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

TL;DR

This paper proves the existence of critical points for a Ginzburg-Landau type energy with semi-stiff boundary conditions in simply connected planar domains, especially for small epsilon and certain degrees, extending previous results.

Contribution

It establishes the existence of critical points under semi-stiff boundary conditions for small epsilon in general domains, with detailed results for degree one and near-disc domains.

Findings

Critical points exist for small epsilon in general domains.

Existence results are refined for degree one boundary conditions.

Critical points are shown to exist in most domains close to a disc.

Abstract

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