Uniform Energy Distribution for Minimizers of an Isoperimetric Problem Containing Long-Range Interactions

We study minimizers of a nonlocal variational problem. The problem is a mathematical paradigm for the ubiquitous phenomenon of energy-driven pattern formation induced by competing short and long-range interactions. The short range interaction is attractive and comes from an interfacial energy, the long range interaction is repulsive and comes from a nonlocal energy contribution. In particular, the problem is the sharp interface version of a functional used to model microphase separation of diblock copolymers. A natural conjecture is that in all space dimensions, minimizers are essentially periodic on an intrinsic scale determined solely by the competition between the short and long-range interactions. Surprisingly, this turns out to be a formidable task to prove in dimensions larger than one. In this paper, we address a weaker statement concerning the distribution of energy for minimizers. We prove in any space dimension, that each component of the energy (interfacial and nonlocal) of any minimizer is uniformly distributed on cubes which are sufficiently large with respect to the intrinsic length scale. Moreover, we also prove an L∞ bound on the optimal potential associated with the long-range interactions which allows to conclude further, that the optimal pattern has less large scale density variations than a random checker-board pattern with strongly decaying correlations.