Uniform energy distribution in a pattern-forming system of surface charges

We consider a variational model for charge density $u\in{-1,1}$ on (hyper)plane, with short-range attraction coming from the interfacial energy and long-range repulsion electrostatic energy. This competition leads to pattern formation. prove that is (asymptotically) equidistributed at scales large compared scale of pattern. follow strategy laid out by Alberti, Choksi Otto (2009). The challenge comes reduced screening capabilities surface charges volume considered in aforementioned work.