  We consider a variational model for a charge density $u\in\{-1,1\}$ on a
(hyper)plane, with a short-range attraction coming from the interfacial energy
and a long-range repulsion coming from the electrostatic energy. This
competition leads to pattern formation. We prove that the interfacial energy
density is (asymptotically) equidistributed at scales large compared to the
scale of the pattern. We follow the strategy laid out in [G. Alberti, R.
Choksi, F. Otto, Uniform energy distribution for an isoperimetric problem with
long-range interactions, J. A.M.S.]. The challenge comes from the reduced
screening capabilities of surface charges compared to the volume charges
considered in that paper.
