A minimization problem related to the Ginzburg–Landau model

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 u_λ when λ goes to zero, for d=0 and d≠0. When d=0, we show the convergence of u_λ in various norms to a smooth harmonic map u_*  :Ω→S¹. When d≠0 (d>0) we show that u_λ converges to a smooth harmonic map u₀  :Ω\{a₁,a₂,…,a_d}→S¹, where a₁,a₂,…,a_d are the vortices where the energy of a minimizer ∫_Ω|∇u_λ|² concentrates, when λ→0.