Functions of bounded variation and free discontinuity problems

Cover ......Page 1
Title ......Page 4
CONTENTS ......Page 12
Basic terminology and notation ......Page 15
1.1 Abstract measure theory ......Page 18
1.2 Weak convergence in Lp spaces ......Page 32
1.3 Measures in metric spaces ......Page 35
1.4 Outer measures and weak* convergence ......Page 38
1.5 Operations on measures ......Page 46
1.6 Exercises ......Page 52
2.1 Convolution ......Page 57
2.2 Sobolev spaces ......Page 59
2.3 Lipschitz functions ......Page 62
2.4 Covering and derivation of measures ......Page 65
2.5 Disintegration ......Page 73
2.6 Functional defined on measures ......Page 79
2.7 Tangent measures ......Page 86
2.8 Hausdorff measures ......Page 89
2.9 Rectifiable sets ......Page 96
2.10 Area formula ......Page 102
2.11 Approximate tangent space ......Page 109
2.12 Coarea formula ......Page 117
2.13 Minkowski content ......Page 125
2.14 Exercises ......Page 130
3 Functions of bounded variation ......Page 133
3.1 The space BV ......Page 134
3.2 BV functions of one variable ......Page 151
3.3 Sets of finite perimeter ......Page 160
3.4 Embedding theorems and isoperimetric inequalities ......Page 165
3.5 Structure of sets of finite perimeter ......Page 170
3.6 Approximate continuity and differentiability ......Page 177
3.7 Fine properties of BV functions ......Page 184
3.8 Decomposability of BV and boundary trace theorems ......Page 194
3.9 Decomposition of derivative and rank one properties ......Page 201
3.10 The chain rule in BV ......Page 205
3.11 One-dimensional restrictions of BV functions ......Page 210
3.12 A brief historical note on BV functions ......Page 221
3.13 Exercises ......Page 225
4 Special functions of bounded variation ......Page 228
4.1 The space SBV ......Page 229
4.2 Proof of the closure and compactness theorems ......Page 234
4.3 Poincare inequality in SBV ......Page 242
4.4 Caccioppoli partitions ......Page 244
4.5 Generalised functions of bounded variation ......Page 252
4.6 Introduction to free discontinuity problems ......Page 260
4.6.2 Optimal partitions ......Page 261
4.6.3 The Mumford-Shah image segmentation problem ......Page 262
4.6.4 A problem related to the theory of liquid crystals ......Page 263
4.6.5 Vector valued and higher order problems ......Page 264
4.6.6 Connexions with plasticity theory ......Page 266
4.6.7 Brittle fracture ......Page 267
4.7 Exercises ......Page 268
5 Semicontinuity in BV ......Page 271
5.1 Isotropic functional in BV ......Page 272
5.2 Convex volume energies ......Page 281
5.3 Surface energies for partitions ......Page 286
5.4 Lower semicontinuous functionals in SBV ......Page 298
5.5 Functionals with linear growth in BV ......Page 315
5.6 Exercises ......Page 333
6 The Mumford-Shah functional ......Page 336
6.1 Weak and strong solutions ......Page 337
6.2 Regularity theory: the state of the art ......Page 340
6.3 Local and global minimisers ......Page 342
6.4 Variational approximation and discrete models ......Page 348
7 Minimisers of free discontinuity problems ......Page 354
7.1 Limit behaviour of sequences in SBV ......Page 356
7.2 The density lower bound ......Page 364
7.3 First variation of the area and mean curvature ......Page 371
7.4 The Euler-Lagrange equation ......Page 377
7.5 Harmonic functions ......Page 383
7.6 Regularity of solutions of the Neumann problem ......Page 387
7.7 Equations of mean curvature type ......Page 393
7.8 Exercises ......Page 396
8 Regularity of the free discontinuity set ......Page 398
8.1 Limit behaviour of sequences of quasi-minimisers ......Page 400
8.2 Lipschitz approximation ......Page 408
8.3 Flatness improvement ......Page 419
8.4 Energy improvement ......Page 423
8.5 Proof of the decay theorem ......Page 431
References ......Page 436
Index ......Page 448