2025/09/11 by Hadiji, Rejeb, Han, Jongmin
#35B40 #35J60 #35Q60 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2509.09231
Let (uε) be a family of solutions of the Ginzburg--Landau equation with boundary condition uε = 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 have (1)/(2) ∫Ω|∇ u_\ve|2 dx ≤ C1 ≤ (1)/(2) ∫Ω|∇ u0|2 dx, then C1 = (1)/(2) ∫Ω|∇ u0|2 dx and u_\ve ~→ ~ u0 \qin H1(\Om). We also prove that if there is a constant C2 such that for \ve small enough we have \frac12 ∫_\Om |∇ u_\ve|2 dx ≥ C2 > \frac12 ∫_\Om |∇ u0|2 dx, then |u\ve| does not converge uniformly to 1 on \Om . We obtain analogous results for both symmetric and non-symmetric two-component Ginzburg--Landau systems.