Uniform energy distribution for an isoperimetric problem with 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, and the long-range interaction is repulsive and comes from a nonlocal energy contribution. In particular, the problem is the sharp interface version of a problem 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. However, proving any periodicity result turns out to be a formidable task 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 $L^\infty$ bound on the optimal potential associated with the long-range interactions. This bound allows for an interesting interpretation: Note that the average volume fraction of the optimal pattern in a subsystem of size $R$ fluctuates around the system average $m$. The bound on the potential yields a rate of decay of these fluctuations as $R$ tends to $+\infty$. This rate of decay is stronger than the one for a random checkerboard pattern. In this sense, the optimal pattern has less large-scale variations of the average volume fraction than a pattern with a finite correlation length.