Rotationally symmetric solutions to the Cahn-Hilliard equation

TL;DR

This paper constructs new rotationally symmetric, periodic solutions to the Cahn-Hilliard equation in high dimensions, with zero level sets approaching Delaunay unduloids, using an improved Lyapunov-Schmidt reduction method.

Contribution

It introduces a refined Lyapunov-Schmidt reduction technique to construct solutions with prescribed geometric properties for the Cahn-Hilliard equation.

Findings

Solutions are periodic and rotationally symmetric.

Zero level sets approximate Delaunay unduloids as epsilon approaches zero.

Method simplifies previous technical constructions.

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.