Advertisement

Mathematische Annalen

, Volume 328, Issue 1–2, pp 1–25 | Cite as

Non trivial L q solutions to the Ginzburg-Landau equation

  • Susana GutiérrezEmail author
Article

Abstract

It is shown that the contour problem for the stationary Ginzburg-Landau equation where ◯=x/r with r=|x|, is well posed in L 4 (ℝ n ) for a class of small data fL 2 (S n−1 ).

Keywords

Small Data Contour Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agmon, S., Hörmander, L.: Asymptotic properties of solutions of differential equations with simple characteristics. J. Anal. Math. 30, 1–38 (1976)zbMATHGoogle Scholar
  2. 2.
    Bergh, J., Löfström, J.: Interpolation of spaces, An introduction. Springer-Verlag, Berlin-Heidelberg-New York, 1976Google Scholar
  3. 3.
    Bethuel, F., Brezis, H., Hélein, H.: Ginzburg-Landau Vortices. Birkhäuser, 1993Google Scholar
  4. 4.
    Bennet, C., Sharpley, R.: Interpolation of operators. Academic Press Inc., Orlando, Florida, 1988Google Scholar
  5. 5.
    Bak, J.-G., McMichael, D., Oberlin, D.: L p-L q Estimates off the line of duality. J. Austral. Math. Soc. 58, 154–166 (1995)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Brezis, H., Merle, F., Rivière, T.: Quantization effects for -Δu=u(1-|u|)2 in ℝ2. Arch. Rational Mech. Anal. 126, 35–58 (1994)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Börjeson, L.: Estimates for the Bochner-Riesz operator with negative index. Indiana U. Math. J., 35 (2), 225–233 (1986)Google Scholar
  8. 8.
    Brezis, H.: Análisis funcional Teoría y aplicaciones. Alianza Editorial, Madrid, 1984Google Scholar
  9. 9.
    Carbery, A., Soria, F.: Almost-everywhere convergence of Fourier integrals for functions in Sobolev spaces, and an L 2-localisation principle. Rev. Mat. Ibero. 4, 319–337 (1988)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chen, X., Elliot, C., Tang, Q.: Shooting method for vortex solutions of a complex valued Ginzburg-Landau equation. Proc. Roy. Soc. Edin. 124A, 1075–1088 (1994)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Gutiérrez, S.: A note on restricted weak-type estimates for Bochner-Riesz operators with negative index in ℝn, n≥ 2. Proc. Amer. Math. Soc. 128, 495–501 (2000)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Gutiérrez, S.: Un problema de contorno para la ecuación de Ginzburg-Landau. PhD. Thesis, Basque Country University, 2000Google Scholar
  13. 13.
    Hervé, R-M., Hervé, M.: Etude qualitative des solutions réelles d’une équation différentielle liée à l’équation de Ginzburg-Landau. Ann. Inst. Henri Poincaré 11, 427–440 (1994)Google Scholar
  14. 14.
    Hörmander, L.: The Análysis of linear partial differential operators I. A series of comprensive studies in mathematics 256, Springer-Verlag, Berlin-Heidelberg-New York, 1983Google Scholar
  15. 15.
    Kenig, C.E., Ruiz, A., Sogge, D.: Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators. Duke Math. J. 55, 329–347 (1987)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ruiz, A., Vega, L.: On local regularity of Schrödinger equations. Internat. Math. Res. Notices, 13–27 (1993)Google Scholar
  17. 17.
    Sogge, C.: Oscillatory integrals and spherical harmonics. Duke Math. J. 53, 43–65 (1986)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Stein, E.M.: Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals. Princeton University Press, Princeton, New Jersey, 1993Google Scholar
  19. 19.
    Stein, E.M., Weiss, G.: An introduction to Fourier analysis on Euclidean spaces. Princeton University Press, Princeton, New Jersey, 1975Google Scholar
  20. 20.
    Tinkham, M.: Introduction to Superconductivity. 2nd edn. MacGraw-Hill, New York, 1990Google Scholar
  21. 21.
    Tychonov, A.N., Samarski, A.A.: Partial differential equations of mathematical physics. Holden-Day Inc. San Francisco, California, 1967Google Scholar
  22. 22.
    Tomas, P.: A restriction theorem for Fourier transform. Bull. Amer. Math. Soc. 81, 477–478 (1975)zbMATHGoogle Scholar
  23. 23.
    Tomas, P.A.: Restriction theorems for the Fourier transform. Harmonic Analysis in Euclidean Spaces (2 volumes), G. Weiss and W.Wainger (eds.), Proc. Symp. Pure Math. # 35, Amer. Math. Soc. 1979, part 1, 111–114Google Scholar
  24. 24.
    Vladimirov, V.S.: Equations of mathematical physics. Mir Publishers, Moscow, 1981Google Scholar
  25. 25.
    Watson, G.W.: Theory of Bessel functions. Cambridge University Press, Cambridge, 1944Google Scholar
  26. 26.
    Zhang, B.: Radiation condition and limiting amplitude principle for acoustic propagators with two unbounded media. Proc. Roy. Soc Edin. 128A, 173–192 (1998)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Departamento de MatemáticasFacultad de Ciencias, Universidad del País Vasco (UPV-EHU)BilbaoSpain

Personalised recommendations