On the Existence of High Multiplicity Interfaces

In many singularly perturbed Ginzburg–Landau type partial differential equations, such as the Allen–Cahn equation, the nonlocal Allen–Cahn equation, and the Cahn–Hilliard equation, the question arises whether or not the limiting interfaces can have high multiplicity. In other words, do there exist solutions of these PDE’s with many transition layers (where the solution passes rapidly between $\pm 1$) which are so close to each other that they collapse to one interface in the limit. In this paper we prove that there exist interfaces with arbitrarily high multiplicity by studying the radially symmetric Allen-Cahn equation. We adapt the energy method of Bronsard-Kohn [BK].

Full Text (PDF format)

Published 1 January 1996