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Topological methods for the Ginzburg-Landau equations. (English) Zbl 0904.35023

This paper is concerned with the Dirichlet problem for the Ginzburg-Landau equation for complex-valued function \(u:\Omega\to\mathbb{C}\), \[ -\Delta u= {1\over\varepsilon^2} u(1-| u|^2)\quad\text{in }\Omega,\quad u= g\quad\text{on }\partial\Omega, \] where \(\Omega\) is a smooth, bounded and simply connected domain in \(\mathbb{R}^2\), \(g\) is a smooth map from \(\partial\Omega\) to \(S^1= \{z\in \mathbb{C};| z|= 1\}\), and where \(\varepsilon> 0\) is a small parameter. Solutions to this problem are critical points of the functional: \[ E_\varepsilon(u)= {1\over 2} \int_\Omega \Biggl\{|\nabla u|^2+{1\over 2\varepsilon^2} (1-| u|^2)^2\Biggr\}. \] \(E_\varepsilon\) is a \(C^1\)-functional (satisfying the Palais-Smale condition) on the Sobolev space \(H^1_g(\Omega;\mathbb{R}^2)= \{u\in H^1(\Omega; \mathbb{R}^2)\); \(u= g\) on \(\partial\Omega\}\). Minimizers of this functional have been studied extensively in the book of F. Bethuel, H. Brezis and F. Hélein [Ginzburg-Landau Vortices, Birkhäuser (1994; Zbl 0802.35142)], where an asymptotic analysis as \(\varepsilon\to 0\) was carried out. The aim of the present paper is to develop a variational framework in order to find non-minimizing solutions. The main result is:
Theorem 1. Assume that \(\text{deg}(g,\partial\Omega)\geq 2\). Then there is some \(\varepsilon_0> 0\) such that if \(\varepsilon\leq\varepsilon_0\), then at least three distinct solutions exist and one of these is not minimizing.
The proof of Theorem 1 is based on the study of the topology of the level sets of \(E_\varepsilon\), and this study is reduced to a similar problem for the renormalized energy (introduced in the above book) defined on a finite-dimensional space.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
82B26 Phase transitions (general) in equilibrium statistical mechanics

Citations:

Zbl 0802.35142
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References:

[1] Almeida, L., The regularity problem for generalized harmonic maps into homogeneous spaces, Calc. Var., 3, 193-242 (1995) · Zbl 0820.58013
[2] Almeida, L.; Bethuel, F., Méthodes topologiques pour l’équation de Ginzburg-Landau, C.R. Acad. Sci. Paris, 320, 935-938 (1995) · Zbl 0826.35036
[4] Almgreen, F.; Browder; Lieb, E. H., Coarea, liquid crystals and minimal surfaces, in DDT-a selection of papers (1987), Springer
[5] Berger, M. S.; Chen, Y. Y., Symmetric vortices for the nonlinear Ginzburg-Landau equation of superconductivity and the nonlinear desingularization phenomenon, J. Funct. Analysis, 82, 259-295 (1989) · Zbl 0685.46051
[6] Bethuel, F.; Brezis, H.; Coron, J. M., Relaxed energies for harmonic maps, (Berestycki; Coron; Ekeland, Variational Methods (1992)), Birhäuser · Zbl 0793.58011
[7] Bethuel, F.; Brezis, H.; Hélein, F., Ginzburg-Landau Vortices (1994), Birkhäuser · Zbl 0802.35142
[8] Bethuel, F.; Brezis, H.; Hélein, F., Asymptotics for the minimization of a Ginzburg-Landau functional, Calculus of variations and PDE’s, 1, 123-148 (1993) · Zbl 0834.35014
[9] Bethuel, F.; Coron, J. M.; Ghidaglia, J. M.; Soyeur, A., Heat-flows and relaxed energies for harmonic maps, (Nonlinear Diffusion Equations and Their Equilibrium State (1992)), 99-109, Gregynog, Birkhäuser · Zbl 0795.35053
[10] Bethuel, F.; Rivière, T., Vortices for a minimization problem related to superconductivity, Annales IHP, Analyse Non Linéaire, 12, 243-303 (1995) · Zbl 0842.35119
[11] Brezis, H.; Coron, J. M.; Leib, E. H., Harmonic maps with defects, Comm. Math. Phys., 107, 649-705 (1986) · Zbl 0608.58016
[12] Brezis, H.; Nirenberg, L., Degree theory and BMO 1: Compact manifolds without boundaries, Selecta Math., 1, 197-263 (1995) · Zbl 0852.58010
[13] Coron, J. M., Topologie et cas limites des injections de Sobolev, C.R. Acad. Sci. Paris, 299, 209-212 (1984) · Zbl 0569.35032
[15] Kelley, J. L., (General Topology (1955), Springer) · Zbl 0066.16604
[16] Kikuchi, N., An approach to the construction of Morse flows for variational functionals, (Coron; Hélein; Ghidaglia, Nematics (1991), Kluwer) · Zbl 0850.76043
[17] Lin, F. H., Solutions of Ginzburg-Landau equations and critical points of the renormalized energy, Annales IHP, Analyse Non Linéaire, 12, 599-622 (1995) · Zbl 0845.35052
[19] Mac Duff, D., Configuration spaces of positive and negative particles, Topology, 14, 91-107 (1974)
[20] Schoen, R.; Uhlenbeck, K., Boundary regularity and the Dirichlet problem for harmonic maps, J. Diff. Geom., 18, 253-286 (1983) · Zbl 0547.58020
[21] Sibner, L.; Sibner, R.; Uhlenbeck, K., Solutions to Yang-Mills that are not selfdual, (Proc. Nat. Acad. Sci., 86 (1982)), 257-298 · Zbl 0731.53031
[23] Struwe, M., J. Diff. Int. Equ., 8, 224 (1995) · Zbl 0817.35029
[24] Struwe, M., (Variational Methods (1990), Springer)
[25] Taubes, C., Min-Max theory for the Yang-Mills-Higgs Equations, Comm. Math. Phys., 97, 473-540 (1985) · Zbl 0585.58016
[26] White, B., Infima of energy functional in homotopy classes of mappings, J. Diff. Geom., 23, 127-142 (1986) · Zbl 0588.58017
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