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

Zbl 0802.35142
Full Text:

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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.