On the Convergence of Solutions for the Ginzburg-Landau Equation and System

Let $(u_{\epsilon })$ be a family of solutions of the Ginzburg–Landau equation with boundary condition $u_{\epsilon }=g$ on $\mathrm{\partial }\mathrm{\Omega }$ and of degree $0$. Let $u_0$ denote the harmonic map satisfying $u_0=g$ on $\mathrm{\partial }\mathrm{\Omega }$. We show that, if there exists a constant $C_1>0$ such that for $\epsilon$ sufficiently small we have $\frac{1}{2}{\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }{u}_{\text{ve}}{\mathrm{\mid }}^{2}dx\le C_1\le \frac{1}{2}{\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }{u}_0{\mathrm{\mid }}^{2}dx,$ then $C_1=\frac{1}{2}{\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }{u}_0{\mathrm{\mid }}^{2}dx$ and $u_{\text{ve}}\to u_0\text{qin}{H}^1(\text{Om})$. We also prove that if there is a constant $C_2$ such that for $\text{ve}$ small enough we have $\frac{1}{2}{\int }_{\text{Om}}\mathrm{\mid }\mathrm{\nabla }{u}_{\text{ve}}{\mathrm{\mid }}^{2}dx\ge C_2>\frac{1}{2}{\int }_{\text{Om}}\mathrm{\mid }\mathrm{\nabla }{u}_0{\mathrm{\mid }}^{2}dx,$ then $\mathrm{\mid }{u}_{\text{ve}}\mathrm{\mid }$ does not converge uniformly to $1$ on $\overline{\text{Om}}$. We obtain analogous results for both symmetric and non-symmetric two-component Ginzburg–Landau systems.