Periodic Minimizers of a Ternary Non-Local Isoperimetric Problem

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 a two-component isoperimetric problem: compared to binary systems, not only does it lack an explicit formula, but, more crucially, it can be neither concave nor convex on parts of its domain.

Automated summary

This study investigates a two-dimensional ternary inhibitory system derived from the sharp-interface limit of the Nakazawa-Ohta density functional theory of triblock copolymers. In this limit, two energy levels are distinguished to encode information on local structures and spatial distribution of components in global minimizers. The main challenge lies in solving a two-component isoperimetric problem, which lacks an explicit formula and exhibits non-concave or non-convex behavior in parts of its domain.