I will discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts. We will see 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 will be also discussed. As a byproduct of the quantitative estimate, I will show new results concerning periodic local minimizers of the area functional. A further application will concern the global and local minimality of certain lamellar configurations.

Abstract