01188nas a2200181 4500008004100000022001400041245007200055210007100127300001600198490000800214520059300222653002900815653001900844653003300863653002100896100001800917856007100935 2017 eng d a0022-039600aClifford Tori and the singularly perturbed Cahn–Hilliard equation0 aClifford Tori and the singularly perturbed Cahn–Hilliard equatio a5306 - 53620 v2623 a
In this paper we construct entire solutions uε to the Cahn–Hilliard equation −ε2Δ(−ε2Δu+W′(u))+W″(u)(−ε2Δu+W′(u))=ε4λε(1−uε), under the volume constraint ∫R3(1−uε)2dx=82π2cε, with cε→1 as ε→0, whose nodal set approaches the Clifford Torus, that is the Torus with radii of ratio 1/2 embedded in R3, as ε→0. It is crucial that the Clifford Torus is a Willmore hypersurface and it is non-degenerate, up to conformal transformations. The proof is based on the Lyapunov–Schmidt reduction and on careful geometric expansions of the Laplacian.
10aCahn–Hilliard equation10aClifford Torus10aLyapunov–Schmidt reduction10aWillmore surface1 aRizzi, Matteo uhttp://www.sciencedirect.com/science/article/pii/S0022039617300530