Homogenization of Multiple Integrals

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Andrea Braides (Professor, Professor, SISSA, Trieste, Italy) Anneliese Defranceschi (Professor, Professor, Parma University, Italy)

Homogenization of Multiple Integrals

Andrea Braides (Professor, Professor, SISSA, Trieste, Italy), Anneliese Defranceschi (Professor, Professor, Parma University, Italy)

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English

Oxford University Press
01 August 1998

Functional analysis & transforms; Integral calculus & equations; Numerical analysis; Maths for scientists; Analytical mechanics

Series: Oxford Lecture Series in Mathematics and Its Applications

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.

By: Andrea Braides (Professor Professor SISSA Trieste Italy), Anneliese Defranceschi (Professor, Professor, Parma University, Italy)
Imprint: Oxford University Press
Country of Publication: United Kingdom
Volume: 12
Dimensions: Height: 241mm, Width: 163mm, Spine: 24mm
Weight: 642g
ISBN: 9780198502463
ISBN 10: 019850246X
Series: Oxford Lecture Series in Mathematics and Its Applications
Pages: 312
Publication Date: 01 August 1998
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

Preface Contents Introduction Notation Part I: Lower Semicontinuity 1: Lower semicontinuity and coerciveness 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 6: The structure of quasiconvex functions 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