Minimality via second variation for microphase separation of diblock copolymer melts

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 quan- titative second order minimality condition. We show that critical configurations with positive second variation are indeed strict local minimizers of the problem. Moreover we provide, via a suitable quantitative inequality of isoperimetric type, an estimate of the deviation from mini- mality for configurations close to the minimum in the L1-topology.