A minimization problem related to the Ginzburg–Landau model

Article type: Research Article

Authors: Khenissy, Saïma

Affiliations: Institut National des Sciences Appliquées et de Technologie, Centre Urbain Nord, BP 676, 1080 Tunis cedex, Tunisie E‐mail: saima.khenissy@insat.rnu.tn

Abstract: We consider the minimization of the Dirichlet integral ∫Ω|∇u|2 with the constraint ∫Ω(1−|u|2)2≤λ, for maps u∈H1(Ω;$\mathbb{R} ^{2})$, where Ω⊂$\mathbb{R} ^{2}$ is a smooth, bounded and simply connected domain, u=g on ∂Ω with g :∂Ω→S1 unit circle in $\mathbb{R} ^{2}$, and λ is a positive small parameter. Denoting by d the topological degree of g, we study the asymptotic behavior of a minimizer uλ when λ goes to zero, for d=0 and d≠0. When d=0, we show the convergence of uλ in various norms to a smooth harmonic map u*  :Ω→S1. When d≠0 (d>0) we show that uλ converges to a smooth harmonic map u0  :Ω\{a1,a2,…,ad}→S1, where a1,a2,…,ad are the vortices where the energy of a minimizer ∫Ω|∇uλ|2 concentrates, when λ→0.

Journal: Asymptotic Analysis, vol. 38, no. 3-4, pp. 241-291, 2004