Periodic solutions to the Cahn-Hilliard equation in the plane

Abstract

In this paper we construct entire solutions to the Cahn-Hilliard equation $-\Delta (-\Delta u+W^{{}^{\prime }}(u))+W^{"}(u)(-\Delta u+W^{{}^{\prime }}(u))=0$ 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.

References in corpus (1)

• Saddle-shaped solutions of bistable diffusion equations in all of ${\mathbb{R}}^{2m}$