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Vortices for a variational problem related to superconductivity. (English) Zbl 0842.35119
Summary: We study minimizers of Ginzburg-Landau functionals, which depend on a parameter \(\varepsilon\). These functionals appear in superconductivity and two dimensional abelian Higgs models. We study the asymptotic limit, as \(\varepsilon \to 0\), of minimizers and show that the limiting configuration has vortices, which have topological degree one.

MSC:
35Q60 PDEs in connection with optics and electromagnetic theory
35A15 Variational methods applied to PDEs
82D55 Statistical mechanical studies of superconductors
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[1] Bethuel, F.; Brezis, H.; Hélein, F., Asymptotics for the minimization of a Ginzburg-Landau functional, Calculus of Variations, Vol. I, 123-148, (1993) · Zbl 0834.35014
[2] Bethuel, F.; Brezis, H.; Hélein, F., Ginzburg-Landau vortices, (1993), Birkhäuser · Zbl 0783.35014
[3] Bethuel, F.; Brezis, H.; Hélhn, F., Limite singulière pour la minimisation de fonctionnelles du type Ginzburg-Landau, C. R. Acad. Sci. Paris, Vol. 314, 891-895, (1992) · Zbl 0773.49003
[4] F. Bethuel, H. Brezis and F. Hélein, Tourbillons de Ginzburg-Landau et energies renormahsées, to appear in C. R. Acad. Sci. Paris, 1993.
[5] H. Brezis, F. Merle and T. Rivière, Quantization effects for -Δu = u(1 -|u|^2) in to appear in Arch, for ratio. Mech., 1993.
[6] H. Brezis, F. Merle and T. Rivière, Quantifications pour les solutions de -Δu = u(1 -|u|^2) dans , to appear in C. R. Acad. Sci. Paris, 1993.
[7] Boutetde Monvel-Berthier, A.; Georgescu, V.; Purice, R., Sur un problème aux limites de la théorie de Ginzburg-Landau, C. R. Acad. Sci. Paris, Vol. 307, 55-58, (1988) · Zbl 0696.35058
[8] Comtet, A.; Gibbons, G. W., Bogomol’nyi bounds for cosmic strings, Nucl. Phys. B, Vol. 299, 719-733, (1988)
[9] Du, Q.; Gunzburger, M.; Peterson, J., Analysis and approximation of the Ginzburg-Landau model of superconductivity, SIAM Review, Vol. 34, 45-81, (1992) · Zbl 0787.65091
[10] Grisvard, P., Elliptic problems in non-smooth domains, (1985), Pitman Marshfields, Mass · Zbl 0695.35060
[11] Jaffe, A.; Taubes, C., Vortices and monopoles, (1980), Birkhäuser · Zbl 0457.53034
[12] Saint-James, D.; Sarma, G.; Thomas, E. J., Type II superconductivity, (1969), Pergamon Press
[13] J. Spruck and Y. Yang, Cosmic string solutions of the Einstein Matter gauge equations, to appear 1993.
[14] Spruck, J.; Yang, Y., On multivortices in the electroweak theory II: existence of bogomol’nyi solutions in , Comm. Math. Phys., Vol. 144, 215-234, (1992) · Zbl 0748.53060
[15] G. Stampacchia, Equations elliptiques du second ordre à coefficients discontinus, Presses Univ. de Montreal, 1966. · Zbl 0151.15501
[16] Yang, Y., Boundary value problems of the Ginzburg-Landau equations, Proc. Roy. Soc. Edinburgh, Vol. 114 A, 355-365, (1990) · Zbl 0708.35074
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