Gamma-convergence for beginners:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Oxford [u.a.]
Oxford Univ. Press
2005
|
| Ausgabe: | Reprint. |
| Schriftenreihe: | Oxford lecture series in mathematics and its applications
22 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014571812&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XII, 218 S. graph. Darst. |
| ISBN: | 0198507844 |
Internformat
MARC
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| 100 | 1 | |a Braides, Andrea |d 1961- |e Verfasser |0 (DE-588)120341735 |4 aut | |
| 245 | 1 | 0 | |a Gamma-convergence for beginners |c Andrea Braides |
| 246 | 1 | 3 | |a G-convergence for beginners |
| 250 | |a Reprint. | ||
| 264 | 1 | |a Oxford [u.a.] |b Oxford Univ. Press |c 2005 | |
| 300 | |a XII, 218 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Oxford lecture series in mathematics and its applications |v 22 | |
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| 830 | 0 | |a Oxford lecture series in mathematics and its applications |v 22 |w (DE-604)BV009910017 |9 22 | |
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Datensatz im Suchindex
| DE-BY-TUM_call_number | 0102 MAT 494 2001 A 8500(1,2005) |
|---|---|
| DE-BY-TUM_katkey | 1554613 |
| DE-BY-TUM_location | 01 |
| DE-BY-TUM_media_number | 040010252150 |
| _version_ | 1821933959741702144 |
| adam_text | CONTENTS
Preface
vu
Introduction
1
Why a variational convergence?
1
Parade of examples
4
A maieutic approach to
Г-соп
vergence.
Direct methods
15
1
F-convergence by numbers
19
1.1
Some preliminaries
19
1.1.1
Lower and upper limits
19
1.1.2
Lower semicontinuity
21
1.2
F-convergence
22
1.3
Some examples on the real line
25
1.4
The many definitions of F-convergence
26
1.5
Convergence of minima
28
1.6
Upper and lower F-limits
30
1.7
The importance of being lower semicontinuous
32
1.7.1
Lower semicontinuity of F-limits
32
1.7.2
The lower-semicontinuous envelope. Relaxation
32
1.7.3
Approximation of lower-semicontinuous functions
33
1.7.4
The direct method
34
1.8
More properties of F-limits
34
1.8.1
Г
-limits
of monotone sequences
35
1.8.2
Compactness of F-convergence
35
1.8.3
F-convergence by subsequences
36
1.9
Г
-limits
indexed by a continuous parameter
37
1.10
Development by F-convergence
37
1.11
Exercises
38
Comments on Chapter
1 39
2
Integral problems
40
2.1
Problems on Lebesgue spaces
40
2.1.1
Weak convergences
41
2.1.2
Weak-coerciveness conditions
43
2.2
Weak lower semicontinuity conditions: convexity
44
2.3
Relaxation and F-convergence in IP spaces
47
2.4
Problems on Sobolev spaces
50
2.4.1
Weak convergence in Sobolev spaces
50
χ
Contents
2.4.2 Integral
funcţionale
on Sobolev spaces.
Coerciveness conditions
51
2.5
Weak lower semicontinuity conditions
52
2.6
F-convergence and convex analysis
54
2.7
Addition of boundary data
57
2.8
Some examples with degenerate growth conditions
58
2.8.1
Degeneracy of lower bounds: discontinuities
58
2.8.2
Degeneracy of upper bounds: functional of the
sup norm
59
2.9
Exercises
61
Comments on Chapter
2 62
3
Some homogenization problems
63
3.1
A direct approach
63
3.2
Different homogenization formulas
66
3.3
Limits of oscillating Riemannian metrics
68
3.4
Homogenization of Hamilton Jacobi equations
71
3.5
Exercises
74
Comments on Chapter
3 75
4
From discrete systems to integral functionals
76
4.1
Discrete functionals
77
4.2
Continuous limits
78
4.2.1
Nearest-neighbour interactions: a convexification
principle
78
4.2.2
Next-to-nearest neighbour interactions: non-convex
relaxation
80
4.2.3
Long-range interactions: homogenization
82
4.2.4
Convergence of minimum problems
84
4.3
Exercises
84
Comments on Chapter
4 84
5
Segmentation problems
85
5.1
Model problems
86
5.2
The space of piecewise-constant functions
87
5.2.1
Coerciveness conditions
87
5.2.2
Functionals on piecewise-constant functions
88
5.3
Lower semicontinuity conditions: subadditivity
88
5.4
Relaxation and F-convergence
91
5.4.1
Translation-invariant functionals
91
5.4.2
Properties of
subadditive
functions on
R
92
5.4.3
Relaxation:
subadditive
envelopes
93
5.4.4
F-convergence
97
5.4.5
Boundary values
98
5.5
Exercises
99
Contents xi
Comments on Chapter
5 100
Caccioppoli partitions
100
6
Phase-transition problems
102
6.1
Phase transitions as segmentation problems
102
6.2
Gradient theory for phase-transition problems
103
6.3
Gradient theory as a development by T-convergence
109
Comments on Chapter
6 112
7
Free-discontinuity problems
114
7.1
Piecewise-Sobolev functions
114
7.2
Some model problems
114
7.2.1
Signal reconstruction: the Mumford-Shah functional
115
7.2.2
Fracture mechanics: the Griffith functional
115
7.3
Functionals on piecewise-SoboIev functions
116
7.4
Examples of existence results
117
Comments on Chapter
7 119
Special functions of bounded variation
120
8
Approximation of free-discontinuity problems
121
8.1
The
Ambrosio Tortorelli
approximation
121
8.2
Approximation by convolution problems
124
8.2.1
Convolution integral functionals
125
8.2.2
Limits of convolution functionals
126
8.3
Finite-difference approximation
130
Comments on Chapter
8 131
9
More homogenization problems
132
9.1
Oscillations and phase transitions
132
9.2
Phase accumulation
135
9.3
Homogenization of free-discontinuity problems
137
Comments on Chapter
9 138
10
Interaction between elliptic problems and partition
problems
139
10.1
Quantitative conditions for lower semicontinuity
139
10.2
Existence without lower semicontinuity
142
10.3
Relaxation by interaction
143
10.4
Exercises
148
Comments on Chapter
10 148
Structured deformations
149
11
Discrete systems and free-discontinuity problems
150
11.1
Interpolation with piecewise-Sobolev functions
151
11.2
Equivalent energies on piecewise-Sobolev functions
153
11.3
Softening and fracture problems as limits of discrete models
154
xii Contents
11.4
Fracture
as a phase transition
156
11.5
Malik
Perona
approximation of free-discontinuity problems
159
11.6
Exercises
159
Comments on Chapter
11 160
12
*Some comments on
vectorial
problems
161
12.1
Lower semicontinuity conditions
162
12.1.1
Quasiconvexity
163
12.1.2
Convexity and polyconvexity
164
12.2
Homogenization and convexity conditions
165
12.2.1
Instability of polyconvexity
166
12.2.2
Density of
isotropie
quadratic forms
168
Comments on Chapter
12 169
13
*Dirichlet problems in perforated domains
171
13.1
Statement of the F-convergence result
172
13.2
A joining lemma on varying domains
174
13.3
Proof of the
lim inf
inequality
177
13.4
Proof of the
lim sup
inequality
178
Comments on Chapter
13 181
14
*Dimension-reduction problems
182
14.1
Convex energies
182
14.2
Non-convex vector-valued problems
185
Comments on Chapter
14 186
15
*The slicing method
187
15.1
A lower inequality by the slicing method
188
15.2
An upper inequality by density
191
Comments on Chapter
15 193
16
*An introduction to the localization method of
F-convergence
194
Appendices
197
A Some quick recalls
197
A.I Convexity
197
A.
2
Sobolev spaces
198
A.3 *Sets of finite perimeter
200
В
Characterization of F-convergence for ID integral problems
203
List of symbols
207
References
209
Index
217
|
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| id | DE-604.BV021250397 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T12:31:33Z |
| institution | BVB |
| isbn | 0198507844 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014571812 |
| oclc_num | 315619911 |
| open_access_boolean | |
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| owner_facet | DE-355 DE-BY-UBR DE-384 DE-91G DE-BY-TUM DE-573 DE-11 DE-20 DE-703 DE-19 DE-BY-UBM |
| physical | XII, 218 S. graph. Darst. |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Oxford Univ. Press |
| record_format | marc |
| series | Oxford lecture series in mathematics and its applications |
| series2 | Oxford lecture series in mathematics and its applications |
| spellingShingle | Braides, Andrea 1961- Gamma-convergence for beginners Oxford lecture series in mathematics and its applications Gamma-Konvergenz Gamma-Konvergenz (DE-588)4311219-5 gnd |
| subject_GND | (DE-588)4311219-5 |
| title | Gamma-convergence for beginners |
| title_alt | G-convergence for beginners |
| title_auth | Gamma-convergence for beginners |
| title_exact_search | Gamma-convergence for beginners |
| title_full | Gamma-convergence for beginners Andrea Braides |
| title_fullStr | Gamma-convergence for beginners Andrea Braides |
| title_full_unstemmed | Gamma-convergence for beginners Andrea Braides |
| title_short | Gamma-convergence for beginners |
| title_sort | gamma convergence for beginners |
| topic | Gamma-Konvergenz Gamma-Konvergenz (DE-588)4311219-5 gnd |
| topic_facet | Gamma-Konvergenz |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014571812&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009910017 |
| work_keys_str_mv | AT braidesandrea gammaconvergenceforbeginners AT braidesandrea gconvergenceforbeginners |
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Teilbibliothek Mathematik & Informatik
| Signatur: |
0102 MAT 494 2001 A 8500(1,2005) Lageplan |
|---|---|
| Exemplar 1 | Ausleihbar Am Standort |