Solutions for the Cahn-Hilliard Equation With Many Boundary Spike Layers

Abstract

In this paper we construct new classes of stationary solutions for the Cahn-Hilliard equation by a novel approach.

One of the results is as follows: Given a positive integer K and a (not necessarily nondegenerate) local minimum point of the mean curvature of the boundary then there are boundary K-spike solutions whose peaks all approach this point. This implies that for any smooth and bounded domain there exist boundary K-spike solutions.

The central ingredient of our analysis is the novel derivation and exploitation of a reduction of the energy to finite dimensions (Lemma 3.5), where the variables are closely related to the peak loations.