*Preprint*

**Inserted:** 16 may 2017

**Last Updated:** 16 may 2017

**Year:** 2017

**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.

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