Introduction to the calculus of variations:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English French |
| Published: |
London
Imperial College Press
2015
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| Edition: | 3. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (pages 301-308) and index |
| Physical Description: | X, 311 S. |
| ISBN: | 9781783265510 1783265515 9781783265527 1783265523 |
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| 240 | 1 | 0 | |a Introduction au calcul des variations |
| 245 | 1 | 0 | |a Introduction to the calculus of variations |c Bernard Dacorogna |
| 250 | |a 3. ed. | ||
| 264 | 1 | |a London |b Imperial College Press |c 2015 | |
| 300 | |a X, 311 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 500 | |a Includes bibliographical references (pages 301-308) and index | ||
| 650 | 4 | |a Calculus of variations | |
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Record in the Search Index
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| adam_text | Titel: Introduction to the calculus of variations
Autor: Dacorogna, Bernard
Jahr: 2015
Contents
Preface to the Third English Edition ix
0 Introduction 1
0.1 Brief historical comments............................................1
0.2 Model problem and some examples..................................3
0.3 Presentation of the content of the monograph......................7
1 Preliminaries 13
1.1 Introduction............................................................13
1.2 Continuous and Holder continuous functions........................14
1.2.1 Space of continuous functions and notations................14
1.2.2 Holder continuous functions..................................16
1.2.3 Exercises......................................................18
1.3 Lp spaces..............................................................20
1.3.1 Basic definitions and properties..............................20
1.3.2 Weak convergence and Riemann-Lebcsgue theorem .... 22
1.3.3 The fundamental lemma of the calculus of variations ... 26
1.3.4 Exercises......................................................27
1.4 Sobolev spaces........................................................30
1.4.1 Definitions and first properties..............................30
1.4.2 Some further properties......................................34
1.4.3 Imbeddings and compact imbeddings........................39
1.4.4 Poincare inequality............................................43
1.4.5 Exercises......................................................45
1.5 Convex analysis........................................................49
1.5.1 Exercises......................................................52
2 Classical methods 55
2.1 Introduction............................................................55
2.2 Euler-Lagrange equation..............................................57
v
vi Contents
2.2.1 The main theorem............................................57
2.2.2 Some important special cases................................50
2.2.3 Lavrentiev phenomenon ..............................57
2.2.4 Exercises ......................................................68
2.3 Second form of the Euler-Lagrange equation........................70
2.3.1 Exercises ......................................................73
2.4 Hamiltonian formulation..............................................73
2.4.1 A technical lemma............................................74
2.4.2 The main theorem and some examples......................77
2.4.3 Exercises......................................................80
2.5 Hamilton-Jacobi equation............................................81
2.5.1 Exercises ......................................................85
2.G Fields theories ........................................................8G
2.6.1 A simple case..................................................8G
2.G.2 Exact fields and Hilbert theorem............................88
2.6.3 Exercises ......................................................91
3 Direct methods: existence 93
3.1 Introduction............................................................93
3.2 The model case: Dirichlct integral..................................95
3.2.1 Exercises ......................................................98
3.3 A general existence theorem..........................................99
3.3.1 The main theorem and some examples......................99
3.3.2 Proof of the main theorem..................103
3.3.3 Exercises...........................106
3.4 Euler-Lagrange equation.......................107
3.4.1 The main theorem and its proof..............107
3.4.2 Some examples........................110
3.4.3 Exercises...........................113
3.5 The vectorial case ..........................114
3.5.1 The main theorem......................114
3.5.2 Weak continuity of determinants..............117
3.5.3 Proof of the main theorem..................120
3.5.4 Exercises...........................122
3.G Relaxation theory...........................124
3.6.1 The relaxation theorem...................124
3.6.2 Some examples........................126
3.6.3 Exercises...........................127
Contents vii
4 Direct methods: regularity 131
4.1 Introduction..............................131
4.2 The one dimensional case......................132
4.2.1 A simple case.........................133
4.2.2 Two general theorems....................134
4.2.3 Exercises...........................138
4.3 The difference quotient method: interior regularity........140
4.3.1 Preliminaries.........................140
4.3.2 The Dirichlet integral....................141
4.3.3 A more general case.....................144
4.3.4 Exercises...........................149
4.4 The difference quotient method: boundary regularity.......150
4.4.1 Exercise............................154
4.5 Higher regularity for the Dirichlet integral.............154
4.5.1 Exercises ...........................157
4.6 Weyl lemma..............................159
4.6.1 Exercise............................161
4.7 Some general results.........................162
4.7.1 Exercises...........................164
5 Minimal surfaces 165
5.1 Introduction..............................165
5.2 Generalities about surfaces .....................168
5.2.1 Main definitions and some examples............168
5.2.2 Minimal surfaces and surfaces of minimal area......173
5.2.3 Isothermal coordinates....................175
5.2.4 Exercises...........................177
5.3 The Douglas-Courant-Tonelli method................178
5.3.1 Exercise............................184
5.4 Regularity, uniqueness and non-uniqueness.............184
5.5 Nonparametric minimal surfaces ..................185
5.5.1 General remarks.......................185
5.5.2 Korn-Miintz theorem.....................187
5.5.3 Exercise............................191
6 Isoperimetric inequality 193
6.1 Introduction..............................193
6.2 The case of dimension 2.......................194
6.2.1 Wirtinger inequality.....................194
6.2.2 The isoperimetric inequality.................198
6.2.3 Exercises...........................200
6.3 The case of dimension n.......................201
viii Contents
6.3.1 Minkowski-Steiner formula .................201
6.3.2 Brunn-Minkowski theorem..................203
6.3.3 Proof of the isoperimetric inequality............204
6.3.4 Proof of Brunn-Minkowski theorem.............205
6.3.5 Exercises...........................208
7 Solutions to the exercises 211
7.1 Chapter 1. Preliminaries.......................211
7.1.1 Continuous and Holder continuous functions .......211
7.1.2 Lp spaces...........................215
7.1.3 Sobolev spaces........................222
7.1.4 Convex analysis........................234
7.2 Chapter 2. Classical methods....................242
7.2.1 Euler-Lagrange equation...................242
7.2.2 Second form of the Euler-Lagrange equation........249
7.2.3 Hamiltonian formulation...................250
7.2.4 Hamilton-Jacobi equation..................251
7.2.5 Fields theories........................254
7.3 Chapter 3. Direct methods: existence ...............256
7.3.1 The model case: Dirichlet integral.............256
7.3.2 A general existence theorem.................259
7.3.3 Euler-Lagrange equation...................261
7.3.4 The vectorial case......................263
7.3.5 Relaxation theory ......................271
7.4 Chapter 4. Direct methods: regularity...............274
7.4.1 The one dimensional case..................274
7.4.2 The difference quotient method: interior regularity .... 278
7.4.3 The difference quotient method: boundary regularity . . . 280
7.4.4 Higher regularity for the Dirichlet integral.........281
7.4.5 Weyl lemma..........................284
7.4.6 Some general results.....................286
7.5 Chapter 5. Minimal surfaces.....................289
7.5.1 Generalities about surfaces.................289
7.5.2 The Douglas-Courant-Tonelli method ...........292
7.5.3 Nonparametric minimal surfaces..............293
7.6 Chapter 6. Isoperimetric inequality.................293
7.6.1 The case of dimension 2...................293
7.6.2 The case of dimension n...................297
Bibliography 301
Index 309
|
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| spelling | Dacorogna, Bernard 1953- Verfasser (DE-588)133791920 aut Introduction au calcul des variations Introduction to the calculus of variations Bernard Dacorogna 3. ed. London Imperial College Press 2015 X, 311 S. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references (pages 301-308) and index Calculus of variations Mathematical analysis Variationsrechnung (DE-588)4062355-5 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Variationsrechnung (DE-588)4062355-5 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028038380&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dacorogna, Bernard 1953- Introduction to the calculus of variations Calculus of variations Mathematical analysis Variationsrechnung (DE-588)4062355-5 gnd |
| subject_GND | (DE-588)4062355-5 (DE-588)4123623-3 |
| title | Introduction to the calculus of variations |
| title_alt | Introduction au calcul des variations |
| title_auth | Introduction to the calculus of variations |
| title_exact_search | Introduction to the calculus of variations |
| title_full | Introduction to the calculus of variations Bernard Dacorogna |
| title_fullStr | Introduction to the calculus of variations Bernard Dacorogna |
| title_full_unstemmed | Introduction to the calculus of variations Bernard Dacorogna |
| title_short | Introduction to the calculus of variations |
| title_sort | introduction to the calculus of variations |
| topic | Calculus of variations Mathematical analysis Variationsrechnung (DE-588)4062355-5 gnd |
| topic_facet | Calculus of variations Mathematical analysis Variationsrechnung Lehrbuch |
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