 | Ambrosio L. Functions of bounded variation and free discontinuity problems / L.Ambrosio, N.Fusco, D.Pallara. - Oxford; New York: Clarendon Press, 2000. - xviii, 434 p.: ill. - (Oxford mathematical monographs). - Ref.: p.420-430. - Ind.: p.431-434. - ISBN 0-19-850245-1
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1 Measure theory ............................................... 1
1.1 Abstract measure theory ................................. 1
1.2 Weak convergence in Lp spaces .......................... 15
1.3 Measures in metric spaces .............................. 18
1.4 Outer measures and weak* convergence ................... 21
1.5 Operations on measures ................................. 29
1.6 Exercises .............................................. 35
2 Basic geometric measure theory .............................. 40
2.1 Convolution ............................................ 40
2.2 Sobolev spaces ......................................... 42
2.3 Lipschitz functions .................................... 45
2.4 Covering and derivation of measures .................... 48
2.5 Disintegration ......................................... 56
2.6 Functionals denned on measures ......................... 62
2.7 Tangent measures ....................................... 69
2.8 Hausdorff measures ..................................... 72
2.9 Rectifiable sets ....................................... 79
2.10 Area formula ........................................... 85
2.11 Approximate tangent space .............................. 92
2.12 Coarea formula ........................................ 100
2.13 Minkowski content ..................................... 108
2.14 Exercises ............................................. 113
3 Functions of bounded variation ............................. 116
3.1 The space BV .......................................... 117
3.2 BV functions of one variable .......................... 134
3.3 Sets of finite perimeter .............................. 143
3.4 Embedding theorems and isoperimetric inequalities ..... 148
3.5 Structure of sets of finite perimeter ................. 153
3.6 Approximate continuity and differentiability .......... 160
3.7 Fine properties of BV functions ....................... 167
3.8 Decomposability of ВV and boundary trace theorems ..... 177
3.9 Decomposition of derivative and rank one properties ... 184
3.10 The chain rale in ВV .................................. 188
3.11 One-dimensional restrictions of ВV functions .......... 193
3.12 A brief historical note on ВV functions ............... 204
3.13 Exercises ............................................. 208
4 Special functions of bounded variation ..................... 211
4.1 The space SBV ......................................... 212
4.2 Proof of the closure and compactness theorems ......... 217
4.3 Poincare inequality in SBV ............................ 225
4.4 Caccioppoli partitions ................................ 227
4.5 Generalised functions of bounded variation ............ 235
4.6 Introduction to free discontinuity problems ........... 243
4.6.1 Sets with prescribed mean curvature ............ 244
4.6.2 Optimal partitions ............................. 244
4.6.3 The Mumford-Shah image segmentation problem .... 245
4.6.4 A problem related to the theory of liquid
crystals ....................................... 246
4.6.5 Vector valued and higher order problems ........ 247
4.6.6 Connexions with plasticity theory .............. 249
4.6.7 Brittle fracture ............................... 250
4.6.8 Structured deformations ........................ 251
4.7 Exercises ............................................. 251
5 Semicontinuity in ВV ....................................... 254
5.1 Isotropic functionals in ВV ........................... 255
5.2 Convex volume energies ................................ 264
5.3 Surface energies for partitions ....................... 269
5.4 Lower semicontinuous functionals in SBV ............... 281
5.5 Functionals with linear growth in ВV .................. 298
5.6 Exercises ............................................. 316
6 The Mumford-Shah functional ................................ 319
6.1 Weak and strong solutions ............................. 320
6.2 Regularity theory: the state of the art ............... 323
6.3 Local and global minimisers ........................... 325
6.4 Variational approximation and discrete models ......... 331
7 Minimisers of free discontinuity problems .................. 337
7.1 Limit behaviour of sequences in SBV ................... 339
7.2 The density lower bound ............................... 347
7.3 First variation of the area and mean curvature ........ 354
7.4 The Euler-Lagrange equation ........................... 360
7.5 Harmonic functions .................................... 366
7.6 Regularity of solutions of the Neumann problem ........ 370
7.7 Equations of mean curvature type ...................... 376
7.8 Exercises ............................................. 379
8 Regularity of the free discontinuity set ................... 381
8.1 Limit behaviour of sequences of quasi-minimisers ...... 383
8.2 Lipschitz approximation ............................... 391
8.3 Flatness improvement .................................. 402
8.4 Energy improvement .................................... 406
8.5 Proof of the decay theorem ............................ 414
References ................................................. 419
Index ......................................................... 431
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