Asymptotic behavior of critical points of an energy involving a loop-well potential

Abstract

We describe the asymptotic behavior of critical points of ${\int }_{\Omega }[(1/2)|\nabla u{|}^2+W(u)/{\epsilon }^2]$ when $\epsilon \to 0$. Here, $W$ is a Ginzburg-Landau type potential, vanishing on a simple closed curve $\Gamma$. Unlike the case of the standard Ginzburg-Landau potential $W(u)=(1-|u{|}^2{)}^2/4$, studied by Bethuel, Brezis and Hélein, we do not assume any symmetry on $W$ or $\Gamma$. In order to overcome the difficulties due to the lack of symmetry, we develop new tools which might be of independent interest.