On the transition to the normal phase for superconductors surrounded by normal conductors

Journal of Differential Equations

Volumn 247, Issue 6, 2009, Pages 1637-1672

  Fournais, Søren ^a   Kachmar, Ayman ^a  

^a AARHUS UNIVERSITY   (Denmark)

Author keywords

Ginzburg Landau type equations; Schr dinger operator with magnetic field; Semiclassical analysis

Indexed keywords

EID: 67650998829     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2009.04.012     Document Type: Article

References (31)

1. Agmon S., Douglis A., and Nirenberg L. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 12 (1959) 623-727
2. Avron J., Herbst I., and Simon B. Schrödinger operators with magnetic fields. I. General interactions. Duke Math. J. 45 (1978) 847-883
3. Bonnaillie-Noël V., and Fournais S. Superconductivity in domains with corners. Rev. Math. Phys. 19 6 (2007) 607-637
4. Chapman S.J., Du Q., and Gunzburger M.D. A Ginzburg-Landau type model of superconducting/normal junctions including Josephson junctions. European J. Appl. Math. 6 2 (1996) 97-114
5. Du Q., Gunzburger M., and Peterson J. Analysis and approximation of the Ginzburg-Landau model of superconductivity. SIAM Rev. 34 (1992) 45-81
6. Evans L.C. Partial Differential Equations. Grad. Stud. Math. vol. 19 (1998), Amer. Math. Soc., Providence, RI
7. S. Fournais, B. Helffer, Spectral Methods in Surface Superconductivity, submitted for publication
8. Fournais S., and Helffer B. On the third critical field in Ginzburg-Landau theory. Comm. Math. Phys. 226 1 (2006) 153-196
9. Fournais S., and Helffer B. Strong diamagnetism in general domains and applications. Ann. Inst. Fourier 52 7 (2007) 2389-2400
10. Fournais S., and Helffer B. Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian. Ann. Inst. Fourier (2006)
11. S. Fournais, A. Kachmar, Nucleation of bulk superconductivity close to critical magnetic field, preprint
12. de Gennes P.G. Superconductivity of Metals and Alloys (1966), Benjamin
13. de Gennes P.G. Boundary effects in superconductors. Rev. Modern Phys. (January 1964)
14. de Gennes P.G., and Hurault J.P. Proximity effects under magnetic fields, II. Interpretation of 'breakdown'. Phys. Lett. 17 3 (1965) 181-182
15. Giorgi T. Superconductors surrounded by normal materials. Proc. Roy. Soc. Edinburgh Sect. A 135 (2005) 331-356
16. Giorgi T., and Phillips D. The breakdown of superconductivity due to strong fields for the Ginzburg-Landau model. SIAM J. Math. Anal. 30 (1999) 341-359
17. Helffer B., and Morame A. Magnetic bottles in connection with superconductivity. J. Funct. Anal. 185 (2001) 604-680
18. Helffer B., and Morame A. Magnetic bottles for the Neumann problem: Curvature effect in the case of dimension 3 (general case). Ann. Sci. Ecole Norm. Sup. 37 (2004) 105-170
19. Helffer B., and Pan X.B. Upper critical field and location of surface nucleation of superconductivity. Ann. Inst. H. Poincaré Anal. Non Linéaire 20 1 (2003) 145-181
20. Kachmar A. On the ground state energy for a magnetic Schrödinger operator and the effect of the de Gennes boundary condition. J. Math. Phys. 47 (2006) 072106 (32 pp.)
21. Kachmar A. On the perfect superconducting solution for a generalized Ginzburg-Landau equation. Asymptot. Anal. 54 3-4 (2007)
22. Kachmar A. On the stability of normal states for a generalized Ginzburg-Landau model. Asymptot. Anal. 55 3-4 (2007) 145-201
23. A. Kachmar, Problèmes aux limites issus de la supraconductivité, estimations semi-classiques et comportement asymptotique des solutions, PhD thesis, Université Paris-Sud, 2007
24. Kato T. Perturbation Theory for Linear Operators (1976), Springer-Verlag, Berlin
25. Lu K., and Pan X.B. Surface nucleation of superconductivity in 3-dimensions. J. Differential Equations 168 2 (2000) 386-452
26. Lu K., and Pan X.B. Estimates of the upper critical field of the Ginzburg-Landau equations of superconductivity. Phys. D 127 1-2 (1999) 73-104
27. c 2. Comm. Math. Phys. 228 2 (2002) 327-370
28. Sandier E., and Serfaty S. Vortices in the Magnetic Ginzburg-Landau Model. Progr. Nonlinear Differential Equations Appl. vol. 70 (2007), Birkhäuser
29. Serfaty S. Local minimizers for the Ginzburg-Landau energy near critical magnetic field. Part I. Commun. Contemp. Math. 1 2 (1999) 213-254
30. Serfaty S. Stable configurations in superconductivity: Uniqueness, multiplicity and vortex-nucleation. Arch. Ration. Mech. Anal. 149 (1999) 329-365
31. Temam R. Navier-Stokes Equation. Theory and Numerical Analysis. (revised version) (1984), Elsevier, Amsterdam