Periodic solutions to a cahn hilliard willmore equation in the plane

Abstract

In this paper we construct entire solutions to the phase field equation of Willmore type in the Euclidean plane, where W(u) is the standard double-well potential . Such solutions have a non-trivial profile that shadows a Willmore planar curve, and converge uniformly to as . 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.