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

Petru Mironescu (1) , Itai Shafrir (2)

1 EDPA - Équations aux dérivées partielles, analyse
2 Technion - Israel Institute of Technology [Haifa]

We describe the asymptotic behavior of critical points of $\int_\Omega [|\nabla u|^2+W(u)/\varepsilon^2]$ when $\varepsilon\to 0$. 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.

Mots clés: Singular potentials; asymptotic analysis

Domaines: Equations aux dérivées partielles [math.AP]

Petru Mironescu, Itai Shafrir. Asymptotic behavior of critical points of an energy involving a loop-well potential. Communications in Partial Differential Equations, 2017, 42 (12), pp.1837-1870. ⟨hal-01532476v3⟩