[Résultats de convergence pour l’équation de Ginzburg–Landau et ses généralisations aux systèmes]
Let $(u_\varepsilon )$ be a family of solutions of the Ginzburg–Landau equation with boundary condition $u_\varepsilon = g$ on $\partial \Omega $ and of degree $0$. Let $u_0$ denote the harmonic map satisfying $u_0 = g$ on $\partial \Omega $. We show that, if there exists a constant $C_1 > 0$ such that for $\varepsilon $ sufficiently small we have $\frac{1}{2} \int _\Omega \vert {\nabla u_\varepsilon }\vert ^2 \, \mathrm{d}x \le C_1 \le \frac{1}{2} \int _\Omega \vert {\nabla u_0}\vert ^2 \, \mathrm{d}x,$ then $C_1 = \frac{1}{2} \int _\Omega \vert {\nabla u_0}\vert ^2 \, \mathrm{d}x$ and $u_\varepsilon \rightarrow u_0$ in $H^1(\Omega )$. We also prove that if there is a constant $C_2$ such that for $\varepsilon $ small enough we have $\frac{1}{2} \int _\Omega \vert {\nabla u_\varepsilon }\vert ^2 \, \mathrm{d}x \ge C_2 > \frac{1}{2} \int _\Omega \vert {\nabla u_0}\vert ^2 \, \mathrm{d}x,$ then $\vert {u_{\varepsilon }}\vert $ does not converge uniformly to $1$ on $\overline{\Omega } $. We obtain analogous results for both symmetric and non-symmetric two-component Ginzburg–Landau systems.
Soit $(u_\varepsilon )$ une famille de solutions de l’équation de Ginzburg–Landau avec la condition au bord $u_\varepsilon = g$ sur $\partial \Omega $ et le degré de $g$ est $0$. On note $u_0$ l’application harmonique vérifiant $u_0 = g$ sur $\partial \Omega $. Nous montrons que, s’il existe une constante $C_1 > 0$ telle que, pour $\varepsilon $ suffisamment petit, $ \frac{1}{2} \int _\Omega \vert {\nabla u_\varepsilon }\vert ^2 \, \mathrm{d}x \le C_1 \le \frac{1}{2} \int _\Omega \vert {\nabla u_0}\vert ^2 \, \mathrm{d}x, $ alors $ C_1 = \frac{1}{2} \int _\Omega \vert {\nabla u_0}\vert ^2 \, \mathrm{d}x $ et $ u_\varepsilon \rightarrow u_0$ dans $H^1(\Omega )$. Nous prouvons également que s’il existe une constante $C_2$ telle que, pour $\varepsilon $ suffisamment petit, $ \frac{1}{2} \int _\Omega \vert {\nabla u_\varepsilon }\vert ^2 \, \mathrm{d}x \ge C_2 > \frac{1}{2} \int _\Omega \vert {\nabla u_0}\vert ^2 \, \mathrm{d}x, $ alors $\vert {u_{\varepsilon }}\vert $ ne converge pas uniformément vers $1$ sur $\overline{\Omega }$. Nous obtenons des résultats analogues pour des systèmes de Ginzburg–Landau à deux composantes, symétriques et non symétriques.
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Keywords: Two component Ginzburg–Landau equations, non-symmetric potential, asymptotic behavior of solutions
Mots-clés : Équations de Ginzburg–Landau à deux composantes, potentiel non symétrique, comportement asymptotique des solutions
Rejeb Hadiji 1 ; Jongmin Han 2
CC-BY 4.0
@article{CRMATH_2026__364_G1_45_0,
author = {Rejeb Hadiji and Jongmin Han},
title = {On the convergence of solutions for the {Ginzburg{\textendash}Landau} equation and system},
journal = {Comptes Rendus. Math\'ematique},
pages = {45--58},
year = {2026},
publisher = {Acad\'emie des sciences, Paris},
volume = {364},
doi = {10.5802/crmath.812},
language = {en},
}
Rejeb Hadiji; Jongmin Han. On the convergence of solutions for the Ginzburg–Landau equation and system. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 45-58. doi: 10.5802/crmath.812
[1] Asymptotics for the minimization of a Ginzburg–Landau functional, Calc. Var. Partial Differ. Equ., Volume 1 (1993) no. 2, pp. 123-148 | DOI | MR | Zbl
[2] Ginzburg–Landau vortices, Progress in Nonlinear Differential Equations and their Applications, 13, Birkhäuser, 1994 | DOI | MR | Zbl
[3] Quantization effects for in , Arch. Ration. Mech. Anal., Volume 126 (1994) no. 1, pp. 35-58 | DOI | MR | Zbl
[4] On a system of multi-component Ginzburg–Landau vortices, Adv. Nonlinear Anal., Volume 12 (2023) no. 1, 20220315, 18 pages | DOI | MR | Zbl
[5] Two-component Ginzburg–Landau vortices with a non-symmetric potential. Part I: Zero degree case (2025) (Submitted) | HAL
[6] Minimization of a Ginzburg–Landau type energy with potential having a zero of infinite order, Differ. Integral Equ., Volume 19 (2006) no. 10, pp. 1157-1176 | MR | Zbl
[7] Existence and stability of semilocal strings, Phys. Rev. Lett., Volume 68 (1992) no. 9, pp. 1263-1266 | DOI | MR | Zbl
[8] Minimax solutions of the Ginzburg–Landau equations, Sel. Math., New Ser., Volume 3 (1997) no. 1, pp. 99-113 | DOI | MR | Zbl
[9] On the basic concentration estimate for the Ginzburg–Landau equation, Differ. Integral Equ., Volume 11 (1998) no. 5, pp. 771-779 | MR | Zbl
[10] On the asymptotic behavior of minimizers of the Ginzburg–Landau model in dimensions, Differ. Integral Equ., Volume 7 (1994) no. 5–6, pp. 1613-1624 | MR | Zbl
[11] Semilocal cosmic strings, Phys. Rev. D, Volume 44 (1991) no. 10, pp. 3067-3071 | DOI | MR
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