Homogenization of Multiple Integrals

Hardcover | November 26, 1998

byAndrea Braides, Anneliese Defranceschi

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The object of homogenization theory is the description of the macroscopic properties of structures with fine microstructure, covering a wide range of applications that run from the study of properties of composites to optimal design. The structures under consideration may model cellularelastic materials, fibred materials, stratified or porous media, or materials with many holes or cracks. In mathematical terms, this study can be translated in the asymptotic analysis of fast-oscillating differential equations or integral functionals. The book presents an introduction to themathematical theory of homogenization of nonlinear integral functionals, with particular regard to those general results that do not rely on smoothness or convexity assumptions. Homogenization results and appropriate descriptive formulas are given for periodic and almost- periodic functionals. Theapplications include the asymptotic behaviour of oscillating energies describing cellular hyperelastic materials, porous media, materials with stiff and soft inclusions, fibered media, homogenization of HamiltonJacobi equations and Riemannian metrics, materials with multiple scales of microstructureand with multi-dimensional structure. The book includes a specifically designed, self-contained and up-to-date introduction to the relevant results of the direct methods of Gamma-convergence and of the theory of weak lower semicontinuous integral functionals depending on vector-valued functions.The book is based on various courses taught at the advanced graduate level. Prerequisites are a basic knowledge of Sobolev spaces, standard functional analysis and measure theory. The presentation is completed by several examples and exercises.

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The object of homogenization theory is the description of the macroscopic properties of structures with fine microstructure, covering a wide range of applications that run from the study of properties of composites to optimal design. The structures under consideration may model cellularelastic materials, fibred materials, stratified or...

Andrea Braides is at SISSA, Trieste. Anneliese Defranceschi is at Parma University.

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Format:HardcoverDimensions:312 pages, 9.21 × 6.14 × 0.94 inPublished:November 26, 1998Publisher:Oxford University Press

The following ISBNs are associated with this title:

ISBN - 10:019850246X

ISBN - 13:9780198502463

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Table of Contents

PrefaceContentsIntroductionNotationPart I: Lower Semicontinuity1. Lower semicontinuity and coerciveness2. Weak convergence3. Minimum problems in sobolev spaces4. Necessary conditions for weak lower semicontinuity5. Sufficient conditions for weak lower semicontinuityPart II: Gamma-convergence6. The structure of quasiconvex functions7. A naive introduction of Gamma-convergence8. The indirect methods of Gamma-convergence9. Direct methods - an integral representation result10. Increasing set functions11. The fundamental estimate12. Integral functionals with standard growth conditionPart III: Basic Homogenization13. A one-dimensional example14. Periodic homogenization15. Almost periodic homogenization16. Two applications17. A closure theorem for the homogenization18. Loss of polyconvexity by homogenizationPart IV: Finer Homogenization Results19. Homogenization of connected media20. Homogenization with stiff and soft inclusions21. Homogenization with non-standard growth conditions22. Iterated homogenization23. Correctors for the homogenization24. Homogenization of multi-dimensional structuresPart V: AppendicesA Almost periodic functionsB Construction of extension operatorsC Some regularity resultsReferencesIndex