论文标题
表面电荷的模式形成系统中的均匀能量分布
Uniform energy distribution in a pattern-forming system of surface charges
论文作者
论文摘要
我们考虑了一个（超级）平面上的电荷密度$ u \ in \ { - 1,1 \} $的变异模型，其短距离吸引来自界面能量，静电能带来了远距离的排斥。这场比赛导致模式形成。我们证明，与模式的尺度相比，（渐近）界面能量密度（渐近）在大的尺度上进行了等分。我们遵循[G. Alberti，R。Choksi，F。Otto，具有远距离相互作用的等速度问题的均匀能量分布，J。A.M.S.]。与该论文中考虑的体积电荷相比，挑战来自表面电荷的筛选功能的降低。
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.
