Asymptotics for the minimization of a Ginzburg-Landau functional

Abstract

LetΩ ⊂ 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 classH ¹_g ={u εH ¹(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 uu _ε be a minimizer for E_ε onH ¹_g . We prove that u_ε → u₀ in$C^{1,\alpha } (\bar \Omega )$ as ε → 0, where u₀ is identified. Moreover$\left\| {u_\varepsilon - u_0 } \right\|_{L^\infty } \leqslant C\varepsilon ^2$.