Multiple Delaunay ends solutions of the Cahn-Hilliard equation

Abstract

Let $Σ$ be a surface of constant mean curvature in ${\mathbb R}^3$ with multiple Delaunay ends. Assuming that $Σ$ is non degenerate in this paper we construct new solutions to the Cahn-Hilliard equation $\varepsilonΔu+\varepsilon^{-1}u(1-u^2)=\ell_\varepsilon$ in ${\mathbb R}^3$ such that as $\varepsilon\to 0$ the zero level set of $u_\varepsilon$ approaches $Σ$. Moreover, on compacts of the connected components of ${\mathbb R}^3\setminus Σ$ we have $1-|u_\varepsilon|\to 0$ uniformly.