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

Abstract

In this paper we construct entire solutions to the Cahn-Hilliard equation −∆(−∆u + W (u)) + W (u)(−∆u + W (u)) = 0 in the Euclidean plane, where W (u) is the standard double-well potential 1 4(1 − u 2 ) 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.

Solutions with multiple interfaces to the generalized parabolic Cahn–Hilliard equation in one and three space dimensions

“…We will get rid of the terms (−1) 𝑖+1 [ Î𝑖,1 (𝑡, 𝑥) + Î𝑖,2 (𝑡, 𝑥) ] with 𝑖 = 1, … , 𝑘 in (2.12) to improve the approximate solution. Inspired by [14,15], we define the modifying functions as the following:…”

“…We will get rid of the terms $$(-1)^{i+1}\left[\, \widehat{\textrm {I}}_{i, 1}(t, x)+\widehat{\textrm {I}}_{i, 2}(t, x)\right]$$ with $$i=1, \ldots, k$$ in () to improve the approximate solution. Inspired by [14, 15], we define the modifying functions as the following: $$$$\begin{equation} \widetilde{\xi }_i(t, x):=\omega ^{\prime }(x)\int _0^x{\left(\omega ^{\prime }(s)\right)}^{-2}\, {\mathrm{d}}s\int _{-\infty }^s\widetilde{\textrm {I}}_{i}(t, \tau)\omega ^{\prime }(\tau)\mathrm{d}\tau, \end{equation}$$$$for all $$i=1, \ldots, k$$, which satisfy $$$$\begin{equation} {\left[\partial _{xx}-W^{\prime \prime }(\omega (x))\right]} \widetilde{\xi }_i(t, x)=\widetilde{\textrm {I}}_i(t, x)...$$…”