The General Theory of Homogenization: A Personalized Introduction (Lecture Notes of the Unione Matematica Italiana Book 7)

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. 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. Luc Tartar. Includes Bibliographical References And Index. Front Matter....Pages i-xxii Why Do I Write?....Pages 1-21 A Personalized Overview of Homogenization I....Pages 23-38 A Personalized Overview of Homogenization II....Pages 39-58 An Academic Question of Jacques-Louis Lions....Pages 59-68 A Useful Generalization by François Murat....Pages 69-74 Homogenization of an Elliptic Equation....Pages 75-87 The Div–Curl Lemma....Pages 89-95 Physical Implications of Homogenization....Pages 97-103 A Framework with Differential Forms....Pages 105-112 Properties of H-Convergence....Pages 113-127 Homogenization of Monotone Operators....Pages 129-136 Homogenization of Laminated Materials....Pages 137-145 Correctors in Linear Homogenization....Pages 147-155 Correctors in Nonlinear Homogenization....Pages 157-165 Holes with Dirichlet Conditions....Pages 167-175 Holes with Neumann Conditions....Pages 177-183 Compensated Compactness....Pages 185-194 A Lemma for Studying Boundary Layers....Pages 195-202 A Model in Hydrodynamics....Pages 203-209 Problems in Dimension N = 2....Pages 211-221 Bounds on Effective Coefficients....Pages 223-233 Functions Attached to Geometries....Pages 235-247 Memory Effects....Pages 249-263 Other Nonlocal Effects....Pages 265-279 The Hashin–Shtrikman Construction....Pages 281-295 Confocal Ellipsoids and Spheres....Pages 297-313 Laminations Again, and Again....Pages 315-323 Wave Front Sets, H-Measures....Pages 325-348 Small-Amplitude Homogenization....Pages 349-359 H-Measures and Bounds on Effective Coefficients....Pages 361-368 H-Measures and Propagation Effects....Pages 369-383 Variants of H-Measures....Pages 385-407 Relations Between Young Measures and H-Measures....Pages 409-429 Conclusion....Pages 431-443 Biographical Information....Pages 445-449 Abbreviations and Mathematical Notation....Pages 451-458 Back Matter....Pages 459-470 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