Approximation of free discontinuity problems:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | German |
| Veröffentlicht: |
Berlin [u.a.]
Springer
1998
|
| Schriftenreihe: | Lecture notes in mathematics
1694 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XI, 148 S. |
| ISBN: | 3540647716 |
Internformat
MARC
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| 245 | 1 | 0 | |a Approximation of free discontinuity problems |c Andrea Braides |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1998 | |
| 300 | |a XI, 148 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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Datensatz im Suchindex
| _version_ | 1804126707284705280 |
|---|---|
| adam_text | CONTENTS
Introduction 1
1 Functions of bounded variation 7
1.1 Measure theory. Basic notation 7
1.1.1 Supremum of a family of measures 9
1.2 Construction of measures. Hausdorff measures 10
1.2.1 Caratheodory s construction 10
1.2.2 Hausdorff measures 11
1.2.3 The De Giorgi and Letta measure criterion 11
1.3 Weak convergence of measures 12
1.3.1 Weak convergence of measures as set functions 13
1.3.2 Reshetnyak s Theorem 14
1.4 BV functions 14
1.4.1 BV functions of one variable 16
1.5 Sets of finite perimeter 16
1.6 Structure of the sets of finite perimeter 18
1.7 Approximate continuity 19
1.8 Structure of BV functions 20
1.8.1 1 dimensional sections of BV functions 21
1.8.2 The chain rule formula 22
1.9 Exercises 23
2 Special functions of bounded variation 27
2.1 SBV functions. A compactness theorem 27
2.2 General lower semicontinuity conditions in one dimension 29
2.2.1 Exercises 34
2.3 A lower semicontinuity theorem in higher dimensions 35
2.4 The Mumford Shah functional 36
2.4.1 GSBV functions 36
2.4.2 The Mumford Shah functional 38
3 Examples of approximation 39
3.1 F convergence: an overview 39
3.2 Elliptic approximations 42
3.2.1 Approximation of the perimeter by elliptic functionals 42
3.2.2 Exercises 46
3.2.3 Approximation of the Mumford Shah functional by ellip¬
tic functionals 47
3.2.4 Approximation of free discontinuity problems by elliptic
functionals 51
X Contents
3.3 Approximations by high order perturbations 56
3.3.1 Surface energies generated by high order singular pertur¬
bation 56
3.3.2 Exercises 63
3.3.3 Approximation of the Mumford Shah functional by high
order perturbations 64
3.3.4 Exercises 67
3.4 Non local approximations 67
3.4.1 Non local approximation of the Mumford Shah functional 67
3.4.2 Exercises 71
3.4.3 Non local approximation of free discontinuity problems 72
3.4.4 Exercises 77
3.5 Finite difference approximation of free discontinuity problems 78
3.5.1 Exercises 85
4 A general approach to approximation 87
4.1 A lower inequality by slicing 87
4.1.1 The slicing method 88
4.1.2 A lower estimate for the perimeter approximation 89
4.1.3 A lower estimate for the elliptic approximation 92
4.1.4 A lower estimate for the approximation by high order per¬
turbations 94
4.2 An upper inequality by density 96
4.2.1 An upper estimate for the perimeter approximation 97
4.2.2 A density result in SBV 99
4.2.3 An upper estimate for the elliptic approximation 100
4.3 Convergence results 101
5 Non local approximation 103
5.1 Non local approximation of the Mumford Shah functional 103
5.1.1 Estimate from below of the volume term 103
5.1.2 Estimate from below of the surface term 108
5.1.3 Estimate from below of the F limit 114
5.1.4 Estimate from above of the F limit 118
5.1.5 Some convergence results 119
5.2 Finite difference approximation of the Mumford Shah functional 124
5.2.1 Compactness 127
5.2.2 Convergence results 129
5.2.3 Exercises 130
Appendix 131
A Some numerical results 131
B Approximation of polyhedral energies 134
C An integral representation result 139
D Gap phenomenon in GSBV 141
Contents XI
Notation 143
References 145
Index 149
|
| any_adam_object | 1 |
| author | Braides, Andrea 1961- |
| author_GND | (DE-588)120341735 |
| author_facet | Braides, Andrea 1961- |
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| author_variant | a b ab |
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| bvnumber | BV012102968 |
| classification_rvk | SI 850 SK 420 SK 660 |
| classification_tum | MAT 410f |
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| dewey-full | 510 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics |
| dewey-raw | 510 |
| dewey-search | 510 |
| dewey-sort | 3510 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV012102968 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:21:44Z |
| institution | BVB |
| isbn | 3540647716 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008195167 |
| oclc_num | 246276039 |
| open_access_boolean | |
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| physical | XI, 148 S. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Braides, Andrea 1961- Verfasser (DE-588)120341735 aut Approximation of free discontinuity problems Andrea Braides Berlin [u.a.] Springer 1998 XI, 148 S. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1694 Approximation - Funktion <Mathematik> - Diskontinuität Funktion von beschränkter Variation (DE-588)4155666-5 gnd rswk-swf Approximation (DE-588)4002498-2 gnd rswk-swf Gamma-Konvergenz (DE-588)4311219-5 gnd rswk-swf Funktion von beschränkter Variation (DE-588)4155666-5 s Gamma-Konvergenz (DE-588)4311219-5 s Approximation (DE-588)4002498-2 s DE-604 Lecture notes in mathematics 1694 (DE-604)BV000676446 1694 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008195167&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Braides, Andrea 1961- Approximation of free discontinuity problems Lecture notes in mathematics Approximation - Funktion <Mathematik> - Diskontinuität Funktion von beschränkter Variation (DE-588)4155666-5 gnd Approximation (DE-588)4002498-2 gnd Gamma-Konvergenz (DE-588)4311219-5 gnd |
| subject_GND | (DE-588)4155666-5 (DE-588)4002498-2 (DE-588)4311219-5 |
| title | Approximation of free discontinuity problems |
| title_auth | Approximation of free discontinuity problems |
| title_exact_search | Approximation of free discontinuity problems |
| title_full | Approximation of free discontinuity problems Andrea Braides |
| title_fullStr | Approximation of free discontinuity problems Andrea Braides |
| title_full_unstemmed | Approximation of free discontinuity problems Andrea Braides |
| title_short | Approximation of free discontinuity problems |
| title_sort | approximation of free discontinuity problems |
| topic | Approximation - Funktion <Mathematik> - Diskontinuität Funktion von beschränkter Variation (DE-588)4155666-5 gnd Approximation (DE-588)4002498-2 gnd Gamma-Konvergenz (DE-588)4311219-5 gnd |
| topic_facet | Approximation - Funktion <Mathematik> - Diskontinuität Funktion von beschränkter Variation Approximation Gamma-Konvergenz |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008195167&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT braidesandrea approximationoffreediscontinuityproblems |