Contents Preface ix Introduction 1 1. Weak Convergence 4 A. Review of Basic Theory 4 B. Convergence of Averages 6 C. Compactness in Sobolev Spaces 6 1. Embeddings 6 2. Compactness Theorems 7 3. Á Refinement of Rellich's Theorem 8 D. Measures of Concentration 9 1. Generalities 9 2. Defect Measures 10 3. Á Refinement of Fatou's Lemma 11 4. Concentration and Sobolev Inequalities 12 E. Measures of Oscillation 14 1. Generalities 14 2. Slicing Measures 14 3. Young Measures 16 2. Convexity 18 A. The Calculus of Variations 18 B. Weak Lower Semicontinuity 19 C. Convergence of Energies and Strong Convergence 21 3. Quasiconvexity 22 A. Definitions 22 1. Rank-One Convexity 22 2. Quasiconvexity 23 B. Weak Lower Semicontinuity 24 C. Convergence of Energies and Strong Convergence 27 D. Partial Regularity of Minimizers 29 E. Examples 31 1. Weak Continuity of Determinants 31 vii
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