Mathematical Problems in Elasticity and Homogenization

Abstract

Some Mathematical Problems of the Theory of Elasticity. Some Functional Spaces and Their Properties. Auxiliary Propositions. Korn's Inequalities. Boundary Value Problems of Linear Elasticity. Perforated Domains with a Periodic Structure. Extension Theorems. Estimates for Solutions of Boundary Value Problems of Elasticity in Perforated Domains. Periodic Solutions of Boundary Value Problems for the System of Elasticity. Saint-Venant's Principle for Periodic Solutions of the Elasticity System. Estimates and Existence Theorems for Solutions of the Elasticity System in Unbounded Domains. Strong G -Convergence of Elasticity Operators. Homogenization of the System of Linear Elasticity. Composites and Perforated Materials. The Mixed Problem in a Perforated Domain with the Dirichlet Boundary Conditions on the Outer Part of the Boundary and the Neumann Conditions on the Surface of the Cavities. The Boundary Value Problem with Neumann Conditions in a Perforated Domain. Asymptotic Expansions for Solutions of Boundary Value Problems of Elasticity in a Perforated Layer. Asymptotic Expansions for Solutions of the Dirichlet Problem for the Elasticity System in a Perforated Domain. Asymptotic Expansions for Solutions of the Dirichlet Problem for the Biharmonic Equation. Some Generalizations for the Case of Perforated Domains with a Non-Periodic Structure. Homogenization of the System of Elasticity with Almost-Periodic Coefficients. Homogenization of Stratified Structures. Estimates for the Rate of G -Convergence of Higher-Order Elliptic Operators. Spectral Problems . Some Theorems from Functional Analysis. Spectral Problems for Abstract Operators. Homogenization of Eigenvalues and Eigenfunctions of Boundary Value Problems for Strongly Non-Homogeneous Elastic Bodies. On the Behaviour of Eigenvalues and Eigenfunctions of the Dirichlet Problem for Second Order Elliptic Equations in Perforated Domains. Third Boundary Value Problem for Second Order Elliptic Equations in Domains with Rapidly Oscillating Boundary. Free Vibrations of Bodies with Concentrated Masses. On the Behaviour of Eigenvalues of the Dirchlet Problem in Domains with Cavities Whose Concentration is Small. Homogenization of Eigenvalues of Ordinary Differential Operators. Asymptotic Expansion of Eigenvalues and Eigenfunctions of the Sturm-Liouville Problem for Equations with Rapidly Oscillating Coefficients. On the Behaviour of the Eigenvalues and Eigenfunctions of a G -Convergent Sequence of Non-Self-Adjoint Operators. References.