Abstract
In this paper we consider the asymptotic behavior of the Ginzburg–Landau model for superconductivity in three dimensions, in various energy regimes. Through an analysis via Γ-convergence, we rigorously derive a reduced model for the vortex density and deduce a curvature equation for the vortex lines. In the companion paper (Baldo et al. Commun. Math. Phys. 2012, to appear) we describe further applications to superconductivity and superfluidity, such as general expressions for the first critical magnetic field H c1, and the critical angular velocity of rotating Bose–Einstein condensates.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alberti G., Baldo S., Orlandi G.: Variational convergence for functionals of Ginzburg–Landau type. Indiana Univ. Math. J. 54, 1411–1472 (2005)
Baldo, S., Orlandi, G., Jerrard, R., Soner, M.: Vortex density models for superconductivity and superfluidity. Commun. Math. Phys. (2012, to appear)
Baldo S., Orlandi G.: A note on the Hodge theory for functionals with linear growth: Manuscr. Math. 97, 453–467 (1998)
Baldo S., Orlandi G., Weitkamp S.: Convergence of minimizers with local energy bounds for the Ginzburg–Landau functionals. Indiana Univ. Math. J. 58, 2369–2407 (2009)
Bethuel F., Brezis H., Hélein F.: Ginzburg–Landau vortices. Progress in Nonlinear Differential Equations and Their Applications, vol. 13. Boston, Birkhäuser (1994)
Bethuel F., Brezis H., Orlandi G.: Asymptotics for the Ginzburg–Landau equation in arbitrary dimensions. J. Funct. Anal. 186, 432–520 (2001)
Bethuel F., Orlandi G., Smets D.: Vortex rings for the Gross-Pitaevskii equation. J. Eur. Math. Soc. 6, 17–94 (2004)
Bethuel F., Orlandi G., Smets D.: Approximations with vorticity bounds for the Ginzburg–Landau functional. Commun. Contemp. Math. 6, 803–832 (2004)
Bethuel, F., Rivière, T.: Vorticité dans les modèles de Ginzburg–Landau pour la supraconductivité. Séminaire X EDP, 1993–1994, Exp. No. XVI, cole Polytechnique, Palaiseau
Bourgain J., Brezis H., Mironescu P.: H 1/2 maps with values into the circle: minimal connections, lifting, and the Ginzburg–Landau equation. Publ. Math. Inst. Hautes Etudes Sci. 99, 1–115 (2004)
Brezis H., Coron J.M., Lieb E.: Harmonic maps with defects. Commun. Math. Phys. 107, 649–705 (1986)
Brezis H., Serfaty S.: A variational formulation for the two-sided obstacle problem with measure data. Commun. Contemp. Math. 4(2), 357–374 (2002)
Chiron D.: Boundary problems for the Ginzburg–Landau equation. Commun. Contemp. Math. 7, 597–648 (2005)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. SIAM, Philadelphia, 2002
De Rham G.: Sur la théorie des formes differentielles harmoniques. Ann. Univ. Grenoble 22, 135–152 (1946)
Federer H.: Geometric Measure Theory. Grundlehren der mathematischen Wissenschaften, 153. Springer, New York (1969)
Giaquinta M., Modica G., Souček J.: Cartesian Currents and the Calculus of Variations vols. 1. and 2. Springer, Berlin (1998)
Giusti E.: BV Functions and Set of Finite Perimeter. Birkhäuser, Berlin (1982)
Iwaniec T., Scott C., Stroffolini B.: Nonlinear Hodge theory on manifolds with boundary. Ann. Mat. Pura Appl. (4) 177, 37–115 (1999)
Jerrard R.: Lower bounds for generalized Ginzburg–Landau functionals. SIAM J. Math. Anal. 30, 721–746 (1999)
Jerrard R., Montero A., Sternberg P.: Local minimizers of the Ginzburg–Landau energy with magnetic field in three dimensions. Commun. Math. Phys. 249, 549–577 (2004)
Jerrard R., Soner H.M.: The Jacobian and the Ginzburg–Landau energy. Calc. Variation PDE 14, 151–191 (2002)
Jerrard R., Soner H.M.: Limiting behavior of the Ginzburg–Landau functional. J. Funct. Anal. 192, 524–561 (2002)
Lin F.H., Rivière T.: Complex Ginzburg–Landau equations in high dimensions and codimension two area minimizing currents. J. Eur. Math. Soc. 1, 237–311 (1999)
Montero A.: Hodge decomposition with degenerate weights and the Gross-Pitaevskii energy. J. Funct. Anal. 254, 1926–1973 (2008)
Montero A., Sternberg P., Ziemer W.: Local minimizers with vortices in the Ginzburg–Landau system in three dimensions. Commun. Pure Appl. Math. 57, 99–125 (2004)
Morrey C.B.: A variational method in the theory of harmonic integrals. II. Am. J. Math. 78, 137–170 (1956)
Sandier, E.: Lower bounds for the energy of unit vector fields and applications. J. Funct. Anal. 152, 379–403 (1998). Erratum: Ibid. 171, 233 (2000)
Sandier E., Serfaty S.: A rigorous derivation of a free-boundary problem arising in superconductivity. Ann. Sci. cole Norm. Sup. (4) 33(4), 561–592 (2000)
Sandier, E., Serfaty, S.: Vortices in the magnetic Ginzburg–Landau model. Progress in Nonlinear Differential Equations and their Applications, 70. Birkhäuser, Boston, 2007
Sandier E., Serfaty S.: A product-estimate for Ginzburg–Landau and corollaries. J. Funct. Anal. 211, 219–244 (2004)
Spanier E.: Algebraic Topology. McGraw-Hill, New York (1966)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by W. E.
Rights and permissions
About this article
Cite this article
Baldo, S., Jerrard, R.L., Orlandi, G. et al. Convergence of Ginzburg–Landau Functionals in Three-Dimensional Superconductivity. Arch Rational Mech Anal 205, 699–752 (2012). https://doi.org/10.1007/s00205-012-0527-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-012-0527-2