Homogenization is not about periodicity, or G-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 FrançoisMurat 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 mustbe introduced, like the author’s H-measures, variants by Patrick Gérard, and others yet to be discovered. INDICE: 1 Why Do I Write?- 2 A Personalized Overview of Homogenization I.-3 A Personalized Overview of Homogenization II.- 4 An Academic Question of Jacques-Louis Lions.- 5 A Useful Generalization by Francois Murat .- 6 Homogenization of an Elliptic Equation .- 7 The Div-Curl Lemma.- 8 Physical Implications of Homogenization.- 9 A Framework with Differential Forms.- 10 Properties ofH-convergence.- 11 Homogenization of Monotone Operators.- 12 Homogenization of Laminated Materials.- 13 Correctors in Linear Homogenization.- 14 Correctorsin Nonlinear Homogenization.- 15 Holes with Dirichlet Conditions.- 16 Holes with Neumann Conditions.- 17 Compensated Compactness.- 18 A Lemma for Studying Boundary Layers.- 19 A Model in Hydrodynamics.- 20 Problems in Dimension N = 2.- 21 Bounds on Eective Coefficients.- 22 Functions Attached to Geometries .- 23 Memory Effects.- 24 Other Nonlocal Effects.- 25 The Hashin-Shtrikman Construction.- 26 Confocal Ellipsoids and Spheres.- 27 Laminations Again, and Again.- 28 Wave Front Sets, H-Measures.- 29 Small-Amplitude Homogenization.- 30 H-Measures and Bounds on Effective Coeffcients.- 31 H-Measures and Propagation Effects.- 32 Variants of H-Measures.- 33 Relations Between Young Measures and H-Measures.- 34 Conclusion.- 35 Biographical Information.- 36 Abbreviations and Mathematical Notation.
- ISBN: 978-3-642-05194-4
- Editorial: Springer
- Encuadernacion: Rústica
- Páginas: 434
- Fecha Publicación: 01/12/2009
- Nº Volúmenes: 1
- Idioma: Inglés