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.
|Autori:||Alberti, Giovanni; Choksi, R.; Otto, F.|
|Titolo:||Uniform energy distribution for an isoperimetric problem with long-range interactions|
|Anno del prodotto:||2009|
|Digital Object Identifier (DOI):||10.1090/S0894-0347-08-00622-X|
|Appare nelle tipologie:||1.1 Articolo in rivista|