SpringerLink
Log in

Uniqueness of vortexless Ginzburg-Landau type minimizers in two dimensions

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

In a simply connected two dimensional domain Ω, we consider Ginzburg-Landau minimizers u with zero degree Dirichlet boundary condition \({g \in H^{1/2}(\partial \Omega; \mathbb{S}^1)}\) . We prove uniqueness of u whenever either the energy or the Ginzburg-Landau parameter are small. This generalizes a result of Ye and Zhou requiring smoothness of g. We also obtain uniqueness when Ω is multiply connected and the degrees of the vortexless minimizer u are prescribed on the components of the boundary, generalizing a result of Golovaty and Berlyand for annular domains. The proofs rely on new global estimates connecting the variation of |u| to the Ginzburg-Landau energy of u. These estimates replace the usual global pointwise estimates satisfied by \({\nabla u}\) when g is smooth, and apply to fairly general potentials. In a related direction, we establish new uniqueness results for critical points of the Ginzburg-Landau energy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berlyand L., Mironescu P.: Ginzburg-Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices. J. Funct. Anal. 239(1), 76–99 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Berlyand L., Mironescu P.: Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain. Netw. Heterog. Media. 3(3), 461–487 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bethuel F., Ghidaglia J.-M.: Improved regularity of solutions to elliptic equations involving Jacobians and applications. J. Math. Pures Appl. (9) 72(5), 441–474 (1993)

    MathSciNet  MATH  Google Scholar 

  4. Bethuel F., Brezis H., Hélein F.: Asymptotics for the minimization of a Ginzburg-Landau functional. Calc. Var. Partial Differ. Equ. 1(2), 123–148 (1993)

    Article  MATH  Google Scholar 

  5. Bethuel F., Brezis H., Hélein F.: Ginzburg-Landau Vortices. Progress in Nonlinear Differential Equations and Their Applications, vol. 13. Birkhäuser Boston Inc., Boston (1994)

    Google Scholar 

  6. Bethuel F., Brezis H., Orlandi G.: Small energy solutions to the Ginzburg-Landau equation. C. R. Acad. Sci. Paris Sér. I Math. 331(10), 763–770 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bourgain J., Brezis H., Mironescu P.: Lifting in Sobolev spaces. J. Anal. Math. 80, 37–86 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brezis, H.: Degree theory: old and new. In Topological Nonlinear Analysis, II (Frascati, 1995), vol. 27 of Progress in Nonlinear Differential Equations Applications, pp. 87–108. Birkhäuser Boston Inc., Boston (1997)

  9. Brezis, H., Mironescu, P.: Sobolev maps with values into the circle. Analytical, Geometrical and Topological Aspects (in preparation)

  10. Brezis H., Nirenberg L.: Degree theory and BMO. I. Compact manifolds without boundaries. Sel. Math. (N.S.). 1(2), 197–263 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Brezis H., Nirenberg L.: Degree Theory and BMO, Part II: Compact manifolds with boundaries. Sel. Math. (N.S.). 2, 309–368 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Comte M., Mironescu P.: Minimizing properties of arbitrary solutions to the Ginzburg-Landau equation. Proc. R. Soc. Edinb. Sect. A. 129(6), 1157–1169 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dos Santos M., Mironescu P., Misiats O.: The Ginzburg-Landau functional with a discontinuous and rapidly oscillating pinning term. Part I: the zero degree case. Commun. Contemp. Math. 13(5), 1–30 (2011)

    Article  MathSciNet  Google Scholar 

  14. Golovaty D., Berlyand L.: On uniqueness of vector-valued minimizers of the Ginzburg-Landau functional in annular domains. Calc. Var. Partial Differ. Equ. 14(2), 213–232 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Krasnosel’skiĭ, M.A.: In: Boron, L.F. (ed.) Positive Solutions of Operator Equations (translated from the Russian by Flaherty, R.E.). P. Noordhoff Ltd., Groningen (1964)

  16. Lassoued L., Mironescu P.: Ginzburg-Landau type energy with discontinuous constraint. J. Anal. Math. 77, 1–26 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lin F.H., Rivière T.: Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents. J. Eur. Math. Soc. (JEMS). 1(3), 237–311 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lin F.H., Rivière T.: A quantization property for static Ginzburg-Landau vortices. Commun. Pure Appl. Math. 54(2), 206–228 (2001)

    Article  MATH  Google Scholar 

  19. Mironescu P.: Explicit bounds for solutions to a Ginzburg-Landau type equation. Rev. Roumaine Math. Pures Appl. 41(3–4), 263–271 (1996)

    MathSciNet  MATH  Google Scholar 

  20. Mironescu P.: Les minimiseurs locaux pour l’équation de Ginzburg-Landau sont à à symétrie radiale. C. R. Acad. Sci. Paris Sér. I Math. 323(6), 593–598 (1996)

    MathSciNet  MATH  Google Scholar 

  21. Murat F.: Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5(3), 489–507 (1978)

    MathSciNet  MATH  Google Scholar 

  22. Pacard, F., Rivière, T.: Linear and nonlinear aspects of vortices. Progress in Nonlinear Differential Equations and Their Applications, vol. 39, The Ginzburg-Landau model. Birkhäuser Boston Inc., Boston, MA (2000)

  23. Rivière, T.: Line vortices in the U(1)-Higgs model. ESAIM Cont. Optim. Calc. Var. 1, 77–167 (electronic) (1995/1996)

  24. Struwe M.: On the asymptotic behavior of minimizers of the Ginzburg-Landau model in 2 dimensions. Differ. Integral Equ. 7(5–6), 1613–1624 (1994)

    MathSciNet  MATH  Google Scholar 

  25. Wente H.C.: An existence theorem for surfaces of constant mean curvature. J. Math. Anal. Appl. 26, 318–344 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ye D., Zhou F.: Uniqueness of solutions of the Ginzburg-Landau problem. Nonlinear Anal. 26(3), 603–612 (1996)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculté de Sciences, LAMFA, CNRS UMR 6140, Université de Picardie Jules Verne, 33, Rue Saint Leu, 80039, Amiens Cedex 1, France

    Alberto Farina

  2. Université de Lyon, Université Lyon 1, INSA de Lyon, Ecole Centrale de Lyon, UMR 5208, Institut Camille Jordan, 43 blvd du 11 novembre 1918, 69622, Villeurbanne Cédex, France

    Petru Mironescu

Authors
  1. Alberto Farina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Petru Mironescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alberto Farina.

Additional information

Communicated by L. Ambrosio.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Farina, A., Mironescu, P. Uniqueness of vortexless Ginzburg-Landau type minimizers in two dimensions. Calc. Var. 46, 523–554 (2013). https://doi.org/10.1007/s00526-012-0492-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-012-0492-5

Mathematics Subject Classification (2000)

Search

Navigation