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.
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. Homogenlóä
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