Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5774202 | Journal of Differential Equations | 2017 | 57 Pages |
Abstract
In this paper we construct entire solutions uε to the Cahn-Hilliard equation âε2Î(âε2Îu+Wâ²(u))+Wâ³(u)(âε2Îu+Wâ²(u))=ε4λε(1âuε), under the volume constraint â«R3(1âuε)2dx=82Ï2cε, with cεâ1 as εâ0, whose nodal set approaches the Clifford Torus, that is the Torus with radii of ratio 1/2 embedded in R3, as εâ0. 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.
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Matteo Rizzi,