# Periodic solutions to the Cahn-Hilliard equation in the plane

## TL;DR

This paper constructs entire solutions to the planar Cahn-Hilliard equation that resemble Willmore curves, converge to ±1 at infinity, and serve as counterexamples to a Gibbons' conjecture analogue for fourth-order equations.

## Contribution

It introduces a novel class of solutions to the Cahn-Hilliard equation with specific asymptotic and geometric properties, challenging existing conjectures.

## Key findings

- Solutions shadow Willmore curves
- Solutions converge to ±1 at infinity
- Counterexample to Gibbons' conjecture for fourth-order equations

## Abstract

In this paper we construct entire solutions to the Cahn-Hilliard equation $-\Delta(-\Delta u+W^{'}(u))+W^{"}(u)(-\Delta u+W^{'}(u))=0$ in the Euclidean plane, where $W(u)$ is the standard double-well potential $\frac{1}{4} (1-u^2)^2$. Such solutions have a non-trivial profile that shadows a Willmore planar curve, and converge uniformly to $\pm 1$ as $x_2 \to \pm \infty$. These solutions give a counterexample to the counterpart of Gibbons' conjecture for the fourth-order counterpart of the Allen-Cahn equation. We also study the $x_2$-derivative of these solutions using the special structure of Willmore's equation.