Uniform Energy Distribution for an Isoperimetric Problem with Long-range Interactions

S(u) denotes the interfacial area associated with the surfaces upon which u jumps, and G(x, y) denotes the Green’s function for − on QL with Neumann boundary conditions. The variational problem consists of competing short-range (S(u)) and long-range (the nonlocal Green’s function term) contributions. The former term is attractive, favoring large domains of pure phases with boundaries of minimal surface area. The latter term is repulsive, favoring small domains which lead to cancellations. The combination of the two leads to pattern formation on a scale determined solely by the competition of the two terms. Mathematically, the natural space for u is BV (QL,±1), functions of bounded variation taking values ±1. The interfacial area is then simply half the total variation measure |∇u| on QL, i.e.,