The Ohta-Kawasaki density functional theory of diblock
copolymers gives rise to a nonlocal free boundary problem. In a proper
parameter range an equilibrium pattern of many droplets is proved to
exist in a general planar domain. A sub-range is identified where the
multiple droplet pattern is stable. Each droplet is close to a round
disc. The boundaries of the droplets satisfy an equation that involves
the curvature of the boundary and a quantity that depends nonlocally on
the whole pattern. The locations of the droplets are determined via a
Green’s function of the domain. In constructing the droplet pattern we
overcome three obstacles: interface oscillation, droplet coarsening, and
droplet translation.