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The General Theory of Homogenization

by Tartar, Luc.
Authors: SpringerLink (Online service) Series: Lecture Notes of the Unione Matematica Italiana, 1862-9113 ; . 7 Physical details: XXII, 471p. online resource. ISBN: 3642051952 Subject(s): Mathematics. | Differential equations, partial. | Mechanics. | Hydraulic engineering. | Mathematics. | Partial Differential Equations. | Mechanics. | Engineering Fluid Dynamics.
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E-Book E-Book AUM Main Library 515.353 (Browse Shelf) Not for loan

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 François 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.

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.

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