NoDEA Nonlinear differ. equ. appl. 5 (1998) 39 – 51 1021-9722/98/010039-13 $ 1.50+0.20/0 c  Birkh¨ auser Verlag, Basel, 1998 Nonlinear Differential Equations and Applications NoDEA On a variant of the Ginzburg-Landau energy Lotfi LASSOUED Laboratoire d’Analyse Num´ erique, Universit´ e Paris VI Tour 55-65, 5-` eme ´ etage, F-75252 Paris 05 e-mail: lassoued@ann.jussieu.fr C˘at˘alinLEFTER Department of Mathematics, University “Al.I.Cuza”-Ia¸si RO-6600 Ia¸ si, Romania e-mail: lefter@uaic.ro 1 Introduction Let G be a smooth, bounded simply connected domain in IR 2 ,g : ∂G → S 1 a smooth boundary data of topological degree deg g = d. In [1],[3], F. Bethuel, H. Brezis, F. H´ elein have studied the behaviour, as ε → 0, of the minimizers u ε of the Ginzburg-Landau energy: E ε (u)= 1 2  G |∇u| 2 + 1 4ε 2  G (1 −|u| 2 ) 2 , (1.1) in the class H 1 g = H 1 g (G, IR 2 )=  u ∈ H 1 (G, C | ); u = g on ∂G  . (1.2) We study in this paper the behaviour of the minimizers u ε of the functional F ε (u)= 1 2  G |∇u| 2 + 1 4ε 2  G |u| 2 (1 −|u| 2 ) 2 , (1.3) in the same class H 1 g . It is easy to see that u ε satisfy the equation: −Δu = 1 ε 2 u(1 −|u| 2 )|u| 2 − 1 2ε 2 u(1 −|u| 2 ) 2 in G (1.4) We obtain similar results, that is, in the case of degree d = 0 we have: Theorem 1.1 If the degree deg g = d =0 then, the minimizers u ε of (1.3) satisfy: u ε → u 0 , in C 1,α ( G), 0 <α< 1, (1.5) ‖u ε − u 0 ‖ L ∞ (G) ≤ Cε 2 , (1.6)