. Mironescu - Les minimiseurs locaux pour l'équation de Ginzburg-Landau sont à symétrie radiale, C.R. Acad. Sci. Paris, Ser.1, 323, 593-598 (1996). Zbl0858.35038MR1411048
  29. [OS] N. Ovchinnikov and M. Sigal - The Ginzburg-Landau equation I. Static vortices, Partial differential equations and their applications (Toronto, ON, 1995), 199-220, CRM Proc. Lecture Notes12, Amer. Math. Soc., Providence, RI (1997). Zbl0912.35078MR1479248
  30. [PR] F. Pacard and T. Rivière - Linear and non-linear aspects of vortices, Birkhäuser (2000). Zbl0948.35003MR1763040
  31. [Qin] J. Qing - Renormalized energy for Ginzburg-Landau vortices on closed surfaces, Math. Z.225, 1-34 (1997). Zbl0871.49035MR1451329
  32. [Ri1] T. Rivière - Asymptotic analysis for the Ginzburg-Landau equations, ETH-Zürich course, January 1997, Boll U.M.I. (1999). MR1719570
  33. [Ri2] T. Rivière - Line vortices in the U(1)-Higgs Model, COCV 1, 77-167 (1995). Zbl0874.53019MR1394302
  34. [Ri3] T. Rivière - Some progress towards Jaffe and Taubes Conjectures, preprint (1999). 
  35. [R] F. Rodrigues - Obstacle problems in mathematical physics, Mathematical Studies, North Holland (1987). Zbl0606.73017MR880369
  36. [SST] D. Saint-James, G. Sarma and J. Thomas - Type-II Superconductivity, Pergamon Press (1969). 
  37. [Sa] E. Sandier - Lower bounds for the energy of unit vector fields and applications, J. Funct. Anal.152, 379-403 (1998). Zbl0908.58004MR1607928
  38. [SS1] E. Sandier and S. Serfaty - Global minimizers for the Ginzburg-Landau functional below the first critical magnetic field, to appear in Annales de l'Inst. Henri Poincaré, Analyse non-linéaire (1999). Zbl0947.49004MR1743433
  39. [SS2] E. Sandier and S. Serfaty - A rigourous derivation of a free-boundary problem arising in superconductivity, Ann. Sci. Éc. Norm. Sup.33 (2000), 561-592. Zbl1174.35552MR1832824
  40. [Se1] S. Serfaty - Étude mathématique de l'équation de Ginzburg-Landau de la supraconductivité, Thèse de l'Université Paris-Sud, Orsay (1999). 
  41. [Se2] S. Serfaty - Local minimizers for the Ginzburg-Landau energy near critical magnetic field I and II, Comm. Contemporary Mathematics1, n° 2 et 3, 213-254 (1999). Zbl0944.49007MR1696100
  42. [Se3] S. Serfaty - Stable configurations in superconductivity: uniqueness, multiplicity and vortex-nucleation, to appear in : Arch. Rat. Mech. Anal. (1999). Zbl0959.35154MR1731999
  43. [St] M. Struwe - On the asymptotic behavior of minimizers of the Ginzburg-Landau model in 2 dimensions, J. Diff. Int. Equations7 (1994), 1613-1624 and Erratum in J. Diff. Int. Zbl0809.35031MR1269674
  44. [Ti] M. Tinkham - Introduction to Superconductivity, 2nd edition, McGraw-Hill (1996). 
  45. [Uh] K. Uhlenbeck - Connections with Lp bounds on curvature, Comm. Math. Phys.83, 31-42 (1982). Zbl0499.58019MR648356

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.