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.
Periodic Solutions to a Cahn–Hilliard–Willmore Equation in the Plane
Rizzi M.
2018-01-01
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.File in questo prodotto:
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