Rotationally symmetric solutions to the Cahn-Hilliard equation

Abstract

This paper is devoted to construction of new solutions to the Cahn-Hilliard equation in $\mathbb R^d$. Staring from a Delaunay unduloid $D_\tau$ with parameter $\tau\in (0,\tau^*)$ we find for each sufficiently small $\varepsilon$ a solution $u$ of this equation which is periodic in the direction of the $x_d$ axis and rotationally symmetric with respect to rotations about this axis. The zero level set of $u$ approaches as $\varepsilon\to 0$ the surface $D_\tau$. We use a refined version of the Lyapunov-Schmidt reduction method which simplifies very technical aspects of previous constructions for similar problems.