论文标题
DONATI代表定理与均化理论有关的周期性函数
Donati representation theorem for periodic functions in relation to homogenization theory
论文作者
论文摘要
本文讨论了周期性功能的属性，重点关注具有周期性边界条件（称为“细胞问题”的）部分微分方程的（系统系统）。这些细胞问题自然来自于迅速振荡系数的PDE的渐近研究。这项研究称为“匀浆理论”。我们认为，本文可能会对众所周知的概念展示新的启示，例如，通过格林的公式，Div-Curl引理和Donati的代表定理之间的隐藏联系。我们陈述并证明了Donati定理的三个扩展，该定理适应了周期性框架，这些框架超出了他们自身的重要性，对于理解应变，压力和位移中细胞问题的变异表述至关重要。第4节介绍了对功能痕迹的性质及其与该功能的周期性特性的关系的独立研究。
This paper discusses properties of periodic functions, focusing on (systems of) partial differential equations with periodicity boundary conditions, called "cellular problems". These cellular problems arise naturally from the asymptotic study of PDEs with rapidly oscillating coefficients; this study is called "homogenization theory". We believe the present paper may shed a new light on well-known concepts, for instance by showing hidden links between Green's formula, the div-curl lemma and Donati's representation theorem. We state and prove three extensions of Donati's Theorem adapted to the periodic framework which, beyond their own importance, are essential for understanding the variational formulations of cellular problems in strain, in stress and in displacement. Section 4 presents a self-contained study of properties of traces of a function and their relations with periodicity properties of that function.
