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 ex-ploitation of a reduction of the energy to finite dimensions (Lemma 3.5), where the variables are closely related to the peak loations. 1