Homogenization of Differential Operators and Integral Functionals

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