Minimality via second variation for a nonlocal isoperimetric problem

TL;DR

This paper establishes conditions under which certain configurations are local minimizers for a nonlocal isoperimetric problem, with implications for microphase separation, and provides new quantitative inequalities and results on minimality of lamellar structures.

Contribution

It introduces a second variation approach to prove local minimality for nonlocal isoperimetric problems and derives new quantitative inequalities and minimality results.

Findings

Critical configurations with positive second variation are local minimizers.

Quantitative isoperimetric inequality with respect to $L^1$-close sets.

New results on periodic local minimizers and lamellar configurations.

Abstract

We discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts. We show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and, in fact, satisfy a quantitative isoperimetric inequality with respect to sets that are $L^1$-close. The link with local minimizers for the diffuse-interface Ohta-Kawasaki energy is also discussed. As a byproduct of the quantitative estimate, we get new results concerning periodic local minimizers of the area functional and a proof, via second variation, of the sharp quantitative isoperimetric inequality in the standard Euclidean case. As a further application, we address the global and local minimality of certain lamellar configurations.