Modern Methods in the Calculus of Variations: L p spaces
Gespeichert in:
Hauptverfasser: | , |
---|---|
Format: | Buch |
Sprache: | English |
Veröffentlicht: |
New York, NY [u.a.]
Springer
2007
|
Schriftenreihe: | Springer monographs in mathematics
|
Schlagworte: | |
Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
Beschreibung: | XIV, 599 S. |
ISBN: | 038735784X 9780387357843 |
Internformat
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100 | 1 | |a Fonseca, Irene |e Verfasser |4 aut | |
245 | 1 | 0 | |a Modern Methods in the Calculus of Variations |b L p spaces |c Irene Fonseca ; Giovanni Leoni |
264 | 1 | |a New York, NY [u.a.] |b Springer |c 2007 | |
300 | |a XIV, 599 S. | ||
336 | |b txt |2 rdacontent | ||
337 | |b n |2 rdamedia | ||
338 | |b nc |2 rdacarrier | ||
490 | 0 | |a Springer monographs in mathematics | |
650 | 4 | |a Calculus of variations | |
650 | 4 | |a Field theory (Physics) | |
650 | 4 | |a Nonlinear theories | |
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Datensatz im Suchindex
_version_ | 1804136261816942592 |
---|---|
adam_text | Contents
Part I Measure Theory and Lp Spaces
Measures
.................................................. 3
1.1
Measures and Integration
................................. 3
1.1.1
Measures and Outer Measures
...................... 3
1.1.2
Radon and
Borei
Measures and Outer Measures
....... 22
1.1.3
Measurable Functions and Lebesgue Integration
....... 37
1.1.4
Comparison Between Measures
...................... 55
1.1.5
Product Spaces
................................... 77
1.1.6
Projection of Measurable Sets
....................... 83
1.2
Covering Theorems and Differentiation of Measures in MN
.... 90
1.2.1
Covering Theorems in RN
.......................... 90
1.2.2
Differentiation Between Radon Measures in K^
.......103
1.3
Spaces of Measures
......................................113
1.3.1
Signed Measures
..................................113
1.3.2
Signed Finitely Additive Measures
...................119
1.3.3
Spaces of Measures as Dual Spaces
..................123
1.3.4
Weak Star Convergence of Measures
.................129
IP Spaces
..................................................139
2.1
Abstract Setting
........................................139
2.1.1
Definition and Main Properties
......................139
2.1.2
Strong Convergence in LP
..........................148
2.1.3
Dual Spaces
......................................156
2.1.4
Weak Convergence in IP
...........................171
2.1.5
Biting Convergence
................................184
2.2
Euclidean Setting
........................................190
2.2.1
Approximation by Regular Functions
................190
2.2.2
Weak Convergence m LP
...........................198
2.2.3
Maximal Functions
................................208
2.3
LP Spaces on Banach Spaces
..............................218
XII Contents
Part II The Direct Method and Lower Semicontinuity
3
The Direct Method and Lower Semicontinuity
.............231
3.1
Lower Semicontinuity
....................................231
3.2
The Direct Method
......................................245
4
Convex Analysis
...........................................247
4.1
Convex Sets
............................................247
4.2
Separating Theorems
....................................254
4.3
Convex Functions
.......................................258
4.4
Lipschitz Continuity in Normed Spaces
.....................262
4.5
Regularity of Convex Functions
...........................266
4.6
Recession Function
......................................288
4.7
Approximation of Convex Functions
.......................293
4.8
Convex Envelopes
.......................................300
4.9
Star-Shaped Sets
........................................318
Part III Functional^ Defined on Lp
5
Integrands
ƒ = ƒ
(z)
.......................................325
5.1
Well-Posedness
..........................................326
5.2
Sequential Lower Semicontinuity
..........................331
5.2.1
Strong Convergence in LP
..........................331
5.2.2
Weak Convergence and Weak Star Convergence in LP
. . 334
5.2.3
Weak Star Convergence in the Sense of Measures
......340
5.2.4
Weak Star Convergence in (Cb
(Έ;
R 1))
.............350
5.3
Integral Representation
..................................354
5.4
Relaxation
..............................................364
5.4.1
Weak Convergence and Weak Star Convergence in LP,
l<p<oo
........................................365
5.4.2
Weak Star Convergence in the Sense of Measures
......369
5.5
Minimization
...........................................373
6
Integrands
ƒ =
f(x,z)
.....................................379
6.1
Multifunctions
..........................................380
6.1.1
Measurable Selections
..............................380
6.1.2
Continuous Selections
..............................395
6.2
Integrands
..............................................401
6.2.1
Equivalent Integrands
..............................401
6.2.2
Normal and
Carathéodory
Integrands
................404
6.2.3
Convex Integrands
.................................413
6.3
Well-Posedness
..........................................428
6.3.1
Well-Posedness,
1 <
ρ
<
oo
.........................428
6.3.2
Well-Posedness,
ρ
=
oo
.............................435
Contents XIII
6.4
Sequential Lower Semicontinuity
..........................436
6.4.1
Strong Convergence in Lp,
1 <
ρ
<
oo
................436
6.4.2
Strong Convergence in L°°
.........................442
6.4.3
Weak Convergence in Lp,
1 <
ρ
<
oo
.................445
6.4.4
Weak Star Convergence in L°°
......................448
6.4.5
Weak Star Convergence in the Sense of Measures
......449
6.5
Integral Representation in IP
.............................464
6.6
Relaxation in U>
........................................473
6.6.1
Weak Convergence and Weak Star Convergence in Lp,
1 <
Ρ
<
oo
........................................473
6.6.2
Weak Star Convergence in the Sense of Measures in L1
. 478
7
Integrands
ƒ = ƒ
(аз, и,
ζ)
..................................485
7.1
Convex Integrands
.......................................485
7.2
Well-Posedness
..........................................489
7.3
Sequential Lower Semicontinuity
..........................491
7.3.1
Strong-Strong Convergence
.........................491
7.3.2
Strong-Weak Convergence I
<
p, q
<
oo
.............492
7.4
Relaxation
..............................................511
8
Young Measures
...........................................517
8.1
The Fundamental Theorem for Young Measures
.............518
8.2
Characterization of Young Measures
.......................532
8.2.1
The Homogeneous Case
............................533
8.2.2
The Inhomogeneous Case
...........................538
8.3
Relaxation
..............................................540
Part IV Appendix
A Functional Analysis and Set Theory
.......................549
A.I Some Results from Functional Analysis
.....................549
A.I.I Topological Spaces
................................549
A.I.
2
Metric Spaces
.....................................552
A.
1.3
Topological Vector Spaces
..........................554
A.1.4 Normed Spaces
...................................558
A.1.5 Weak Topologies
..................................560
A.1.6 Dual Pairs
........................................563
A.I.? Hilbert Spaces
....................................566
A.2 Wellorderings, Ordinals, and Cardinals
.....................567
В
Notes and Open Problems
.................................573
Notation and List of Symbols
..................................581
Acknowledgments
.............................................585
XIV Contents
References
.....................................................587
Index
..........................................................595
|
adam_txt |
Contents
Part I Measure Theory and Lp Spaces
Measures
. 3
1.1
Measures and Integration
. 3
1.1.1
Measures and Outer Measures
. 3
1.1.2
Radon and
Borei
Measures and Outer Measures
. 22
1.1.3
Measurable Functions and Lebesgue Integration
. 37
1.1.4
Comparison Between Measures
. 55
1.1.5
Product Spaces
. 77
1.1.6
Projection of Measurable Sets
. 83
1.2
Covering Theorems and Differentiation of Measures in MN
. 90
1.2.1
Covering Theorems in RN
. 90
1.2.2
Differentiation Between Radon Measures in K^
.103
1.3
Spaces of Measures
.113
1.3.1
Signed Measures
.113
1.3.2
Signed Finitely Additive Measures
.119
1.3.3
Spaces of Measures as Dual Spaces
.123
1.3.4
Weak Star Convergence of Measures
.129
IP Spaces
.139
2.1
Abstract Setting
.139
2.1.1
Definition and Main Properties
.139
2.1.2
Strong Convergence in LP
.148
2.1.3
Dual Spaces
.156
2.1.4
Weak Convergence in IP
.171
2.1.5
Biting Convergence
.184
2.2
Euclidean Setting
.190
2.2.1
Approximation by Regular Functions
.190
2.2.2
Weak Convergence 'm LP
.198
2.2.3
Maximal Functions
.208
2.3
LP Spaces on Banach Spaces
.218
XII Contents
Part II The Direct Method and Lower Semicontinuity
3
The Direct Method and Lower Semicontinuity
.231
3.1
Lower Semicontinuity
.231
3.2
The Direct Method
.245
4
Convex Analysis
.247
4.1
Convex Sets
.247
4.2
Separating Theorems
.254
4.3
Convex Functions
.258
4.4
Lipschitz Continuity in Normed Spaces
.262
4.5
Regularity of Convex Functions
.266
4.6
Recession Function
.288
4.7
Approximation of Convex Functions
.293
4.8
Convex Envelopes
.300
4.9
Star-Shaped Sets
.318
Part III Functional^ Defined on Lp
5
Integrands
ƒ = ƒ
(z)
.325
5.1
Well-Posedness
.326
5.2
Sequential Lower Semicontinuity
.331
5.2.1
Strong Convergence in LP
.331
5.2.2
Weak Convergence and Weak Star Convergence in LP
. . 334
5.2.3
Weak Star Convergence in the Sense of Measures
.340
5.2.4
Weak Star Convergence in (Cb
(Έ;
R"1))'
.350
5.3
Integral Representation
.354
5.4
Relaxation
.364
5.4.1
Weak Convergence and Weak Star Convergence in LP,
l<p<oo
.365
5.4.2
Weak Star Convergence in the Sense of Measures
.369
5.5
Minimization
.373
6
Integrands
ƒ =
f(x,z)
.379
6.1
Multifunctions
.380
6.1.1
Measurable Selections
.380
6.1.2
Continuous Selections
.395
6.2
Integrands
.401
6.2.1
Equivalent Integrands
.401
6.2.2
Normal and
Carathéodory
Integrands
.404
6.2.3
Convex Integrands
.413
6.3
Well-Posedness
.428
6.3.1
Well-Posedness,
1 <
ρ
<
oo
.428
6.3.2
Well-Posedness,
ρ
=
oo
.435
Contents XIII
6.4
Sequential Lower Semicontinuity
.436
6.4.1
Strong Convergence in Lp,
1 <
ρ
<
oo
.436
6.4.2
Strong Convergence in L°°
.442
6.4.3
Weak Convergence in Lp,
1 <
ρ
<
oo
.445
6.4.4
Weak Star Convergence in L°°
.448
6.4.5
Weak Star Convergence in the Sense of Measures
.449
6.5
Integral Representation in IP
.464
6.6
Relaxation in U>
.473
6.6.1
Weak Convergence and Weak Star Convergence in Lp,
1 <
Ρ
<
oo
.473
6.6.2
Weak Star Convergence in the Sense of Measures in L1
. 478
7
Integrands
ƒ = ƒ
(аз, и,
ζ)
.485
7.1
Convex Integrands
.485
7.2
Well-Posedness
.489
7.3
Sequential Lower Semicontinuity
.491
7.3.1
Strong-Strong Convergence
.491
7.3.2
Strong-Weak Convergence I
<
p, q
<
oo
.492
7.4
Relaxation
.511
8
Young Measures
.517
8.1
The Fundamental Theorem for Young Measures
.518
8.2
Characterization of Young Measures
.532
8.2.1
The Homogeneous Case
.533
8.2.2
The Inhomogeneous Case
.538
8.3
Relaxation
.540
Part IV Appendix
A Functional Analysis and Set Theory
.549
A.I Some Results from Functional Analysis
.549
A.I.I Topological Spaces
.549
A.I.
2
Metric Spaces
.552
A.
1.3
Topological Vector Spaces
.554
A.1.4 Normed Spaces
.558
A.1.5 Weak Topologies
.560
A.1.6 Dual Pairs
.563
A.I.? Hilbert Spaces
.566
A.2 Wellorderings, Ordinals, and Cardinals
.567
В
Notes and Open Problems
.573
Notation and List of Symbols
.581
Acknowledgments
.585
XIV Contents
References
.587
Index
.595 |
any_adam_object | 1 |
any_adam_object_boolean | 1 |
author | Fonseca, Irene Leoni, Giovanni 1967- |
author_GND | (DE-588)139069348 |
author_facet | Fonseca, Irene Leoni, Giovanni 1967- |
author_role | aut aut |
author_sort | Fonseca, Irene |
author_variant | i f if g l gl |
building | Verbundindex |
bvnumber | BV022261516 |
callnumber-first | Q - Science |
callnumber-label | QA316 |
callnumber-raw | QA316 |
callnumber-search | QA316 |
callnumber-sort | QA 3316 |
callnumber-subject | QA - Mathematics |
classification_rvk | SK 660 |
classification_tum | MAT 280f MAT 490f |
ctrlnum | (OCoLC)255569643 (DE-599)BVBBV022261516 |
dewey-full | 515.64 |
dewey-hundreds | 500 - Natural sciences and mathematics |
dewey-ones | 515 - Analysis |
dewey-raw | 515.64 |
dewey-search | 515.64 |
dewey-sort | 3515.64 |
dewey-tens | 510 - Mathematics |
discipline | Physik Mathematik |
discipline_str_mv | Physik Mathematik |
format | Book |
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illustrated | Not Illustrated |
index_date | 2024-07-02T16:42:59Z |
indexdate | 2024-07-09T20:53:36Z |
institution | BVB |
isbn | 038735784X 9780387357843 |
language | English |
oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015472154 |
oclc_num | 255569643 |
open_access_boolean | |
owner | DE-703 DE-824 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-11 DE-188 DE-91G DE-BY-TUM DE-20 DE-83 |
owner_facet | DE-703 DE-824 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-11 DE-188 DE-91G DE-BY-TUM DE-20 DE-83 |
physical | XIV, 599 S. |
publishDate | 2007 |
publishDateSearch | 2007 |
publishDateSort | 2007 |
publisher | Springer |
record_format | marc |
series2 | Springer monographs in mathematics |
spelling | Fonseca, Irene Verfasser aut Modern Methods in the Calculus of Variations L p spaces Irene Fonseca ; Giovanni Leoni New York, NY [u.a.] Springer 2007 XIV, 599 S. txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics Calculus of variations Field theory (Physics) Nonlinear theories Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Lp-Raum (DE-588)4168195-2 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 s Lp-Raum (DE-588)4168195-2 s 1\p DE-604 Leoni, Giovanni 1967- Verfasser (DE-588)139069348 aut text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2820406&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015472154&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
spellingShingle | Fonseca, Irene Leoni, Giovanni 1967- Modern Methods in the Calculus of Variations L p spaces Calculus of variations Field theory (Physics) Nonlinear theories Variationsrechnung (DE-588)4062355-5 gnd Lp-Raum (DE-588)4168195-2 gnd |
subject_GND | (DE-588)4062355-5 (DE-588)4168195-2 |
title | Modern Methods in the Calculus of Variations L p spaces |
title_auth | Modern Methods in the Calculus of Variations L p spaces |
title_exact_search | Modern Methods in the Calculus of Variations L p spaces |
title_exact_search_txtP | Modern Methods in the Calculus of Variations L p spaces |
title_full | Modern Methods in the Calculus of Variations L p spaces Irene Fonseca ; Giovanni Leoni |
title_fullStr | Modern Methods in the Calculus of Variations L p spaces Irene Fonseca ; Giovanni Leoni |
title_full_unstemmed | Modern Methods in the Calculus of Variations L p spaces Irene Fonseca ; Giovanni Leoni |
title_short | Modern Methods in the Calculus of Variations |
title_sort | modern methods in the calculus of variations l p spaces |
title_sub | L p spaces |
topic | Calculus of variations Field theory (Physics) Nonlinear theories Variationsrechnung (DE-588)4062355-5 gnd Lp-Raum (DE-588)4168195-2 gnd |
topic_facet | Calculus of variations Field theory (Physics) Nonlinear theories Variationsrechnung Lp-Raum |
url | http://deposit.dnb.de/cgi-bin/dokserv?id=2820406&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015472154&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
work_keys_str_mv | AT fonsecairene modernmethodsinthecalculusofvariationslpspaces AT leonigiovanni modernmethodsinthecalculusofvariationslpspaces |