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

Xavier Lamy (1) , Petru Mironescu (1)

1 EDPA - Équations aux dérivées partielles, analyse

Let $\Omega$ be a smooth bounded simply connected domain in $R^2$. We investigate the existence of critical points of the energy $E_\varepsilon(u)=1/2\int_\Omega |\nabla u|^2+1/(4\varepsilon^2)\int_\Omega (1-|u|^2)^2$, where the complex map $u$ has modulus one and prescribed degree $d$ on the boundary. Under suitable nondegeneracy assumptions on $\Omega$, we prove existence of critical points for small $\varepsilon$. More can be said when the prescribed degree equals one. First, we obtain existence of critical points in domains close to a disc. Next, we prove that critical points exist in ''most'' of the domains.

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

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. Journal de Mathématiques Pures et Appliquées, 2014, 102 (2), pp.385-418. ⟨hal-00784904v4⟩