Asymptotics for the minimization of a Ginzburg-Landau functional

AbstractLetΩ ⊂ ℝ2 be a smooth bounded simply connected domain. Consider the functional $$E_\varepsilon (u) = \frac{1}{2}\int\limits_\Omega {\left| {\nabla u} \right|^2 + \frac{1}{{4\varepsilon ^2 }}} \int\limits_\Omega {(|u|^2 - 1)^2 } $$ on the classHg1={u εH1(Ω; ℂ);u=g on ∂Ω} whereg:∂Ω∂ → ℂ is a prescribed smooth map with ¦g¦=1 on ∂Ω∂ and deg(g, ∂Ω)=0. Let uuε be a minimizer for Eε onHg1. We prove that uε → u0 in $$C^{1,\alpha } (\bar \Omega )$$ as ε → 0, where u0 is identified. Moreover $$\left\| {u_\varepsilon - u_0 } \right\|_{L^\infty } \leqslant C\varepsilon ^2 $$ .