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Homogenization of Differential Operators and Integral Functionals

  • Book
  • © 1994

Overview

Authors:
  1. V. V. Jikov

    1. Department of Mathematics, Pedagogical Institute of Vladimir, Vladimir, Russia

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  2. S. M. Kozlov

    1. Université Aix-Provence I, Marseille, France

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  3. O. A. Oleinik

    1. Department of Mathematics and Mechanics, Moscow State University, Moscow, Russia

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Table of contents (18 chapters)

  1. Front Matter

    Pages i-xi
    Download chapter PDF

  2. Homogenization of Second Order Elliptic Operators with Periodic Coefficients

    • V. V. Jikov, S. M. Kozlov, O. A. Oleinik
    Pages 1-54

  3. An Introduction to the Problems of Diffusion

    • V. V. Jikov, S. M. Kozlov, O. A. Oleinik
    Pages 55-85

  4. Elementary Soft and Stiff Problems

    • V. V. Jikov, S. M. Kozlov, O. A. Oleinik
    Pages 86-132

  5. Homogenization of Maxwell Equations

    • V. V. Jikov, S. M. Kozlov, O. A. Oleinik
    Pages 133-148

  6. G-Convergence of Differential Operators

    • V. V. Jikov, S. M. Kozlov, O. A. Oleinik
    Pages 149-186

  7. Estimates for the Homogenized Matrix

    • V. V. Jikov, S. M. Kozlov, O. A. Oleinik
    Pages 187-221

  8. Homogenization of Elliptic Operators with Random Coefficients

    • V. V. Jikov, S. M. Kozlov, O. A. Oleinik
    Pages 222-249

  9. Homogenization in Perforated Random Domains

    • V. V. Jikov, S. M. Kozlov, O. A. Oleinik
    Pages 250-297

  10. Homogenization and Percolation

    • V. V. Jikov, S. M. Kozlov, O. A. Oleinik
    Pages 298-322

  11. Some Asymptotic Problems for a Non-Divergent Parabolic Equation with Random Stationary Coefficients

    • V. V. Jikov, S. M. Kozlov, O. A. Oleinik
    Pages 323-337

  12. Spectral Problems in Homogenization Theory

    • V. V. Jikov, S. M. Kozlov, O. A. Oleinik
    Pages 338-366

  13. Homogenization in Linear Elasticity

    • V. V. Jikov, S. M. Kozlov, O. A. Oleinik
    Pages 367-390

  14. Estimates for the Homogenized Elasticity Tensor

    • V. V. Jikov, S. M. Kozlov, O. A. Oleinik
    Pages 391-414

  15. Elements of the Duality Theory

    • V. V. Jikov, S. M. Kozlov, O. A. Oleinik
    Pages 415-437

  16. Homogenization of Nonlinear Variational Problems

    • V. V. Jikov, S. M. Kozlov, O. A. Oleinik
    Pages 438-459

  17. Passing to the Limit in Nonlinear Variational Problems

    • V. V. Jikov, S. M. Kozlov, O. A. Oleinik
    Pages 460-491

  18. Basic Properties of Abstract Γ-Convergence

    • V. V. Jikov, S. M. Kozlov, O. A. Oleinik
    Pages 492-501

  19. Limit Load

    • V. V. Jikov, S. M. Kozlov, O. A. Oleinik
    Pages 502-535

  20. Back Matter

    Pages 536-572
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Keywords

About this book

It was mainly during the last two decades that the theory of homogenization or averaging of partial differential equations took shape as a distinct mathe­ matical discipline. This theory has a lot of important applications in mechanics of composite and perforated materials, filtration, disperse media, and in many other branches of physics, mechanics and modern technology. There is a vast literature on the subject. The term averaging has been usually associated with the methods of non­ linear mechanics and ordinary differential equations developed in the works of Poincare, Van Der Pol, Krylov, Bogoliubov, etc. For a long time, after the works of Maxwell and Rayleigh, homogeniza­ tion problems for· partial differential equations were being mostly considered by specialists in physics and mechanics, and were staying beyond the scope of mathematicians. A great deal of attention was given to the so called disperse media, which, in the simplest case, are two-phase media formed by the main homogeneous material containing small foreign particles (grains, inclusions). Such two-phase bodies, whose size is considerably larger than that of each sep­ arate inclusion, have been discovered to possess stable physical properties (such as heat transfer, electric conductivity, etc.) which differ from those of the con­ stituent phases. For this reason, the word homogenized, or effective, is used in relation to these characteristics. An enormous number of results, approximation formulas, and estimates have been obtained in connection with such problems as electromagnetic wave scattering on small particles, effective heat transfer in two-phase media, etc.

Authors and Affiliations

  • Department of Mathematics, Pedagogical Institute of Vladimir, Vladimir, Russia

    V. V. Jikov

  • Université Aix-Provence I, Marseille, France

    S. M. Kozlov

  • Department of Mathematics and Mechanics, Moscow State University, Moscow, Russia

    O. A. Oleinik

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