Multiple Delaunay ends solutions of the Cahn-Hilliard equation

Abstract

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