论文标题
Neumann-Poincaré操作员的几何系列扩展：复合材料的应用
Geometric series expansion of the Neumann-Poincaré operator: application to composite materials
论文作者
论文摘要
Neumann-Poincaré操作员是一个域边界上的单数积分算子，当人们通过边界积分公式解决电导率传输问题时，自然会出现。最近，基于几何函数理论，在二维中开发了Neumann-Poincaré操作员的一系列表达。在本文中，我们使用此系列扩展研究了复合材料的几何特性。特别是，就相关的外部保形映射而言，我们获得了极化张量的显式公式，以及具有最大电导率的任意形状包含或任意形状的周期性阵列的有效电导率。此外，我们通过数值计算观察到，诺伊曼（Neumann）的频谱 - poincaré操作员在包含的形状变形方面具有单调行为。此外，我们通过使用与该域相对应的偏振张量的特性来得出任意Lipschitz域的Riemann映射系数的不平等关系。
The Neumann-Poincaré operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the Neumann-Poincaré operator was developed in two dimensions based on geometric function theory. In this paper, we investigate geometric properties of composite materials by using this series expansion. In particular, we obtain explicit formulas for the polarization tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the Neumann--Poincaré operator has a monotonic behavior with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain by using the properties of the polarization tensor corresponding to the domain.
