Rigorously Computing Symmetric Stationary States of the Ohta--Kawasaki Problem in Three Dimensions

In this paper we develop a symmetry preserving method for the rigorous computation of stationary states of the Ohta--Kawasaki partial differential equation in three space dimensions. By preserving the relevant symmetries we achieve an enormous reduction in computational cost. This makes it feasible to construct computer-assisted proofs of complex three-dimensional structures. In particular, we provide the first existence proofs for both the double gyroid and body centered cubic packed sphere solutions to this problem.