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

Abstract

Let $\Omega$ be a smooth bounded simply connected domain in ${\mathbb{R}}^2$. We investigate the existence of critical points of the energy $E_{\epsilon }(u)=1/2{\int }_{\Omega }|\nabla u{|}^2+1/(4{\epsilon }^2){\int }_{\Omega }(1-|u{|}^2{)}^2$, 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 $\epsilon$. 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.