On a Quaternary Non-Local Isoperimetric Problem

TL;DR

This paper investigates a two-dimensional quaternary inhibitory system combining interface energy and Coulomb-type interactions, analyzing the limiting behavior of micro-domains and their geometric configurations.

Contribution

It introduces a novel limit analysis for a quaternary system, linking micro-domain geometry with global component distribution in a non-local isoperimetric problem.

Findings

Derived geometrical descriptions of limit configurations.

Identified two energy levels governing local structures and global distributions.

Established a connection between micro-domain shapes and energy minimization.

Abstract

We study a two-dimensional quaternary inhibitory system. 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 three 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 three-component isoperimetric problem, while the second level describes the spatial distribution of components in global minimizers. Geometrical descriptions of limit configurations are derived.