Minimality via second variation for microphase separation of diblock copolymer melts

TL;DR

This paper analyzes the stability of configurations in a nonlocal isoperimetric problem modeling diblock copolymer microphase separation, establishing conditions for local minimality and quantifying deviations from minimality.

Contribution

It introduces a second order variational analysis to identify strict local minimizers and provides quantitative estimates of deviation from minimality for near-minimal configurations.

Findings

Critical configurations with positive second variation are strict local minimizers.

A quantitative inequality bounds deviation from minimality in the $L^1$ topology.

The analysis applies to the sharp interface limit of the Ohta-Kawasaki free energy.

Abstract

We consider a non local isoperimetric problem arising as the sharp interface limit of the Ohta-Kawasaki free energy introduced to model microphase separation of diblock copolymers. We perform a second order variational analysis that allows us to provide a quantitative second order minimality condition. We show that critical configurations with positive second variation are indeed strict local minimizers of the nonlocal perimeter. Moreover we provide, via a suitable quantitative inequality of isoperimetric type, an estimate of the deviation from minimality for configurations close to the minimum in the $L^1$ topology .