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V. V. Zhikov

Other affiliations: Moscow State University
Bio: V. V. Zhikov is an academic researcher from Pedagogical University. The author has contributed to research in topics: Nonlinear system & Parabolic partial differential equation. The author has an hindex of 14, co-authored 37 publications receiving 3664 citations. Previous affiliations of V. V. Zhikov include Moscow State University.

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Book
19 Aug 1994
TL;DR: In this article, the problem of homogenizing a two-dimensional matrix has been studied in the context of Diffusion problems, where the homogenization problem is formulated as a set of problems of diffusion.
Abstract: 1 Homogenization of Second Order Elliptic Operators with Periodic Coefficients.- 1.1 Preliminaries.- 1.2 Setting of the Homogenization Problem.- 1.3 Problems of Justification Further Examples.- 1.4 The Method of Asymptotic Expansions.- 1.5 Explicit Formulas for the Homogenized Matrix in the Two-Dimensional Case.- 1.6 Estimates and Approximations for the Homogenized Matrix.- 1.7 The Rayleigh-Maxwell Formulas.- Comments.- 2 An Introduction to the Problems of Diffusion.- 2.1 Homogenization of Parabolic Operators.- 2.2 Homogenization and the Central Limit Theorem.- 2.3 Stabilization of Solutions of Parabolic Equations.- 2.4 Diffusion in a Solenoidal Flow.- 2.5 Diffusion in an Arbitrary Periodic Flow.- 2.6 Spectral Approach to the Asymptotic Problems of Diffusion.- 2.7 Diffusion with Absorption.- Comments.- 3 Elementary Soft and Stiff Problems.- 3.1 Homogenization of Soft Inclusions.- 3.2 Homogenization of Stiff Inclusions.- 3.3 Virtual Mass.- 3.4 The Method of Asymptotic Expansions.- 3.5 On a Dense Cubic Packing of Balls.- 3.6 The Dirichlet Problem in a Perforated Domain.- Comments.- 4 Homogenization of Maxwell Equations.- 4.1 Preliminary Results.- 4.2 A Lemma on Compensated Compactness.- 4.3 Homogenization.- 4.4 The Problem of an Artificial Dielectric.- Comments.- 5 G-Convergence of Differential Operators.- 5.1 Basic Properties of G-Convergence.- 5.2 A Sufficient Condition of G-Convergence.- 5.3 G-Convergence of Abstract Operators.- 5.4 Compactness Theorem and Its Implications.- 5.5 G-Convergence and Duality.- 5.6 Stratified Media.- 5.7 G-Convergence of Divergent Elliptic Operators of Higher Order.- Comments.- 6 Estimates for the Homogenized Matrix.- 6.1 The Hashin-Shtrikman Bounds.- 6.2 Attainability of Bounds. The Hashin Structure.- 6.3 Extremum Principles.- 6.4 The Variational Method.- 6.5 G-Limit Media Attainment of the Bounds on Stratified Composites.- 6.6 The Method of Quasi-Convexity.- 6.7 The Method of Null Lagrangians.- 6.8 The Method of Integral Representation.- Comments.- 7 Homogenization of Elliptic Operators with Random Coefficients.- 7.1 Probabilistic Description of Non-Homogeneous Media.- 7.2 Homogenization.- 7.3 Explicit Formulas in Two-Dimensional Problems.- 7.4 Homogenization of Almost-Periodic Operators.- 7.5 The General Theorem of Individual Homogenization.- Comments.- 8 Homogenization in Perforated Random Domains.- 8.1 Homogenization.- 8.2 Remarks on Positive Definiteness of the Homogenized Matrix.- 8.3 Central Limit Theorem.- 8.4 Disperse Media.- 8.5 Criterion of Pointwise Stabilization A Refinement of the Central Limit Theorem.- 8.6 Stiff Problem for a Random Spherical Structure.- 8.7 Random Spherical Structure with Small Concentration.- Comments.- 9 Homogenization and Percolation.- 9.1 Existence of the Effective Conductivity.- 9.2 Random Structure of Chess-Board Type.- 9.3 The Method of Percolation Channels.- 9.4 Conductivity Threshold for a Random Cubic Structure in ?3.- 9.5 Resistance Threshold for a Random Cubic Structure in ?3.- 9.6 Central Limit Theorem for Random Motion in an Infinite Two-Dimensional Cluster.- Comments.- 10 Some Asymptotic Problems for a Non-Divergent Parabolic Equation with Random Stationary Coefficients.- 10.1 Preliminary Remarks.- 10.2 Auxiliary Equation A*p = 0 on a Probability Space.- 10.3 Homogenization and the Central Limit Theorem.- 10.4 Criterion of Pointwise Stabilization.- Comments.- 11 Spectral Problems in Homogenization Theory.- 11.1 Spectral Properties of Abstract Operators Forming a Sequence.- 11.2 On the Spectrum of G-Convergent Operators.- 11.3 The Sturm-Liouville Problem.- 11.4 Spectral Properties of Stratified Media.- 11.5 Density of States for Random Elliptic Operators.- 11.6 Asymptotics of the Density of States.- Comments.- 12 Homogenization in Linear Elasticity.- 12.1 Some General Facts from the Theory of Elasticity.- 12.2 G-Convergence of Elasticity Tensors.- 12.3 Homogenization of Periodic and Random Tensors.- 12.4 Fourth Order Operators.- 12.5 Linear Problems of Incompressible Elasticity.- 12.6 Explicit Formulas for Two-Dimensional Incompressible Composites.- 12.7 Some Questions of Analysis on a Probability Space.- 13 Estimates for the Homogenized Elasticity Tensor.- 13.1 Basic Estimates.- 13.2 The Variational Method.- 13.3 Two-Phase Media Attainability of Bounds on Stratified Composites.- 13.4 On the Hashin Structure.- 13.5 Disperse Media with Inclusions of Small Concentration.- 13.6 Fourth Order Operators Systems of Stokes Type.- Comments.- 14 Elements of the Duality Theory.- 14.1 Convex Functions.- 14.2 Integral Functionals.- 14.3 On Two Types of Boundary Value Problems.- 14.4 Dual Boundary Value Problems.- 14.5 Extremal Relations.- 14.6 Examples of Regular Lagrangians.- Comments.- 15 Homogenization of Nonlinear Variational Problems.- 15.1 Random Lagrangians.- 15.2 Two Principal Lemmas.- 15.3 Homogenization Theorems.- 15.4 Applications to Boundary Value Problems in Perforated Domains.- 15.5 Chess Lagrangians Dychne's Formula.- Comments.- 16 Passing to the Limit in Nonlinear Variational Problems.- 16.1 Definition of ?-Convergence of Lagrangians Formulation of the Compactness Theorems.- 16.2 Convergence of Energies and Minimizers.- 16.3 Proof of the Compactness Theorems.- 16.4 Two Examples: Ulam's Problem Homogenization Problem.- 16.5 Compactness of Lagrangians in Plasticity Problems Application to Ll-Closedness.- 16.6 Remarks on Non-Convex Functionals.- Comments.- 17 Basic Properties of Abstract ?-Convergence.- 17.1 ?-Convergence of Functions on a Metric Space.- 17.2 ?-Convergence of Functions Defined in a Banach Space.- 17.3 ?-Convergence of Integral Functionals.- Comments.- 18 Limit Load.- 18.1 The Notion of Limit Load.- 18.2 Dual Definition of Limit Load.- 18.3 Equivalence Principle.- 18.4 Convergence of Limit Loads in Homogenization Problems.- 18.5 Surface Loads.- 18.6 Representation of the Functional $$\bar F$$ on BV0.- 18.7 ?-Convergence in BV0.- Comments.- Appendix A. Proof of the Nash-Aronson Estimate.- Appendix C. A Property of Bounded Lipschitz Domains.- References.

2,677 citations

Journal ArticleDOI
TL;DR: In this article, the authors consider the case where the boundedness condition is separated from the coercivity condition, which leads to the loss of uniqueness, regularity, and some other properties of solutions.
Abstract: We study elliptic problems where the boundedness condition is “separated” from the coercivity condition, which leads to the loss of uniqueness, regularity, and some other properties of solutions. We propose new methods allowing us to establish the existence results for such problems, in particular, in situations where a weak solution to the Dirichlet problem is not unique and the energy equality fails. We develop a special techniques of the weak convergence of fluxes to a flux owing to which it is possible to pass to the limit in nonlinear terms. Based on this technique, we establish the solvability of the well-known thermistor problem without any restrictions on the spatial dimension and smallness of the data. Various model examples and counterexamples are also given. Bibliography: 72 titles. Illustrations: 3 figures.

172 citations

Journal ArticleDOI

60 citations

Journal ArticleDOI
Abstract: We consider the problem of passing to the limit in a sequence of nonlinear elliptic problems. The “limit” equation is known in advance, but it has a nonclassical structure; namely, it contains the p-Laplacian with variable exponent p = p(x). Such equations typically exhibit a special kind of nonuniqueness, known as the Lavrent’ev effect, and this is what makes passing to the limit nontrivial. Equations involving the p(x)-Laplacian occur in many problems of mathematical physics. Some applications are included in the present paper. In particular, we suggest an approach to the solvability analysis of a well-known coupled system in non-Newtonian hydrodynamics (“stationary thermo-rheological viscous flows”) without resorting to any smallness conditions.

41 citations


Cited by
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Journal ArticleDOI
TL;DR: Convergence of Probability Measures as mentioned in this paper is a well-known convergence of probability measures. But it does not consider the relationship between probability measures and the probability distribution of probabilities.
Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

5,689 citations

MonographDOI
06 May 2002
TL;DR: Some of the greatest scientists including Poisson, Faraday, Maxwell, Rayleigh, and Einstein have contributed to the theory of composite materials Mathematically, it is the study of partial differential equations with rapid oscillations in their coefficients Although extensively studied for more than a hundred years, an explosion of ideas in the last five decades has dramatically increased our understanding of the relationship between the properties of the constituent materials, the underlying microstructure of a composite, and the overall effective moduli which govern the macroscopic behavior as mentioned in this paper.
Abstract: Some of the greatest scientists including Poisson, Faraday, Maxwell, Rayleigh, and Einstein have contributed to the theory of composite materials Mathematically, it is the study of partial differential equations with rapid oscillations in their coefficients Although extensively studied for more than a hundred years, an explosion of ideas in the last five decades (and particularly in the last three decades) has dramatically increased our understanding of the relationship between the properties of the constituent materials, the underlying microstructure of a composite, and the overall effective (electrical, thermal, elastic) moduli which govern the macroscopic behavior This renaissance has been fueled by the technological need for improving our knowledge base of composites, by the advance of the underlying mathematical theory of homogenization, by the discovery of new variational principles, by the recognition of how important the subject is to solving structural optimization problems, and by the realization of the connection with the mathematical problem of quasiconvexification This 2002 book surveys these exciting developments at the frontier of mathematics

2,455 citations

Journal ArticleDOI
TL;DR: The heterogeneous multiscales method (HMM), a general framework for designing multiscale algorithms, is reviewed and emphasis is given to the error analysis that comes naturally with the framework.
Abstract: The heterogeneous multiscale method (HMM), a general framework for designing multiscale algorithms, is reviewed. Emphasis is given to the error analysis that comes naturally with the framework. Examples of finite element and finite difference HMM are presented. Applications to dynamical systems and stochastic simulation algorithms with multiple time scales, spall fracture and heat conduction in microprocessors are discussed.

675 citations

Journal ArticleDOI
TL;DR: In this article, the convergence of the macroscopic field variables on the selected size of unit cells is studied quantitatively via the computational homogenization method, and the convergence nature of microscopic stress values is quantitatively through the computation homogenisation method.

521 citations

Journal ArticleDOI
TL;DR: In this paper, simple mathematical models for the turbulent diffusion of a passive scalar field are developed with an emphasis on the symbiotic interaction between rigorous mathematical theory (including exact solutions), physical intuition, and numerical simulations.

511 citations