TOC Front Cover 2 Asymptotic Analysis for Periodic Structures 5 Copyright Page 6 Introduction 7 TABLE OF CONTENTS 17 Chapter 1. Elliptic Operators 27 1. Setting of the "model" problem 28 2. Asymptotic expansions 37 3. Energy proof of the homogenization formula 49 4. LP estimates 61 5. Correctors 75 6. Second order elliptic operators with non-uniformly oscillating coefficients 97 7. Complements on boundary conditions 113 8. Reiterated homogenization 122 9. Homogenization of elliptic systems 143 Front Cover 2 Asymptotic Analysis for Periodic Structures 5 Copyright Page 6 Introduction 7 TABLE OF CONTENTS 17 Chapter 1. Elliptic Operators 27 1. Setting of the "model" problem 28 2. Asymptotic expansions 37 3. Energy proof of the homogenization formula 49 4. LP estimates 61 5. Correctors 75 6. Second order elliptic operators with non-uniformly oscillating coefficients 97 7. Complements on boundary conditions 113 8. Reiterated homogenization 122 9. Homogenization of elliptic systems 143 10. Homogenization of the Stokes equation 155 11. Homogenization of equations of Maxwell's type 164 12. Homogenization with rapidly oscillating potentials 184 13. Study of lower order terms 207 14. Singular perturbations and homogenization 214 15. Non-local limits 220 16. Introduction to non-linear problems 226 17. Homogenization of multi-valued operators 233 18. Comments and problems 244 Bibliography 254 Chapter 2. Evolution Operators 259 1. Parabolic operators: Asymptotic expansions 260 2. Convergence of the homogenization of parabolic equations 279 3. Evolution operators of hyperbolic, Petrowsky or Schrodinger type 325 4. Comments and problems 351 Bibliography 368 Chapter 3. Probabilistic Problems and Methods 371 1. Stochastic differential equations and connections with partial differential equations 374 2. Martingale formulation of stochastic differential equations 383 3. Some results from ergodic theory 390 4. Homogenization with a constant coefficients limit operator 409 5. Analytic approach to the problem (4.76) 440 6. Operators with locally periodic coefficients 455 7. Reiterated homogenization 481 8. Problems with potentials 493 9. Homogenization of reflected diffusion processes 501 10. Evolution problems 514 11. Averaging 542 12. Comments and problems 555 Bibliography 560 Chapter 4. High Frequency wave Propagation in Periodic Structures 563 1. Formulation of the problems 564 2. The W. K. B. or geometrical optics method 573 3. Spectral theory for differential operators with periodic coefficients 640 4. Simple applications of the spectral expansion 653 5. The general geometrical optics expansion 697 6. Comments and problems 720 Bibliography 722 Show more