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Clifford Tori and the singularly perturbed Cahn–Hilliard equation

TitleClifford Tori and the singularly perturbed Cahn–Hilliard equation
Publication TypeJournal Article
Year of Publication2017
AuthorsRizzi, M
JournalJournal of Differential Equations
Pagination5306 - 5362
KeywordsCahn–Hilliard equation; Clifford Torus; Lyapunov–Schmidt reduction; Willmore surface

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.


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