Uniqueness theorems for variational problems by the method of transformation groups:
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2004
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| Schriftenreihe: | Lecture Notes in Mathematics
1841 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIII, 152 S. graph. Darst. |
| ISBN: | 3540218394 |
Internformat
MARC
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| 100 | 1 | |a Reichel, Wolfgang |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Uniqueness theorems for variational problems by the method of transformation groups |c Wolfgang Reichel |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2004 | |
| 300 | |a XIII, 152 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture Notes in Mathematics |v 1841 | |
| 650 | 4 | |a Variationsrechnung - Transformationsgruppe - Eindeutigkeitssatz | |
| 650 | 4 | |a Calculus of variations | |
| 650 | 4 | |a Transformation groups | |
| 650 | 4 | |a Transformations (Mathematics) | |
| 650 | 0 | 7 | |a Eindeutigkeitssatz |0 (DE-588)4151250-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Transformationsgruppe |0 (DE-588)4127386-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Variationsrechnung |0 (DE-588)4062355-5 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
| _version_ | 1804132784097198080 |
|---|---|
| adam_text | Contents
1 Introduction 1
1.1 A convex functional 1
1.2 A functional with supercritical growth 2
1.3 Construction of the transformations 5
2 Uniqueness of critical points (I) 9
2.1 One parameter transformation groups 9
2.2 Variational sub symmetries and uniqueness of critical points .. 10
2.3 Uniqueness results for critical points of constrained functionals 12
2.4 First order variational integrals 12
2.5 Classical uniqueness results 13
2.5.1 Convex functionals 13
2.5.2 Uniqueness of a saddle point 14
2.5.3 Strict variational sub symmetry w.r.t. an affine subspace 17
2.5.4 Uniqueness of positive solutions for sublinear problems . 21
2.5.5 Simplicity of the first eigenvalue 24
3 Uniqueness of critical points (II) 27
3.1 Riemannian manifolds 27
3.2 The total space M xRk 30
3.3 One parameter transformation groups on M x I* 31
3.4 Action of transformation groups on functions 32
3.5 Rate of change of derivatives and volume forms 35
3.6 Rate of change of first order variational functionals 39
3.6.1 Partial derivatives of Lagrangians 39
3.6.2 The rate of change formula 41
3.6.3 Noether s formula and Pohozaev s identity 43
3.7 Admissible transformation groups 44
3.8 Rate of change formula for solutions 46
3.9 Variational sub symmetries 48
3.10 Uniqueness of critical points 50
XII Contents
3.11 Uniqueness of critical points for constrained functionals 53
3.11.1 Functional constraints 53
3.11.2 Pointwise constraints 54
3.12 Differentiability of the group orbits 56
4 Variational problems on Riemannian manifolds 59
4.1 Example manifolds and their representations 59
4.2 Supercritical boundary value problems 61
4.2.1 A weak substitute for the vector field x 62
4.2.2 Critical points of a free functional 63
4.2.3 Critical points of constrained functionals 66
4.2.4 Applications 68
4.3 Harmonic maps 71
4.4 Supercritical boundary value problems: revisited 73
4.4.1 A better substitute for the vector field x l 74
4.4.2 Conformal vector fields and conformal maps 76
4.4.3 Yamabe s equation 79
4.4.4 Yamabe s equation with boundary terms 80
4.4.5 Conformal vector fields on conformally flat manifolds . . 80
4.4.6 The bifurcation problem on W §n, Hn 83
4.4.7 The bifurcation problem on rotation surfaces 84
4.5 Harmonic maps into conformally flat manifolds 86
5 Scalar problems in Euclidean space 89
5.1 Extensions of Pohozaev s result to more general domains 89
5.1.1 Nonlinear Neumann boundary conditions 94
5.1.2 Extension to operators of g Laplacian type 97
5.1.3 Extension to the mean curvature operator 99
5.2 Uniqueness of non zero solutions 100
5.3 The subcritical case 106
5.4 Perturbations of conformally contractible domains 107
5.5 Uniqueness in the presence of radial symmetry 110
5.5.1 Radially symmetric problems on R ,S H 112
5.5.2 The radially symmetric q Laplarian 119
5.5.3 Partial radial symmetry 121
5.6 Notes on further results 123
6 Vector problems in Euclidean space 127
6.1 The Emden Fowler system 127
6.2 Boundary displacement problem in nonlinear elasticity 130
6.2.1 Uniqueness for the boundary displacement problem
(compressible case) 132
6.2.2 Uniqueness for the boundary displacement problem
(incompressible case) 134
6.3 A uniqueness result in dimension two 134
Contents XIII
6.4 H. White s uniqueness result for closed surfaces of prescribed
mean uirV;iture . .. ¦ • 137
A Frechet dlffcT«.ntiability 139
B Lipschitz proi erties of g£ and fie 141
References ....¦•.. ¦ • 145
Index • • 151
|
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| indexdate | 2024-07-09T19:58:19Z |
| institution | BVB |
| isbn | 3540218394 |
| language | English |
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| physical | XIII, 152 S. graph. Darst. |
| publishDate | 2004 |
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| publishDateSort | 2004 |
| publisher | Springer |
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| series | Lecture Notes in Mathematics |
| series2 | Lecture Notes in Mathematics |
| spelling | Reichel, Wolfgang Verfasser aut Uniqueness theorems for variational problems by the method of transformation groups Wolfgang Reichel Berlin [u.a.] Springer 2004 XIII, 152 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture Notes in Mathematics 1841 Variationsrechnung - Transformationsgruppe - Eindeutigkeitssatz Calculus of variations Transformation groups Transformations (Mathematics) Eindeutigkeitssatz (DE-588)4151250-9 gnd rswk-swf Transformationsgruppe (DE-588)4127386-2 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 s Transformationsgruppe (DE-588)4127386-2 s Eindeutigkeitssatz (DE-588)4151250-9 s DE-604 Lecture Notes in Mathematics 1841 (DE-604)BV000676446 1841 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012817091&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Reichel, Wolfgang Uniqueness theorems for variational problems by the method of transformation groups Lecture Notes in Mathematics Variationsrechnung - Transformationsgruppe - Eindeutigkeitssatz Calculus of variations Transformation groups Transformations (Mathematics) Eindeutigkeitssatz (DE-588)4151250-9 gnd Transformationsgruppe (DE-588)4127386-2 gnd Variationsrechnung (DE-588)4062355-5 gnd |
| subject_GND | (DE-588)4151250-9 (DE-588)4127386-2 (DE-588)4062355-5 |
| title | Uniqueness theorems for variational problems by the method of transformation groups |
| title_auth | Uniqueness theorems for variational problems by the method of transformation groups |
| title_exact_search | Uniqueness theorems for variational problems by the method of transformation groups |
| title_full | Uniqueness theorems for variational problems by the method of transformation groups Wolfgang Reichel |
| title_fullStr | Uniqueness theorems for variational problems by the method of transformation groups Wolfgang Reichel |
| title_full_unstemmed | Uniqueness theorems for variational problems by the method of transformation groups Wolfgang Reichel |
| title_short | Uniqueness theorems for variational problems by the method of transformation groups |
| title_sort | uniqueness theorems for variational problems by the method of transformation groups |
| topic | Variationsrechnung - Transformationsgruppe - Eindeutigkeitssatz Calculus of variations Transformation groups Transformations (Mathematics) Eindeutigkeitssatz (DE-588)4151250-9 gnd Transformationsgruppe (DE-588)4127386-2 gnd Variationsrechnung (DE-588)4062355-5 gnd |
| topic_facet | Variationsrechnung - Transformationsgruppe - Eindeutigkeitssatz Calculus of variations Transformation groups Transformations (Mathematics) Eindeutigkeitssatz Transformationsgruppe Variationsrechnung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012817091&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT reichelwolfgang uniquenesstheoremsforvariationalproblemsbythemethodoftransformationgroups |