Intro
Foreword
Preliminary Remarks
Contents
1 In Dimension ``Zero''
1.1 A Simplification to Better Understand
1.2 The Periodic Setting
1.3 Energy of Infinite Systems of Particles
1.3.1 Choice of a Model for an Infinite Periodic System
1.3.2 Introduction to (Local) Defects
1.3.3 Toward More General Perturbations
1.4 Periodicity with Defects
1.4.1 Compactly Supported Perturbations
1.4.2 Perturbation in Lp
1.4.3 An Example of Non-local Defects
1.4.4 Formalizing the Link with Systems of Particles
1.4.5 A General Deterministic Framework

1.5 The Quasi- and Almost Periodic Settings
1.5.1 The Quasi-Periodic Setting
1.5.2 The Almost Periodic Setting
1.6 The Random Setting
1.6.1 Basic Elements for the Random Setting
1.6.2 The Notion of Stationarity
1.6.3 The Periodic, Quasi-Periodic and Almost-Periodic Settings as Particular Cases of the Continuous Random Setting
1.6.4 Properties of Stationary Functions
2 Homogenization in Dimension 1
2.1 Our First One-Dimensional Cases
2.1.1 Solution and Limit of the Elliptic Equation
2.1.2 What About Numerics?
2.1.3 The Periodic Case

2.2 The Quality of Approximation: The Corrector
2.3 One-Dimensional Defects
2.4 The 1D Random Case
2.5 Some ``Bad'' Cases
2.5.1 The Homogenized Equation May Take a Different Form
2.5.1.1 A Simple Example
2.5.1.2 Related Examples
2.5.1.3 An Unstable Phenomenon
2.5.2 The Homogenized Equation May Not Exist, and/or be of a Different Nature
2.5.3 A Small Defect in a Specific Nonlinear Equation
3 Dimension ≥2: The ``Simple'' Cases: Abstract or Periodic Settings
3.1 The Abstract Setting and Its Proof
3.1.1 An Abstract Result

3.1.2 Proof of the Abstract Result Using the Compactness Method
3.2 Interlude: When Geometry Comes into Play
3.2.1 A Laminated Material
3.2.2 Checkerboard Materials
3.2.2.1 The Periodic Checkerboard
3.2.2.2 The Random Checkerboard
3.3 Correction in the General Setting
3.3.1 Formal Intuition of the Corrector: Two-Scale Expansion
3.3.2 The Correction Theorem
3.4 Some Possible Proofs in an Explicit Case: The Periodic Setting
3.4.1 Proof in the (Very) Regular Case
3.4.2 Identification of the Homogenized Limit and Convergence via the Div-Curl Lemma

3.4.3 Convergence and Rate of Convergence
3.4.3.1 Some Preliminary Computations
3.4.3.2 A Series of Progressively Less Formal Arguments
3.4.3.3 Interior H1 Convergence
3.4.3.4 Convergence in H1 up to the Boundary
3.4.3.5 Convergence in W1,q (and Other ``Gradient Norms'')
3.4.4 Alternative Methods
4 Dimension ≥2: Some Explicit Cases Beyond the Periodic Setting
4.1 Localized Defects
4.1.1 The Case of a Defect in L2(Rd)
4.1.1.1 Existence (and Uniqueness) of the Corrector
4.1.1.2 Unchanged Homogenized Coefficient
4.1.1.3 Using the Corrector