We consider a class of energy functionals containing a small parameter ε and a long-range interaction. Such functionals arise from models for phase separation in diblock copolymers and from stationary solutions of FitzHugh–Nagumo type systems.
On an interval of arbitrary length, we show that every global minimizer is periodic, and provide asymptotic expansions for the periods.
In 2D, periodic hexagonal structures are observed in experiments in certain di-block
copolymer melts. Using the modular function and an heuristic reduction of a mathematical model, we present a mathematical account of a hexagonal pattern selection observed in di-block copolymer melts.
We also consider the sharp interface problem arising in the singular limit,
and prove the existence and the nondegeneracy of solutions whose interface is a distorted circle in a two-dimensional bounded domain without any assumption on the symmetry of the domain.