  In the study of equilibrium solutions for partial differential equations
there are so many equilibria that one cannot hope to find them all. Therefore
one usually concentrates on finding individual branches of equilibrium
solutions. On the one hand, a rigorous theoretical understanding of these
branches is ideal but not generally tractable. On the other hand, numerical
bifurcation searches are useful but not guaranteed to give an accurate
structure, in that they could miss a portion of a branch or find a spurious
branch where none exists. In a series of recent papers, we have aimed for a
third option. Namely, we have developed a method of computer-assisted proofs to
prove both existence and isolation of branches of equilibrium solutions. In the
current paper, we extend these techniques to the Ohta-Kawasaki model for the
dynamics of diblock copolymers in dimensions one, two, and three, by giving a
detailed description of the analytical underpinnings of the method. Although
the paper concentrates on applying the method to the Ohta-Kawasaki model, the
functional analytic approach and techniques can be generalized to other
parabolic partial differential equations.
