Clifford Tori and the singularly perturbed Cahn-Hilliard equation

Abstract

In this paper we construct entire solutions $u_{\epsilon }$ to the Cahn-Hilliard equation $-{\epsilon }^2\Delta (-{\epsilon }^2\Delta u+W^{{}^{\prime }}(u))+W^{"}(u)(-{\epsilon }^2\Delta u+W^{{}^{\prime }}(u))=0$, under the volume constraint ${\int }_{{\mathbb{R}}^3}(1-u_{\epsilon })dx=4\sqrt{2}{\pi }^2$, whose nodal set approaches the Clifford Torus, that is the Torus with radii of ratio $1/\sqrt{2}$ embedded in ${\mathbb{R}}^3$, as $\epsilon \to 0$. What is crucial is 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.