The general theory of homogenization : a personalized introduction
QA377 .T35 2009
- Unknown QA377 .T35 2009
- Includes bibliographical references and index.
- Why Do I Write?.- A Personalized Overview of Homogenization I.- A Personalized Overview of Homogenization II.- An Academic Question of Jacques-Louis Lions.- A Useful Generalization by Francois Murat.- Homogenization of an Elliptic Equation.- The Div-Curl Lemma.- Physical Implications of Homogenization.- A Framework with Differential Forms.- Properties of H-Convergence.- Homogenization of Monotone Operators.- Homogenization of Laminated Materials.- Correctors in Linear Homogenization.- Correctors in Nonlinear Homogenization.- Holes with Dirichlet Conditions.- Holes with Neumann Conditions.- Compensated Compactness.- A Lemma for Studying Boundary Layers.- A Model in Hydrodynamics.- Problems in Dimension = 2.- Bounds on Effective Coefficients.- Functions Attached to Geometries.- Memory Effects.- Other Nonlocal Effects.- The Hashin-Shtrikman Construction.- Confocal Ellipsoids and Spheres.- Laminations Again, and Again.- Wave Front Sets, H-Measures.- Small-Amplitude Homogenization.- H-Measures and Bounds on Effective Coefficients.- H-Measures and Propagation Effects.- Variants of H-Measures.- Relations Between Young Measures and H-Measures.- Conclusion.- Biographical Information.- Abbreviations and Mathematical Notation.
- (source: Nielsen Book Data)
- Publisher's Summary
- 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 Francois 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 Gerard, and others yet to be discovered.
(source: Nielsen Book Data)
- Publication date
- Luc Tartar.
- Lecture notes of the Unione Matematica Italiana, 1862-9113 ; 7