Archive for Rational Mechanics and Analysis

, Volume 228, Issue 3, pp 821–866 | Cite as

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 \({-\Delta(-\Delta u+W^{\prime}(u))+W^{\prime\prime}(u)(-\Delta u+W^{\prime}(u))=0}\) 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.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Scuola Normale SuperiorePisaItaly
  2. 2.Institut für AnalysisKITKarlsruheGermany
  3. 3.Centro de Modelamiento MatemáticoUniversidad de ChileSantiagoChile

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