Asymptotics for the minimization of a Ginzburg-Landau functional. (English) Zbl 0834.35014

Authors’ abstract: Let \(\Omega \subset \mathbb{R}^2\) be a smooth bounded simply connected domain. Consider the functional \[ E_\varepsilon (u)= {1\over 2} \int_\Omega |\nabla u|^2+ {1\over {4\varepsilon^2}} \int_\Omega (|u|^2- 1)^2 \] on the class \(H^1_g= \{u\in H^1 (\Omega; \mathbb{C})\); \(u=g\) on \(\partial \Omega\}\) where \(g: \partial \Omega\to \mathbb{C}\) is a prescribed smooth map with \(|g|=1\) on \(\partial \Omega\) and \(\deg (g, \partial \Omega) =0\). Let \(u_\varepsilon\) be a minimizer for \(E_\varepsilon\) on \(H^1_g\). We prove that \(u_\varepsilon\to u_0\) in \(C^{1, \alpha} (\overline {\Omega})\) as \(\varepsilon\to 0\), where \(u_0\) is identified. Moreover \(|u_\varepsilon- u_0 |_{L^\infty} \leq C\varepsilon^2\).
Reviewer: J.Q.Liu (Beijing)


35B25 Singular perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35J60 Nonlinear elliptic equations
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
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