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Homogenization of Boundary Layers in the Boltzmann--Poisson System
Multiscale Modeling and Simulation ( IF 1.9 ) Pub Date : 2021-03-23 , DOI: 10.1137/18m1193888 Clemens Heitzinger , José A. Morales E.
Multiscale Modeling and Simulation ( IF 1.9 ) Pub Date : 2021-03-23 , DOI: 10.1137/18m1193888 Clemens Heitzinger , José A. Morales E.
Multiscale Modeling &Simulation, Volume 19, Issue 1, Page 506-532, January 2021.
We homogenize the Boltzmann--Poisson system where the background medium is given by a periodic permittivity and a periodic charge concentration. The domain is the half-space with a periodic charge concentration on the boundary. Hence the domain consists of cells in ${\mathbb{R}}^3$ that are periodically repeated in two dimensions and unbounded in the third dimension. We obtain formal results for this homogenization problem. We prove the existence and uniqueness of the solution of the Laplace and Poisson problems in the fast variables with periodic and surface charge boundary conditions generating an electric field at infinity, obtaining formal solutions for the potential in terms of Magnus expansions for the case where the diagonal permittivity matrix depends on the vertical fast variable. Further on, splitting the potential into a stationary part and a self-consistent part, performing the two-scale homogenization expansions for the Poisson and the Boltzmann equations, and applying a solvability condition, we arrive at the drift-diffusion equations for the boundary-layer problem.
中文翻译：
Boltzmann--Poisson 系统中边界层的均质化
多尺度建模与仿真，第 19 卷，第 1 期，第 506-532 页，2021 年 1 月。
我们将 Boltzmann--Poisson 系统均质化，其中背景介质由周期性介电常数和周期性电荷浓度给出。域是边界上具有周期性电荷集中的半空间。因此，域由 ${\mathbb{R}}^3$ 中的单元组成，这些单元在二维中周期性重复，在第三维中是无界的。我们获得了这个同质化问题的正式结果。我们证明了在具有周期性和表面电荷边界条件的快速变量中拉普拉斯和泊松问题的解的存在性和唯一性，在无穷远处产生电场，在对角线介电常数矩阵取决于垂直快速变量。进一步，
更新日期：2021-03-23
We homogenize the Boltzmann--Poisson system where the background medium is given by a periodic permittivity and a periodic charge concentration. The domain is the half-space with a periodic charge concentration on the boundary. Hence the domain consists of cells in ${\mathbb{R}}^3$ that are periodically repeated in two dimensions and unbounded in the third dimension. We obtain formal results for this homogenization problem. We prove the existence and uniqueness of the solution of the Laplace and Poisson problems in the fast variables with periodic and surface charge boundary conditions generating an electric field at infinity, obtaining formal solutions for the potential in terms of Magnus expansions for the case where the diagonal permittivity matrix depends on the vertical fast variable. Further on, splitting the potential into a stationary part and a self-consistent part, performing the two-scale homogenization expansions for the Poisson and the Boltzmann equations, and applying a solvability condition, we arrive at the drift-diffusion equations for the boundary-layer problem.
中文翻译：
Boltzmann--Poisson 系统中边界层的均质化
多尺度建模与仿真，第 19 卷，第 1 期，第 506-532 页，2021 年 1 月。
我们将 Boltzmann--Poisson 系统均质化，其中背景介质由周期性介电常数和周期性电荷浓度给出。域是边界上具有周期性电荷集中的半空间。因此，域由 ${\mathbb{R}}^3$ 中的单元组成，这些单元在二维中周期性重复，在第三维中是无界的。我们获得了这个同质化问题的正式结果。我们证明了在具有周期性和表面电荷边界条件的快速变量中拉普拉斯和泊松问题的解的存在性和唯一性，在无穷远处产生电场，在对角线介电常数矩阵取决于垂直快速变量。进一步，
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