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

Katarina Bellova,Antoine Julia,Felix Otto

doi:10.4171/rmi/1308

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.

Highlights

We consider a variational model for a charge density u ∈ {−1, 1} on aplane, with a short-range attraction coming from the interfacial energy and a long-range repulsion coming from the electrostatic energy
That paper deals with the popular model where the configuration space consists of characteristic functions u ∈ {−1, 1}, the short-range attractive interaction is the interfacial energy between the two phases, and the long-range repulsive interaction is electrostatic, with the order parameter u playing the role of a charge density
The main challenge in establishing a mesoscopically uniform energy distribution lies in capturing screening effects: On mesoscopic scales, charges arrange themselves in such a way as to reduce the macroscopic part of the electric field b as much as possible

Summary

Statement of the main results

The main results of this paper are the following two theorems: Theorem 1 (Uniform distribution of energy). If (u, b) is a minimizer in Aper(QL) and L ≥ l ≫ 1, there holds (5). We have no reason to believe that the exponent 1/2 in (4) and (5) is optimal in any sense. It comes up naturally through Lemma 14. It should be compared to the (better) exponent 1 in the case of [2], which can be improved by using the first variation (see Proposition 6.1 in that paper). A scaling argument similar to one used in [2, Theorem 1.2] yields: Theorem 2 (Equipartition of the energy). For (u, b) as above, if L ≥ l ≫ 1, there hold

Uniform energy bound

Pointwise estimates on the fields

Findings

Proofs