Functions of bounded variation and free discontinuity problems:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford
Clarendon Press
2006
|
| Edition: | 1. publ., reprint. |
| Series: | Oxford mathematical monographs
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 434 S. graph. Darst. |
| ISBN: | 0198502451 |
Staff View
MARC
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| 100 | 1 | |a Ambrosio, Luigi |d 1963- |e Verfasser |0 (DE-588)133791408 |4 aut | |
| 245 | 1 | 0 | |a Functions of bounded variation and free discontinuity problems |c Luigi Ambrosio ; Nicola Fusco ; Diego Pallara |
| 250 | |a 1. publ., reprint. | ||
| 264 | 1 | |a Oxford |b Clarendon Press |c 2006 | |
| 300 | |a XVIII, 434 S. |b graph. Darst. | ||
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Record in the Search Index
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|---|---|
| adam_text | CONTENTS
Basic terminology and notation
xvi
1
1
15
18
21
29
35
40
40
42
45
48
56
62
69
72
79
85
92
100
108
113
Functions of bounded variation
116
3.1
The space
B V
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
S F
functions
167
3.8
Decomposability of
B V
and boundary trace theorems
177
3.9
Decomposition of derivative and rank one properties
184
3.10
The chain rule in
B V
188
3.11
One-dimensional restrictions of
B V
functions
193
Measure theory
1.1
Abstract measure theory
1.2
Weak convergence in Lp spaces
1.3
Measures in metric spaces
1.4
Outer measures and weak* convergence
1.5
Operations on measures
1.6
Exercises
Basic geometric measure theory
2.1
Convolution
2.2
Sobolev spaces
2.3
lipschitz functions
2.4
Covering and derivation of measures
2.5
Disintegration
2.6
Functionals defined on measures
2.7
Tangent measures
2.8
Hausdorff measures
2.9
Rectifiable sets
2.10
Area formula
2.11
Approximate tangent space
2.12
Coarea
formula
2.13
Minkowski content
2.14
Exercises
xiv CONTENTS
3.12
A brief historical note on
В У
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
Poincaré
inequality in SB V
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
SemicontinuityinBV
254
5.1 Isotropie
funcţionale
in
BV
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
B V
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 SB V
339
7.2
The density lower bound
347
7.3
Krst
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
CONTENTS xv
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
|
| adam_txt |
CONTENTS
Basic terminology and notation
xvi
1
1
15
18
21
29
35
40
40
42
45
48
56
62
69
72
79
85
92
100
108
113
Functions of bounded variation
116
3.1
The space
B V
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
S F
functions
167
3.8
Decomposability of
B V
and boundary trace theorems
177
3.9
Decomposition of derivative and rank one properties
184
3.10
The chain rule in
B V
188
3.11
One-dimensional restrictions of
B V
functions
193
Measure theory
1.1
Abstract measure theory
1.2
Weak convergence in Lp spaces
1.3
Measures in metric spaces
1.4
Outer measures and weak* convergence
1.5
Operations on measures
1.6
Exercises
Basic geometric measure theory
2.1
Convolution
2.2
Sobolev spaces
2.3
lipschitz functions
2.4
Covering and derivation of measures
2.5
Disintegration
2.6
Functionals defined on measures
2.7
Tangent measures
2.8
Hausdorff measures
2.9
Rectifiable sets
2.10
Area formula
2.11
Approximate tangent space
2.12
Coarea
formula
2.13
Minkowski content
2.14
Exercises
xiv CONTENTS
3.12
A brief historical note on
В У
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
Poincaré
inequality in SB V
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
SemicontinuityinBV
254
5.1 Isotropie
funcţionale
in
BV
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
B V
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 SB V
339
7.2
The density lower bound
347
7.3
Krst
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
CONTENTS xv
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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| author | Ambrosio, Luigi 1963- Fusco, Nicola 1956- Pallara, Diego |
| author_GND | (DE-588)133791408 (DE-588)124005721 (DE-588)13630351X |
| author_facet | Ambrosio, Luigi 1963- Fusco, Nicola 1956- Pallara, Diego |
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| author_sort | Ambrosio, Luigi 1963- |
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| id | DE-604.BV022618536 |
| illustrated | Illustrated |
| index_date | 2024-07-02T18:19:12Z |
| indexdate | 2024-07-09T21:01:47Z |
| institution | BVB |
| isbn | 0198502451 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015824636 |
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| physical | XVIII, 434 S. graph. Darst. |
| publishDate | 2006 |
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| publishDateSort | 2006 |
| publisher | Clarendon Press |
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| series2 | Oxford mathematical monographs |
| spelling | Ambrosio, Luigi 1963- Verfasser (DE-588)133791408 aut Functions of bounded variation and free discontinuity problems Luigi Ambrosio ; Nicola Fusco ; Diego Pallara 1. publ., reprint. Oxford Clarendon Press 2006 XVIII, 434 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Oxford mathematical monographs Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Maßtheorie (DE-588)4074626-4 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 s Maßtheorie (DE-588)4074626-4 s DE-604 Fusco, Nicola 1956- Verfasser (DE-588)124005721 aut Pallara, Diego Verfasser (DE-588)13630351X aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015824636&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ambrosio, Luigi 1963- Fusco, Nicola 1956- Pallara, Diego Functions of bounded variation and free discontinuity problems Variationsrechnung (DE-588)4062355-5 gnd Maßtheorie (DE-588)4074626-4 gnd |
| subject_GND | (DE-588)4062355-5 (DE-588)4074626-4 |
| title | Functions of bounded variation and free discontinuity problems |
| title_auth | Functions of bounded variation and free discontinuity problems |
| title_exact_search | Functions of bounded variation and free discontinuity problems |
| title_exact_search_txtP | Functions of bounded variation and free discontinuity problems |
| title_full | Functions of bounded variation and free discontinuity problems Luigi Ambrosio ; Nicola Fusco ; Diego Pallara |
| title_fullStr | Functions of bounded variation and free discontinuity problems Luigi Ambrosio ; Nicola Fusco ; Diego Pallara |
| title_full_unstemmed | Functions of bounded variation and free discontinuity problems Luigi Ambrosio ; Nicola Fusco ; Diego Pallara |
| title_short | Functions of bounded variation and free discontinuity problems |
| title_sort | functions of bounded variation and free discontinuity problems |
| topic | Variationsrechnung (DE-588)4062355-5 gnd Maßtheorie (DE-588)4074626-4 gnd |
| topic_facet | Variationsrechnung Maßtheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015824636&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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