The general theory of homogenization: A personalized introduction

✍ Scribed by Luc Tartar (auth.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 2010
• Tongue: English
• Leaves: 493
• Series: Lecture Notes of the Unione Matematica Italiana 7
• Edition: 1
• Category: Library

✦ Synopsis

Homogenization is not about periodicity, or Gamma-convergence, but about understanding which effective equations to use at macroscopic level, knowing which partial differential equations govern mesoscopic levels, without using probabilities (which destroy physical reality); instead, one uses various topologies of weak type, the G-convergence of Sergio Spagnolo, the H-convergence of François Murat and the author, and some responsible for the appearance of nonlocal effects, which many theories in continuum mechanics or physics guessed wrongly.

For a better understanding of 20th century science, new mathematical tools must be introduced, like the author’s H-measures, variants by Patrick Gérard, and others yet to be discovered.

✦ Table of Contents

Front Matter....Pages i-xxii
Why Do I Write?....Pages 1-21
A Personalized Overview of Homogenization I....Pages 23-38
A Personalized Overview of Homogenization II....Pages 39-58
An Academic Question of Jacques-Louis Lions....Pages 59-68
A Useful Generalization by François Murat....Pages 69-74
Homogenization of an Elliptic Equation....Pages 75-87
The Div–Curl Lemma....Pages 89-95
Physical Implications of Homogenization....Pages 97-103
A Framework with Differential Forms....Pages 105-112
Properties of H-Convergence....Pages 113-127
Homogenization of Monotone Operators....Pages 129-136
Homogenization of Laminated Materials....Pages 137-145
Correctors in Linear Homogenization....Pages 147-155
Correctors in Nonlinear Homogenization....Pages 157-165
Holes with Dirichlet Conditions....Pages 167-175
Holes with Neumann Conditions....Pages 177-183
Compensated Compactness....Pages 185-194
A Lemma for Studying Boundary Layers....Pages 195-202
A Model in Hydrodynamics....Pages 203-209
Problems in Dimension N = 2....Pages 211-221
Bounds on Effective Coefficients....Pages 223-233
Functions Attached to Geometries....Pages 235-247
Memory Effects....Pages 249-263
Other Nonlocal Effects....Pages 265-279
The Hashin–Shtrikman Construction....Pages 281-295
Confocal Ellipsoids and Spheres....Pages 297-313
Laminations Again, and Again....Pages 315-323
Wave Front Sets, H-Measures....Pages 325-348
Small-Amplitude Homogenization....Pages 349-359
H-Measures and Bounds on Effective Coefficients....Pages 361-368
H-Measures and Propagation Effects....Pages 369-383
Variants of H-Measures....Pages 385-407
Relations Between Young Measures and H-Measures....Pages 409-429
Conclusion....Pages 431-443
Biographical Information....Pages 445-449
Abbreviations and Mathematical Notation....Pages 451-458
Back Matter....Pages 459-470

✦ Subjects

Partial Differential Equations;Mechanics;Engineering Fluid Dynamics