2D Phase Diagram for Minimizers of a Cahn-Hilliard Functional with   Long-Range Interactions
  This paper presents a detailed asymptotic and numerical investigation of the phase diagram for global minimizers to a Cahn-Hilliard functional with long-range interactions in two space dimensions. We introduce a small parameter measuring perturbation from the minimal orderdisorder transition, and derive asymptotic estimates for stability regions as the parameter tends to zero. Based upon the H^-1 gradient ow, we introduce a hybrid numerical method to navigate through the complex energy landscape and access the ground state of the functional. We use this method to numerically compute the phase diagram. Our asymptotic predictions show surprisingly good agreement with our numerical results. 
Rustum Choksi, Mirjana Maras and J.F. Williams
math.DS