The general theory of homogenization
Subtitle: a personalized introduction
By: Luc Tartar
Series: Lecture notes of the Unione Matematica Italiana -- 7; Lecture notes of the Unione Matematica Italiana -- 7.
Publisher: Springer; UMI
Published: 2009
Place of publication: Heidelberg; New York
Language: eng
Number of pages: 470
ISBN-13: 9783642051944; 9783642051951
Subjects: Homogenization (Differential equations)
Description: 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.
Notes: Includes bibliographical references and index.
Table of contents: title: Why do I write?; title: A personalized overview of homogenization I; title: A personalized overview of homogenization II; title: An academic question of Jacques-Louis Lions; title: A useful generalization by François Murat; title: Homogenization of an elliptic equation; title: The div-curl lemma; title: Physical implications of homogenization; title: A framework with differential forms; title: Properties of H-convergence; title: Homogenization of monotone operators; title: Homogenization of laminated materials; title: Correctors in linear homogenization; title: Correctors in nonlinear homogenization; title: Holes with Dirichlet conditions; title: Holes with Neumann conditions; title: Compensated compactness; title: A Lemma for studying boundary layers; title: A model in hydrodynamics; title: Problems in dimension N=2; title: Bounds on effective coefficients; title: Functions attached to geometries; title: Memory effects; title: Other nonlocal effects; title: The Hashin-Shtrikman construction; title: Confocal ellipsoids and spheres; title: Laminations again, and again; title: Wave front sets, H-measures; title: Small-amplitude homogenization; title: H-measures and bounds on effective coefficients; title: H-measures and propagation effects; title: Variants of H-measures; title: Relations between young measures and H-measures.