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Table of Contents
Preface; Contents; 1 Motivation and Framework ; 1.1 The Model Problem; 1.2 Variations on the Same Theme; 1.3 Why It Is an Interesting Problem; 1.4 Why It Is a Difficult Problem; 1.5 General Procedure; 1.6 Subrelaxation; 1.7 What is Known; 1.7.1 Homogenization; 1.7.2 Engineering; 1.7.3 Some Brief, Additional Information; 1.8 Structure of the Book; 1.9 Bibliographical Comments; References; 2 Our Approach ; 2.1 The Strategy; 2.2 Young Measures; 2.2.1 Some Practice with Young Measures; 2.3 Relaxation; 2.4 Basic Differential Information: The Div-curl Lemma; 2.5 Subrelaxation
2.6 Bibliographical CommentsReferences; 3 Relaxation Through Moments ; 3.1 The Relaxation Revisited: Constraints; 3.2 The Moment Problem; 3.3 Characterization of Limit Pairs; 3.4 Laminates; 3.5 Characterization of Limit Pairs II; 3.6 Final Form of the Relaxation; 3.7 The Compliance Situation; 3.8 Bibliographical Comments; References; 4 Optimality; 4.1 Descent Method; 4.2 Optimality Conditions; 4.3 Final Remarks; 4.4 Bibliographical Comments; References; 5 Simulation ; 5.1 A Direct Approximation Scheme; 5.2 Some Selected Simulations; 5.2.1 Dependence on Initialization
5.2.2 Dependence on Volume Fraction5.2.3 Dependence on Contrast; 5.3 Some Additional Simulations; 5.4 Bibliographical Comments; References; 6 Some Extensions ; 6.1 A Non-linear Cost Functional; 6.1.1 Young Measures; 6.1.2 Solution of the Mathematical Program; 6.1.3 Subrelaxation; 6.1.4 Optimality; 6.1.5 Some Numerical Simulations; 6.2 A Non-linear State Law; 6.2.1 Reformulation and Young Measures for the Non-linear Situation; 6.2.2 Moments; 6.2.3 Necessary Conditions; 6.2.4 Sufficient Conditions; 6.3 A General Heuristic Approximation Method; 6.3.1 Conductivity; 6.3.2 Elasticity
6.4 Bibliographical CommentsReferences; 7 Some Technical Proofs ; 7.1 Div-Curl Lemma; 7.2 Riemann-Lebesgue Lemma; 7.3 Young Measures; 7.3.1 The Existence Theorem; 7.3.2 Some Results to Identify Young Measures; 7.3.3 Second-Order Laminares; 7.4 A Non-linear Elliptic Equation; 7.5 Covering Lemma; 7.6 Bibliographical Comments; References
2.6 Bibliographical CommentsReferences; 3 Relaxation Through Moments ; 3.1 The Relaxation Revisited: Constraints; 3.2 The Moment Problem; 3.3 Characterization of Limit Pairs; 3.4 Laminates; 3.5 Characterization of Limit Pairs II; 3.6 Final Form of the Relaxation; 3.7 The Compliance Situation; 3.8 Bibliographical Comments; References; 4 Optimality; 4.1 Descent Method; 4.2 Optimality Conditions; 4.3 Final Remarks; 4.4 Bibliographical Comments; References; 5 Simulation ; 5.1 A Direct Approximation Scheme; 5.2 Some Selected Simulations; 5.2.1 Dependence on Initialization
5.2.2 Dependence on Volume Fraction5.2.3 Dependence on Contrast; 5.3 Some Additional Simulations; 5.4 Bibliographical Comments; References; 6 Some Extensions ; 6.1 A Non-linear Cost Functional; 6.1.1 Young Measures; 6.1.2 Solution of the Mathematical Program; 6.1.3 Subrelaxation; 6.1.4 Optimality; 6.1.5 Some Numerical Simulations; 6.2 A Non-linear State Law; 6.2.1 Reformulation and Young Measures for the Non-linear Situation; 6.2.2 Moments; 6.2.3 Necessary Conditions; 6.2.4 Sufficient Conditions; 6.3 A General Heuristic Approximation Method; 6.3.1 Conductivity; 6.3.2 Elasticity
6.4 Bibliographical CommentsReferences; 7 Some Technical Proofs ; 7.1 Div-Curl Lemma; 7.2 Riemann-Lebesgue Lemma; 7.3 Young Measures; 7.3.1 The Existence Theorem; 7.3.2 Some Results to Identify Young Measures; 7.3.3 Second-Order Laminares; 7.4 A Non-linear Elliptic Equation; 7.5 Covering Lemma; 7.6 Bibliographical Comments; References