Periodic Minimizers of a Ternary Non-Local Isoperimetric Problem

TL;DR

This paper rigorously analyzes the asymptotic behavior of a ternary non-local isoperimetric problem modeling triblock copolymers, revealing complex local structures and coexistence phenomena in global energy minimizers.

Contribution

It provides a rigorous derivation of the asymptotic limit and geometrical descriptions of minimizers, including coexistence of single and double bubbles, for a ternary system with non-local interactions.

Findings

Derived the asymptotic limit for minimizers using Gamma-convergence.

Identified coexistence of single and double bubbles in minimizers.

Provided geometrical descriptions of local structures in the limit.

Abstract

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.