Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T01:18:20.615Z Has data issue: false hasContentIssue false
  • English
  • Français

Boundary value problems of the Ginzburg–Landau equations

Published online by Cambridge University Press:  14 November 2011

Yisong Yang
Yisong Yang
Affiliation:
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

For a domain Ω in the Euclidean space Rd (d = 2, 3) existence of weak solutions for both interior and exterior Dirichlet boundary value problems of the Ginzburg-Landau equations are established without any restriction on the range of the coupling constant λ, thesize of Ω, or the boundary data. For the critical choice λ = 1, we prove the existence of confined multivortices in a bounded domain by a constructive monotone iteration method.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 114 , Issue 3-4 , 1990 , pp. 355 - 365
Copyright
Copyright © Royal Society of Edinburgh 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Aharonov, Y. and Bohm, D.. Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115 (1959), 485–491.CrossRefGoogle Scholar
2Aharanov, Y. and Bohm, D.. Further considerations on electromagnetic potentials in the quantum theory. Phys. Rev. 123 (1969), 1511–1524.CrossRefGoogle Scholar
3Bogomol'nyi, E. B.. The stability of classical solutions. SovietJ. Nuclear Phys. 24 (1976), 449–454.Google Scholar
4Carroll, R. W. and Glick, A. J.. On the Ginzburg–Landau equations. Arch. Rational Mech. Anal. 16 (1964), 373–384.CrossRefGoogle Scholar
5Felsager, B.. Geometry, Particles, and Fields, 2nd edn (Odense: Odense University Press, 1983).CrossRefGoogle Scholar
6Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order (Berlin: Springer, 1977).CrossRefGoogle Scholar
7Ginzburg, V. L. and Landau, L. D.. On the theory of superconductivity. In Collected Papers of L. D. Landau, ed. Haar, D. ter, pp. 546–568 (New York: Pergamon, 1965).Google Scholar
8Gorkov, L. P.. Microscopic derivation of the Ginzburg–Landau equations in the theory of superconductivity. Soviet Phys. JETP 9 (1959), 1364–1367.Google Scholar
9Jacobs, L. and Rebbi, C.. Interaction energy of superconducting vortices. Phys. Rev. B(3) 19 (1978), 4486–4494.CrossRefGoogle Scholar
10Jaffe, A. and Taubes, C. H.. Vortices and Monopoles (Boston: Birkhäuser, 1980).Google Scholar
11Klimov, V. S.. Nontrivial solutions of the Ginzburg–Landau equations. Theoret. and Math. Phys. 50 (1982), 383–389.CrossRefGoogle Scholar
12Ladyzhenskaya, O. A.. The Mathematical Theory of Viscous Incompressible Flow (New York: Gordon and Breach, 1969).Google Scholar
13Lohe, M. A.. Generalized noninteracting vortices. Phys. Rev. D(3) 23 (1981), 2335–2339.Google Scholar
14Lohe, M. A. and Hoerk, J. van der. Existence and uniqueness of generalizedvortices. J. Math. Phys. 24 (1983), 148–153.CrossRefGoogle Scholar
15Nielsen, H. B. and Olesen, P.. Vortex-line models for dual strings. Nuclear Phys. B 61 (1973), 45–61.CrossRefGoogle Scholar
16Nirenberg, L.. Variational and topological methods in nonlinear problems. Bull. Amer. Math. Soc. 4 (1981), 267–302.CrossRefGoogle Scholar
17Rabinowitz, P. H.. Minimax Methods in Critical Point Theory with Applications to Differential Equations, Regional Conf. Ser. Math. 65, (Providence, R. I.: American Mathematical Society, 1986).CrossRefGoogle Scholar
18Rebbi, C.. Interaction of superconducting vortices. In Geom. Topo. Methods in Gauge Theory, pp. 96–113. Lecture Notes in Physics 129 (Berlin: Springer, 1980).Google Scholar
19Taubes, C. H.. Arbitrary N-vortex solutions to the first order Ginzburg–Landau equations. Comm. Math. Phys. 72 (1980), 277–292.CrossRefGoogle Scholar
20Taubes, C. H.. On the equivalence of the first and second order equationsfor gauge theories. Comm. Math. Phys. 75 (1980), 207–227.CrossRefGoogle Scholar
21Weinberg, E. J.. Multivortex solutions of the Ginzburg-Landau equations. Phys. Rev. D(3) 19 (1979), 3008–3012.Google Scholar