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.

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

Andrea Malchiodi
;
2018

Abstract

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.
Settore MAT/05 - Analisi Matematica
Cahn-Hilliard equation; Willmore curves; Gibbons conjecture.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11384/68585
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