Scaling laws in microphase separation of diblock copolymers

In this note, we study a nonlocal variational problem modeling microphase separation of diblock copolymers ([22], [3], [21]). We apply certain new tools developed in [5] to determine the principal part of the asymptotic expansion of the minimum free energy. That is, we prove a scaling law for the minimum energy and confirm that it is attained by a simple periodic lamellar structure. A previous result of Ohnishi et al. [23] was for one space dimension. Here, we obtain a similar result for the full three-dimensional problem.