Uniform energy distribution in a pattern-forming system of surface charges. (English) Zbl 1487.49049
Summary: 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 G. Alberti et al. [J. Am. Math. Soc. 22, No. 2, 569–605 (2009; Zbl 1206.49046)]. The challenge comes from the reduced screening capabilities of surface charges compared to the volume charges considered in the aforementioned work.
MSC:
| 49Q10 | Optimization of shapes other than minimal surfaces |
| 35J50 | Variational methods for elliptic systems |
| 49S05 | Variational principles of physics |
| 49N60 | Regularity of solutions in optimal control |
Keywords:
pattern formation; isoperimetric problem; elliptic regularity theory; long-range interactions; uniform energy distributionCitations:
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