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The General Theory of Homogenization [electronic resource] : A Personalized Introduction / by Luc Tartar.

By: Contributor(s): Material type: TextTextSeries: Lecture Notes of the Unione Matematica Italiana ; 7Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg, 2010Description: XXII, 471 p. online resourceContent type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9783642051951
Subject(s): Additional physical formats: Printed edition:: No titleDDC classification:
  • 530.1 23
LOC classification:
  • QC19.2-20.85
Online resources:
Contents:
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 Fran#x00E7;ois Murat -- Homogenization of an Elliptic Equation -- The Div#x2013;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#x2013;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.
In: Springer eBooksSummary: 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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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 Fran#x00E7;ois Murat -- Homogenization of an Elliptic Equation -- The Div#x2013;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#x2013;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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