The General Theory of Homogenization (eBook)

A Personalized Introduction

Luc Tartar (Autor)

eBook Download: PDF

2009 | 2010
XXII, 471 Seiten
Springer Berlin (Verlag)
978-3-642-05195-1 (ISBN)

Infos

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.

Inhalt

The General Theoryof Homogenization 2
1 Why Do I Write? 22
2 A Personalized Overview of Homogenization I 43
3 A Personalized Overview of Homogenization II 59
4 An Academic Question of Jacques-Louis Lions 79
5 A Useful Generalization by François Murat 89
6 Homogenization of an Elliptic Equation 95
7 The Div–Curl Lemma 108
8 Physical Implications of Homogenization 115
9 A Framework with Differential Forms 122
10 Properties of H-Convergence 130
11 Homogenization of Monotone Operators 145
12 Homogenization of Laminated Materials 153
13 Correctors in Linear Homogenization 162
14 Correctors in Nonlinear Homogenization 171
15 Holes with Dirichlet Conditions 180
16 Holes with Neumann Conditions 189
17 Compensated Compactness 196
18 A Lemma for Studying Boundary Layers 206
19 A Model in Hydrodynamics 214
20 Problems in Dimension N = 2 221
21 Bounds on Effective Coefficients 232
22 Functions Attached to Geometries 243
23 Memory Effects 256
24 Other Nonlocal Effects 271
25 The Hashin–Shtrikman Construction 286
26 Confocal Ellipsoids and Spheres 301
27 Laminations Again, and Again 318
28 Wave Front Sets, H-Measures 327
29 Small-Amplitude Homogenization 351
30 H-Measures and Bounds on Effective Coefficients 362
31 H-Measures and Propagation Effects 370
32 Variants of H-Measures 385
33 Relations Between Young Measuresand H-Measures 408
34 Conclusion 429
35 Biographical Information 442
36 Abbreviations and Mathematical Notation 447
References 455
Index 462

Produkt-Details

• Erscheint lt. Verlag: 3.12.2009
• Reihe/Serie: Lecture Notes of the Unione Matematica Italiana
• Reihe/Serie: Lecture Notes of the Unione Matematica Italiana
• Zusatzinfo: XXII, 471 p.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Allgemeines / Lexika
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik
• Themenwelt: Technik ► Bauwesen
• Schlagworte: 35J99, 35K99, 35L99, 35S99, 74Q05, 74Q10, 74Q15, 74Q20, 76A9 • Compactness • Continuum Mechanics • differential equation • H-measures • homogenization • Partial differential equations • Physics
• ISBN-10: 3-642-05195-2 / 3642051952
• ISBN-13: 978-3-642-05195-1 / 9783642051951