Asymptotic Analysis for Periodic Structures

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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

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