Sets of Finite Perimeter and Geometric Variational Problems An Introduction to Geometric Measure Theory

Contents Preface page xiii Notation xvii PART I RADON MEASURES ON R n 1 1 Outer measures 4 1.1 Examples of outer measures 4 1.2 Measurable sets and σ-additivity 7 1.3 Measure Theory and integration 9 2 Borel and Radon measures 14 2.1 Borel measures and Carathéodory's criterion 14 2.2 Borel regular measures 16 2.3 Approximation theorems for Borel measures 17 2.4 Radon measures. Restriction, support, and push-forward 19 3 Hausdorff measures 24 3.1 Hausdorff measures and the notion of dimension 24 3.2 H 1 and the classical notion of length 27 3.3 H n = L n and the isodiametric inequality 28 4 Radon measures and continuous functions 31 4.1 Lusin's theorem and density of continuous functions 31 4.2 Riesz's theorem and vector-valued Radon measures 33 4.3 Weak-star convergence 41 4.4 Weak-star compactness criteria 47 4.5 Regularization of Radon measures 49 5 Differentiation of Radon measures 51 5.1 Besicovitch's covering theorem 52 viii Contents 5.2 Lebesgue–Besicovitch differentiation theorem 58 5.3 Lebesgue points 62 6 Two further applications of differentiation theory 64 6.1 Campanato's criterion 64 6.2 Lower dimensional densities of a Radon measure 66 7 Lipschitz functions 68 7.1 Kirszbraun's theorem 69 7.2 Weak gradients 72 7.3 Rademacher's theorem 74 8 Area formula 76 8.1 Area formula for linear functions 77 8.2 The role of the singular set J f = 0 8 0 8.3 Linearization of Lipschitz immersions 82 8.4 Proof of the area formula 84 8.5 Area formula with multiplicities 85 9 Gauss–Green theorem 89 9.1 Area of a graph of codimension one 89 9.2 Gauss–Green theorem on open sets with C 1-boundary 90 9.3 Gauss–Green theorem on open sets with almost C 1-boundary 93 10 Rectifiable sets and blow-ups of Radon measures 96 10.1 Decomposing rectifiable sets by regular Lipschitz images 97 10.2 Approximate tangent spaces to rectifiable sets 99 10.3 Blow-ups of Radon measures and rectifiability 102 11 Tangential differentiability and the area formula 106 11.1 Area formula on surfaces 106 11.2 Area formula on rectifiable sets 108 11.3 Gauss–Green theorem on surfaces 110 Notes 114 PART II SETS OF FINITE PERIMETER 117 12 Sets of finite perimeter and the Direct Method 122 12.1 Lower semicontinuity of perimeter 125 12.2 Topological boundary and Gauss–Green measure 127 12.3 Regularization and basic set operations 128 12.4 Compactness from perimeter bounds 132 Contents ix 12.5 Existence of minimizers in geometric variational problems 136 12.6 Perimeter bounds on volume …