  We study a two-dimensional ternary inhibitory system derived as a
sharp-interface limit of the Nakazawa-Ohta density functional theory of
triblock copolymers. This free energy functional combines an interface energy
favoring micro-domain growth with a Coulomb-type long range interaction energy
which prevents micro-domains from unlimited spreading. Here we consider a limit
in which two species are vanishingly small, but interactions are
correspondingly large to maintain a nontrivial limit. In this limit two energy
levels are distinguished: the highest order limit encodes information on the
geometry of local structures as a two-component isoperimetric problem, while
the second level describes the spatial distribution of components in global
minimizers. We provide a sharp rigorous derivation of the asymptotic limit,
both for minimizers and in the context of Gamma-convergence. Geometrical
descriptions of limit configurations are derived; among other results, we will
show that, quite unexpectedly, coexistence of single and double bubbles can
arise. The main difficulties are hidden in the optimal solution of
two-component isoperimetric problem: compared to binary systems, not only it
lacks an explicit formula, but, more crucially, it can be neither concave nor
convex on parts of its domain.
