Donati representation theorem for periodic functions in relation to homogenization theory

TL;DR

This paper extends Donati's representation theorem for periodic functions, providing new insights into homogenization theory and cellular problems related to PDEs with periodic boundary conditions.

Contribution

It introduces three new extensions of Donati's Theorem tailored for periodic frameworks, crucial for variational formulations in homogenization theory.

Findings

Established links between Green's formula, div-curl lemma, and Donati's theorem.

Proved three extensions of Donati's Theorem for periodic functions.

Analyzed properties of traces and their relation to periodicity.

Abstract

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.