| We study the asymptotic behavior of the Ohta-Kawasaki energy in dimension 2. In that model, two phases appear, and they interact via a nonlocal Coulomb type energy. We focus on the regime where one of the phases has very small volume fraction, thus creating “droplets” of that phase in a sea of the other phase. We compute the Γ-limit of the leading order energy and yield averaged information for almost minimizers, namely that the density of droplets should be uniform and almost spherical. We then derive a next order Γ-limit energy determines which the geoemtric arrangement of the droplets. Without appealing at all to the Euler-Lagrange equation, we establish here for all configurations which have “almost minimal energy,” the asymptotic roundness and radius of the droplets, and the fact that they asymptotically shrink to points whose arrangement should minimize this energy, in some averaged sense. This leads to expecting to see hexagonal lattices of droplets. In addition, we prove that the density of droplets of non-minimizing critical points of the energy is also uniform and that droplets are spherical in some averaged sense. Next we study a non-local isoperimetric problem in ;2;2... |
