Periodic Solutions to a Cahn–Hilliard–Willmore Equation in the Plane

In this paper we construct entire solutions to the phase field equation of Willmore type - Δ (- Δ u+ W′(u)) + W″(u) (- Δ u+ W′(u)) = 0 in the Euclidean plane, where W(u) is the standard double-well potential 14(1-u2)2. Such solutions have a non-trivial profile that shadows a Willmore planar curve, and converge uniformly to ± 1 as x2→ ± ∞. 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 x2-derivative of these solutions using the special structure of Willmore’s equation.