Multiple Delaunay ends solutions of the Cahn-Hilliard equation

TL;DR

This paper constructs new solutions to the Cahn-Hilliard equation in three-dimensional space, where the zero level set approximates a non-degenerate constant mean curvature surface with multiple Delaunay ends as the parameter approaches zero.

Contribution

It introduces a method to generate solutions whose zero level sets approximate complex CMC surfaces with Delaunay ends, expanding understanding of phase transition models.

Findings

Zero level set approaches the given CMC surface as epsilon tends to zero.

On compact subsets away from the surface, the solution magnitude approaches 1.

The solutions are constructed under the non-degeneracy assumption of the surface.

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 $\varepsilon\Delta u+\varepsilon^{-1}u(1-u^2)=\ell_\varepsilon$ in ${\mathbb R}^3$ such that as $\varepsilon\to 0$ the zero level set of $u_\varepsilon$ approaches $\Sigma$. Moreover, on compacts of the connected components of ${\mathbb R}^3\setminus \Sigma$ we have $1-|u_\varepsilon|\to 0$ uniformly.