On the convergence of solutions for the Ginzburg–Landau equation and system

Let (u_ε) be a family of solutions of the Ginzburg–Landau equation with boundary condition u_ε = g on ∂Ω and of degree 0. Let u₀ denote the harmonic map satisfying u₀ = g on ∂Ω. We show that, if there exists a constant C₁ > 0 such that for ε sufficiently small we [Formula In Abstract]. We also prove that if there is a constant C₂ such that for ε small enough we have [Formula In Abstract] dx, then |u_ε | does not converge uniformly to 1 on Ω. We obtain analogous results for both symmetric and non-symmetric two-component Ginzburg–Landau systems.