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Homogenization of Multiple Integrals

Andrea Braides and Anneliese Defranceschi

Price: £71.00 (hardback)
ISBN-13: 978-0-19-850246-3
Publication date: 26 November 1998
312 pages, 234x156 mm
Series: Oxford Lecture Series in Mathematics and Its Applications number 12
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Description

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 cellular elastic 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 the mathematical 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. The applications 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 microstructure and 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.

Readership: Advanced graduate students, lecturers and postgraduate students of applied mathematics and mechanics.

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

Authors, editors, and contributors

Andrea Braides, Professor, SISSA, Trieste and
Anneliese Defranceschi, Professor, Parma University

Links to web resources and related information

More in the same subject area:
Analytical mechanics
Differential equations
Integral equations
Calculus of variations
Mathematical modelling
Functional analysis

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