Direct methods in the calculus of variations:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Springer
[2008]
|
| Ausgabe: | Second edition |
| Schriftenreihe: | Applied mathematical sciences
volume 78 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | xii, 619 Seiten Illustrationen, Diagramme |
| ISBN: | 9780387357799 |
Internformat
MARC
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| 020 | |a 9780387357799 |q hardcover |9 978-0-387-35779-9 | ||
| 035 | |a (OCoLC)183703350 | ||
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| 100 | 1 | |a Dacorogna, Bernard |d 1953- |e Verfasser |0 (DE-588)133791920 |4 aut | |
| 245 | 1 | 0 | |a Direct methods in the calculus of variations |c Bernard Dacorogna |
| 250 | |a Second edition | ||
| 264 | 1 | |a New York, NY |b Springer |c [2008] | |
| 264 | 4 | |c © 2008 | |
| 300 | |a xii, 619 Seiten |b Illustrationen, Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Applied mathematical sciences |v volume 78 | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 650 | 4 | |a Calcul des variations | |
| 650 | 4 | |a Calculus of variations | |
| 650 | 0 | 7 | |a Variationsrechnung |0 (DE-588)4062355-5 |2 gnd |9 rswk-swf |
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| 689 | 0 | |5 DE-604 | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-0-387-55249-1 |
| 830 | 0 | |a Applied mathematical sciences |v volume 78 |w (DE-604)BV000005274 |9 78 | |
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Datensatz im Suchindex
| _version_ | 1804138494141923328 |
|---|---|
| adam_text | Contents
Preface
χι
1
Introduction
1
1.1
The direct
methods of the calculus of variations
.......... 1
1.2
Convex analysis and the scalar case
................. 3
1.2.1
Convex analysis
........................ 4
1.2.2
Lower semicontinuity and existence results
........ 5
1.2.3
The one dimensional case
.................. 7
1.3
Quasiconvex analysis and the
vectorial
case
............ 9
1.3.1
Quasiconvex functions
.................... 9
1.3.2
Quasiconvex envelopes
.................... 12
1.3.3
Quasiconvex sets
....................... 13
1.3.4
Lower semicontinuity and existence theorems
....... 15
1.4
Relaxation and non-convex problems
................ 17
1.4.1
Relaxation theorems
..................... 18
1.4.2
Some existence theorems for differential inclusions
.... 19
1.4.3
Some existence results for non-quasiconvex
integrands
........................... 20
1.5
Miscellaneous
............................. 23
1.5.1
Holder and Sobolev spaces
.................. 23
1.5.2
Singular values
........................ 23
1.5.3
Some underdetermined partial differential
equations
........................... 24
1.5.4
Extension of Lipschitz maps
................. 25
Convex analysis and the scalar case
29
Convex sets and convex functions
31
2.1
Introduction
.............................. 31
2.2
Convex sets
.............................. 32
2.2.1
Basic definitions and properties
............... 32
2.2.2
Separation theorems
..................... 34
VI
CONTENTS
2.2.3
Convex
hull and Carathéodory
theorem
.......... 38
2.2.4
Extreme points and Minkowski theorem
.......... 42
2.3
Convex functions
........................... 44
2.3.1
Basic definitions and properties
............... 44
2.3.2
Continuity of convex functions
............... 46
2.3.3
Convex envelope
....................... 52
2.3.4
Lower semicontinuous envelope
............... 56
2.3.5
Legendre transform and duality
............... 57
2.3.6
Subgradients
and differentiability
of convex functions
...................... 61
2.3.7
Gauges and their
polars
................... 68
2.3.8
Choquet function
....................... 70
Lower semicontinuity and existence theorems
73
3.1
Introduction
.............................. 73
3.2
Weak lower semicontinuity
...................... 74
3.2.1
Preliminaries
......................... 74
3.2.2
Some approximation lemmas
................ 77
3.2.3
Necessary condition: the case without lower
order terms
.......................... 82
3.2.4
Necessary condition: the general case
........... 84
3.2.5
Sufficient condition: a particular case
........... 94
3.2.6
Sufficient condition: the general case
............ 96
3.3
Weak continuity and invariant integrals
.............. 101
3.3.1
Weak continuity
....................... 101
3.3.2
Invariant integrals
...................... 103
3.4
Existence theorems and Euler-Lagrange equations
........ 105
3.4.1
Existence theorems
...................... 105
3.4.2
Euler-Lagrange equations
.................. 108
3.4.3
Some regularity results
.................... 116
The one dimensional case
119
4.1
Introduction
.............................. 119
4.2
An existence theorem
........................ 120
4.3
The Euler-Lagrange equation
.................... 125
4.3.1
The classical and the weak forms
.............. 125
4.3.2
Second form of the Euler-Lagrange equation
........ 129
4.4
Some inequalities
........................... 132
4.4.1
Poincaré-Wirtinger
inequality
................ 132
4.4.2
Wirtinger inequality
..................... 132
4.5
Hamiltonian formulation
....................... 137
4.6
Regularity
............................... 143
4.7
Lavrentiev phenomenon
....................... 148
CONTENTS
vii
II
Quasiconvex
analysis
and the
vectorial
case
153
5
Polyconvex, quasiconvex and rank one convex functions
155
5.1
Introduction
..............................155
5.2
Definitions and main properties
...................156
5.2.1
Definitions and notations
..................156
5.2.2
Main properties
........................158
5.2.3
Further properties of polyconvex functions
.........163
5.2.4
Further properties of quasiconvex functions
........171
5.2.5
Further properties of rank one convex functions
......174
5.3
Examples
...............................178
5.3.1 Quasiaffine
functions
.....................179
5.3.2
Quadratic case
........................191
5.3.3
Convexity of SO
(τι) χ
SO
(n)
and
O (N) x O (n)
invariant
functions......................
197
5.3.4
Polyconvexity and
rank one convexity of SO
(n) x
SO
(n)
and
O (N) x O
(τι)
invariant
functions
...........202
5.3.5
Functions depending
on a
quasiaffine
function
.......212
5.3.6
The area type case
......................215
5.3.7
The example of Sverak
....................219
5.3.8
The example of Alibert-Dacorogna-Marcellmi
.......221
5.3.9
Quasiconvex functions with subquadratic growth
......237
5.3.10
The case of homogeneous functions of degree one
.....239
5.3.11
Some more examples
.....................245
5.4
Appendix: some basic properties of determinants
.........249
6
Polyconvex, quasiconvex and rank one convex envelopes
265
6.1
Introduction
.............................. 265
6.2
The polyconvex envelope
....................... 266
6.2.1
Duality for polyconvex functions
.............. 266
6.2.2
Another representation formula
............... 269
6.3
The quasiconvex envelope
...................... 271
6.4
The rank one convex envelope
.................... 277
6.5
Some more properties of the envelopes
............... 280
6.5.1
Envelopes and sums of functions
.............. 280
6.5.2
Envelopes and
invariances
.................. 282
6.6
Examples
............................... 285
6.6.1
Duality for SO
(n) x
SO (n) and
O (N) x O (n)
invariant
functions
...................... 285
6.6.2
The case of singular values
................. 291
6.6.3
Functions depending on
a
quasiaffine
function
....... 296
6.6.4
The area type case
...................... 298
6.6.5
The Kohn-Strang example
.................. 300
6.6.6
The Saint Venant-Kirchhoff energy function
........ 305
6.6.7
The case of a norm
...................... 309
viii CONTENTS
7
Polyconvex,
quasiconvex and rank one convex sets
313
7.1
Introduction
..............................313
7.2
Polyconvex, quasiconvex and rank one convex sets
........315
7.2.1
Definitions and main properties
............... 315
7.2.2
Separation theorems for polyconvex sets
.......... 321
7.2.3
Appendix: functions with finitely many gradients
..... 322
7.3
The different types of convex hulls
................. 323
7.3.1
The different convex hulls
..................323
7.3.2
The different convex finite hulls
...............331
7.3.3
Extreme points and Minkowski type theorem
for polyconvex, quasiconvex and rank one convex sets
. . 335
7.3.4
Gauges for polyconvex sets
.................342
7.3.5
Choquet functions for polyconvex and rank one
convex sets
..........................344
7.4
Examples
...............................347
7.4.1
The case of singular values
................. 348
7.4.2
The case of potential wells
.................. 355
7.4.3
The case of a quasiaffine function
.............. 362
7.4.4
A problem of optimal design
................ 364
8
Lower semi continuity and existence theorems in the
vectorial
case
367
8.1
Introduction
.............................. 367
8.2
Weak lower semiconthmity
...................... 368
8.2.1
Necessary condition
..................... 368
8.2.2
Lower semicontinuity for quasiconvex functions
without lower order terms
.................. 369
8.2.3
Lower semicontinuity for general quasiconvex
functions for
ρ
=
oo
..................... 377
8.2.4
Lower semicontinuity for general quasiconvex
functions for
1 <
ρ
<
oo
................... 381
8.2.5
Lower semicontinuity for polyconvex functions
...... 391
8.3
Weak Continuity
........................... 393
8.3.1
Necessary condition
..................... 393
8.3.2
Sufficient condition
...................... 394
8.4
Existence theorems
.......................... 403
8.4.1
Existence theorem for quasiconvex functions
........ 403
8.4.2
Existence theorem for polyconvex functions
........ 404
8.5
Appendix: some properties of Jacobians
.............. 407
III Relaxation and non-convex problems
413
9
Relaxation theorems
415
9.1
Introduction
..............................415
9.2
Relaxation Theorems
.........................416
CONTENTS ix
9.2.1
The case without lower order terms
............ 416
9.2.2
The general case
....................... 424
10
Implicit partial differential equations
439
10.1
Introduction
.............................. 439
10.2
Existence theorems
.......................... 440
10.2.1
An abstract theorem
..................... 440
10.2.2
A sufficient condition for the relaxation property
..... 444
10.2.3
Appendix: Baire one functions
............... 449
10.3
Examples
............................... 451
10.3.1
The scalar case
........................ 451
10.3.2
The case of singular values
................. 459
10.3.3
The case of potential wells
.................. 461
10.3.4
The case of a qnasiaffine function
.............. 462
10.3.5
A problem of optimal design
................ 463
11
Existence of minima for non-quasiconvex integrands
465
11.1
Introduction
.............................. 465
11.2
Sufficient conditions
......................... 467
11.3
Necessary conditions
......................... 472
11.4
The scalar case
............................ 483
11.4.1
The case of single integrals
................. 483
11.4.2
The case of multiple integrals
................ 485
11.5
The
vectorial
case
.......................... 487
11.5.1
The case of singular values
................. 488
11.5.2
The case of
quasiaffine
functions
.............. 490
11.5.3
The Saint Venant-Kirchhoff energy
............. 492
11.5.4
A problem of optimal design
................ 493
11.5.5
The area type case
...................... 494
11.5.6
The case of potential wells
.................. 498
IV Miscellaneous
501
12
Function spaces
503
12.1
Introduction
.............................. 503
12.2
Main notation
............................. 503
12.3
Some properties of Holder spaces
.................. 506
12.4
Some properties of Sobolev spaces
................. 509
12.4.1
Definitions and notations
.................. 510
12.4.2
Imbeddings and compact imbeddings
............ 510
12.4.3
Approximation by smooth and piecewise
affine
functions
. 512
x
CONTENTS
13 Singular
values
515
13.1
Introduction
..............................515
13.2 Definition
and basic properties
...................515
13.3
Signed singular values and
von
Neumann type inequalities
.... 519
14
Some underdetermined partial differential equations
529
14.1
Introduction
.............................. 529
14.2
The equations
divu
= ƒ
and curl-u
= ƒ............... 529
14.2.1
A preliminary lemma
.................... 529
14.2.2
The case
diviť
= ƒ...................... 531
14.2.3
The case curl
u
=
f......................
533
14.3
The equation
det
Vu
= ƒ ...................... 535
14.3.1
The main theorem and some corollaries
.......... 535
14.3.2
A deformation argument
................... 539
14.3.3
A proof under a smallness assumption
........... 541
14.3.4
Тгто
proofs of the main theorem
.............. 543
15
Extension of Lipschitz functions on Banach spaces
549
15.1
Introduction
..............................549
15.2
Preliminaries and notation
......................549
15.3
Norms induced by an inner product
................551
15.4
Extension from a general subset of
E
to
E
.............558
15.5
Extension from a convex subset of
E
to
E
.............565
Bibliography
569
Notation
611
Index
615
|
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| id | DE-604.BV035227962 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:29:05Z |
| institution | BVB |
| isbn | 9780387357799 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017033960 |
| oclc_num | 183703350 |
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| physical | xii, 619 Seiten Illustrationen, Diagramme |
| publishDate | 2008 |
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| publisher | Springer |
| record_format | marc |
| series | Applied mathematical sciences |
| series2 | Applied mathematical sciences |
| spelling | Dacorogna, Bernard 1953- Verfasser (DE-588)133791920 aut Direct methods in the calculus of variations Bernard Dacorogna Second edition New York, NY Springer [2008] © 2008 xii, 619 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Applied mathematical sciences volume 78 Hier auch später erschienene, unveränderte Nachdrucke Calcul des variations Calculus of variations Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Direkte Methode (DE-588)4705893-6 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 s Direkte Methode (DE-588)4705893-6 s DE-604 Erscheint auch als Online-Ausgabe 978-0-387-55249-1 Applied mathematical sciences volume 78 (DE-604)BV000005274 78 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017033960&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dacorogna, Bernard 1953- Direct methods in the calculus of variations Applied mathematical sciences Calcul des variations Calculus of variations Variationsrechnung (DE-588)4062355-5 gnd Direkte Methode (DE-588)4705893-6 gnd |
| subject_GND | (DE-588)4062355-5 (DE-588)4705893-6 |
| title | Direct methods in the calculus of variations |
| title_auth | Direct methods in the calculus of variations |
| title_exact_search | Direct methods in the calculus of variations |
| title_full | Direct methods in the calculus of variations Bernard Dacorogna |
| title_fullStr | Direct methods in the calculus of variations Bernard Dacorogna |
| title_full_unstemmed | Direct methods in the calculus of variations Bernard Dacorogna |
| title_short | Direct methods in the calculus of variations |
| title_sort | direct methods in the calculus of variations |
| topic | Calcul des variations Calculus of variations Variationsrechnung (DE-588)4062355-5 gnd Direkte Methode (DE-588)4705893-6 gnd |
| topic_facet | Calcul des variations Calculus of variations Variationsrechnung Direkte Methode |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017033960&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005274 |
| work_keys_str_mv | AT dacorognabernard directmethodsinthecalculusofvariations |