In this talk we discuss a rigorous numerical method to study pattern formation in the Ohta-Kawasaki model in both two and three dimensions. This model appears in the study of di-block copolymers. The model describes, roughly speaking, the balance between long range attraction and short range repulsion.
For the two dimensional case we compare local minimizers of the Ohta-Kawasaki functional. In particular, we validate part of the phase diagram, identifying regions of parameter space where rolls are favorable, where hexagonally packed spots have lowest energy and finally where the constant mixed state does. This includes determining optimal domain sizes, which are unknown a priori. In terms of the practical realities of applying the computer assisted theorems ideas, this work represents a step forward past clean-cut test problems to more elaborate variational problems in pattern formation.
For the three dimensional case we illustrate the work in progress by fixing a cubical domain and finding several highly symmetric periodic stationary states. This is based on joint work with JF Williams (SFU).