Rotationally symmetric solutions to the Cahn-Hilliard equation

Abstract

This paper is devoted to construction of new solutions to the Cahn-Hilliard equation in ℝd. Staring from the Delaunay unduloid Dô with parameter τ ∈ (0, τ∗) we find for each sufficiently small ε a solution u of this equation which is periodic in the direction of the xd axis and rotationally symmetric with respect to rotations about this axis. The zero level set of u approaches as ε → 0 the surface Dτ. We use a refined version of the Lyapunov-Schmidt reduction method which simplifies very technical aspects of previous constructions for similar problems.