Ginzburg-Landau equations and stable solutions in a rotational domain

SIAM Journal on Mathematical Analysis

Volumn 27, Issue 5, 1996, Pages 1360-1385

  Jimbo, Shuichi ^a   Morita, Yoshihisa ^b  

^a HOKKAIDO UNIVERSITY   (Japan)

^b RYUKOKU UNIVERSITY   (Japan)

Author keywords

Ginzburg Landau equation; Rotational domain; Stable solutions

Indexed keywords

EID: 0030487176     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/0527075     Document Type: Article

References (26)

1. H. AMANN, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J., 21 (1971), pp. 125-146.
2. P. BAUMAN, N. N. CARLSON, AND D. PHILLIPS, On the zeros of solutions to Ginzburg-Landau type systems, SIAM J. Math. Anal., 24 (1993), pp. 1283-1293.
3. M. BERGER AND Y. CHEN, Symmetric vortices for the Ginzburg-Landau equations of super-conductivity and the nonlinear desingularization phenomenon, J. Funct. Anal., 82 (1989), pp. 259-295.
4. F. BETHUEL, H. BREZIS, AND F. HELEIN, Limite singulière pour la minimisation de fonctionnelles du type Ginzburg-Landau, C. R. Acad. Sci. Paris Sér. I Math., 314 (1992), pp. 891-896.
5. N. A. BOBYLEV, Topological index of extremals of multidimensional variational problems, Funct. Anal. Appl., 20 (1986), pp. 89-93.
6. R. W. CARROLL AND A. J. GLICK, On the Ginzburg-Landau equations, Arch. Rational Mech. Anal., 16 (1968), pp. 373-384.
7. Y. CHEN, Nonsymmetric vortices for the Ginzburg-Landau equations on the bounded domain, J. Math. Phys., 30 (1989), pp. 1942-1950.
8. Q. DU, M. GUNZBERGER, AND J. PETERSON, Analysis and approximation of the Ginzburg-Landau model of superconductivity, SIAM Review, 34 (1992), pp. 54-81.
9. C. ELLIOT, H. MATANO, AND Q. TANG, Zeros of a complex Ginzburg-Landau order parameter with application to superconductivity, European J. Appl. Math., 5 (1994), pp. 431-448.
10. D. GILBARG AND N. TRUDINGER, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1977.
11. V. GINZBURG AND L. LANDAU, On the theory of superconductivity, Zh. Éksper. Teoret. Fiz., 20 (1950), pp. 1064-1082.
12. A. JAFFE AND C. TAUBES, Vortices and Monopoles, Birkhäuser, Boston, 1980.
13. S. JIMBO AND Y. MORITA, Stability of non-constant steady state solutions to a Ginzburg-Landau equation in higher space dimensions, Nonlinear Anal., 22 (1994), pp. 753-770.
14. V. S. KLIMOV, Nontrivial solutions of the Ginzburg-Landau equation, Theort. Math. and Phys., 50 (1982), pp. 383-389.
15. S. KOBAYASHI AND K. NOMIZU, Foundations of Differential Geometry, Vol. II, Interscience, New York, 1963.
16. O. A. LADYZENSKAYA, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Beach, New York, 1969.
17. H. MATANO AND M. MIMURA, Pattern formation in competion diffusion systems in nonconvex domains, Publ. Res. Inst. Math. Sci., 19 (1983), pp. 1049-1079.
18. H. MATANO, Existence of nontrivial unstable sets for equilibriums of strongly order preserving systems, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30 (1984), pp. 645-673.
19. A. B. MONVEL-BERTHIER, V. GEORGESCU, AND R. PRUCE, A boundary value problem related with the Ginzburg-Landau model, Comm. Math. Phys., 142 (1991), pp. 1-23.
20. F. ODEH, Existence and bifurcation theorems for the Ginzburg-Landau equations, J. Math. Phys., 8 (1967), pp. 2351-2356.
21. M. H. PROTTER AND H. F. WEINBERGER, Maximum Principle in Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1967.
22. M. REED AND B. SIMON, Methods of Mathematical Physics IV, Academic Press, New York, 1978.
23. D. SATTINGER, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1972), pp. 979-1000.
24. R. TEMAM, Navier-Stokes Equations, North-Holland, Amsterdam, 1979.
25. 3, Comm. Math. Phys., 123 (1989), pp. 147-161.
26. _, Boundary value problems of the Ginzburg-Landau equations, Proc. Roy. Soc. Edinburgh, 114A (1990), pp. 355-365.