Applied Mathematics Seminar-Periodic Minimizers of a Ternary Non-Local Isoperimetric Problem

Title: Periodic Minimizers of a Ternary Non-Local Isoperimetric Problem
Speaker:Chong Wang
Affiliation: McMaster University
Date and Time:Tuesday, October 8, 3:30pm-4:30pm
Place: Phillips 730

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.