Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Gauge Invariant Eigenvalue Problems
in ${\mathbb{R}}^{2}$ and in ${\mathbb{R}}^{2}_{+}$

Authors: Kening Lu and Xing-Bin Pan
Journal: Trans. Amer. Math. Soc. 352 (2000), 1247-1276
MSC (1991): Primary 82D55
Published electronically: October 6, 1999
MathSciNet review: 1675206
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is devoted to the study of the eigenvalue problems for the Ginzburg-Landau operator in the entire plane ${\mathbb{R}}^{2}$ and in the half plane ${\mathbb{R}}^{2}_{+}$. The estimates for the eigenvalues are obtained and the existence of the associate eigenfunctions is proved when $curl\ A$ is a non-zero constant. These results are very useful for estimating the first eigenvalue of the Ginzburg-Landau operator with a gauge-invariant boundary condition in a bounded domain, which is closely related to estimates of the upper critical field in the theory of superconductivity.

References [Enhancements On Off] (What's this?)

  • [BBH] Fabrice Bethuel, Haïm Brezis, and Frédéric Hélein, Ginzburg-Landau vortices, Progress in Nonlinear Differential Equations and their Applications, vol. 13, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1269538
  • [CHO] S. J. Chapman, S. D. Howison and J. R. Ockendon, Macroscopic models for superconductivity, SIAM Review, 34(1992), 529-560. MR 94b:84037
  • [DGP] Qiang Du, Max D. Gunzburger, and Janet S. Peterson, Analysis and approximation of the Ginzburg-Landau model of superconductivity, SIAM Rev. 34 (1992), no. 1, 54–81. MR 1156289,
  • [dG] P. G. De Gennes, Superconductivity of Metals and Alloys, Benjamin, New York and Amsterdam, (1966).
  • [GL] V. Ginzburg and L. Landau, On the theory of superconductivity, Zh. Ekaper. Teoret. Fiz. 20(1950), 1064-1082; English transl., Collected Papers of L. D. Landau, Gordon and Breach, New York, 1965, pp. 546-568.
  • [JT] Arthur Jaffe and Clifford Taubes, Vortices and monopoles, Progress in Physics, vol. 2, Birkhäuser, Boston, Mass., 1980. Structure of static gauge theories. MR 614447
  • [L] Fang-Hua Lin, Solutions of Ginzburg-Landau equations and critical points of the renormalized energy, Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995), no. 5, 599–622 (English, with English and French summaries). MR 1353261
  • [LP1] Kening Lu and Xing-Bin Pan, Ginzburg-Landau equation with DeGennes boundary condition, J. Differential Equations 129 (1996), no. 1, 136–165. MR 1400799,
  • [LP2] Kening Lu and Xing-Bin Pan, Eigenvalue problems for Ginzburg-Landau operator in bounded domains, J. Math. Phys., 40 (1999), 2647-2670. CMP 99:13
  • [LP3] Kening Lu and Xing-Bin Pan, Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity, Physica D, 127 (1999), 73-104. CMP 99:10
  • [SG] D. Saint-James and P. G. De Gennes, Onset of superconductivity in decreasing fields, Phys. Letters, 7(1963), 306-308.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 82D55

Retrieve articles in all journals with MSC (1991): 82D55

Additional Information

Kening Lu
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602

Xing-Bin Pan
Affiliation: Center for Mathematical Sciences, Zhejiang University, Hangzhou 310027, P.R. China; Department of Mathematics, National University of Singapore, Singapore

Keywords: Superconductivity, Ginzburg-Landau operator, eigenvalue
Received by editor(s): November 1, 1996
Received by editor(s) in revised form: December 18, 1997
Published electronically: October 6, 1999
Article copyright: © Copyright 1999 American Mathematical Society