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

Abstract

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 by Alberti, Choksi and Otto (2009). The challenge comes from the reduced screening capabilities of surface charges compared to the volume charges considered in the aforementioned work.