Existence of critical points with semi-stiff boundary conditions for singular perturbation problems in simply connected planar domains

Xavier Lamy (1) , Petru Mironescu (2)

1 EDPA - Équations aux dérivées partielles, analyse
2 ICJ - Institut Camille Jordan

Let $\Omega$ be a smooth bounded simply connected domain in ${\mathbb R}^2$. We investigate the existence of critical points of the energy $E_\varepsilon(u) = 1/2\int|\nabla u|^2+1/(4\varepsilon^2)^2\int (1-|u|^2)^2$, where the complex map $u$ has modulus one and pre- scribed degree $d$ on the boundary. Under suitable nondegeneracy assumptions on $\Omega$, we prove existence of critical points for small $\varepsilon$. In particular, we prove existence of critical points of prescribed degree one in domains close to a disc.

Mots clés: Ginzburg-Landau; prescribed degrees; singular perturbation

Domaines: Equations aux dérivées partielles [math.AP]

Xavier Lamy, Petru Mironescu. Existence of critical points with semi-stiff boundary conditions for singular perturbation problems in simply connected planar domains. 2013. ⟨hal-00784904v1⟩