Homogenization of Multiple Integrals

by Anneliese Defranceschi, Andrea Braides

Overview

Homogenization theory describes the macroscopic properties of structures with fine microstructure. Its applications are diverse and include optimal design and the study of composites. The theory relies on the asymptotic analysis of fast-oscillating differential equations or integral functionals. This book is an introduction to the homogenization of nonlinear integral functionals. It emphasizes general results that do not rely on smoothness or convexity assumptions. The book presents a rigorous mathematical description of the overall properties of such functionals, with various applications that range from cellular elastic materials to Riemannian metrics and Hamilton-Jacobi equations. The book also includes self-contained introductions to the theories of gamma-convergence and weak lower semicontinuous functionals.

Product Details

ISBN-13:	9780198502463
Publisher:	Oxford University Press, USA
Publication date:	03/28/1999
Series:	Oxford Lecture Series in Mathematics and Its Applications Series , #12
Pages:	312
Product dimensions:	6.30(w) x 9.30(h) x 0.90(d)

About the Author

SISSA, Trieste

Parma University

Table of Contents

Preface
Contents
Introduction
Notation
Part I: Lower Semicontinuity
2. Weak convergence
3. Minimum problems in sobolev spaces
4. Necessary conditions for weak lower semicontinuity
5. Sufficient conditions for weak lower semicontinuity
Part II: Gamma-convergence
7. A naive introduction of Gamma-convergence
8. The indirect methods of Gamma-convergence
9. Direct methods - an integral representation result
10. Increasing set functions
11. The fundamental estimate
12. Integral functionals with standard growth condition
Part III: Basic Homogenization
13. A one-dimensional example
14. Periodic homogenization
15. Almost periodic homogenization
16. Two applications
17. A closure theorem for the homogenization
18. Loss of polyconvexity by homogenization
Part IV: Finer Homogenization Results
19. Homogenization of connected media
20. Homogenization with stiff and soft inclusions
21. Homogenization with non-standard growth conditions
22. Iterated homogenization
23. Correctors for the homogenization
24. Homogenization of multi-dimensional structures
Part V: Appendices
A Almost periodic functions
B Construction of extension operators
C Some regularity results
References
Index