Variational methods in mathematical physics: a unified approach
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Heidelberg ; New York ; London ; Paris ; Tokyo ; Hong Kong ; Barcelona ; Budapest
Springer
1992
|
| Series: | Texts and monographs in physics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Dt. Ausg. u.d.T.: Blanchard, Philippe: Direkte Methoden der Variationsrechnung |
| ISBN: | 3540161902 0387161902 |
Staff View
MARC
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Record in the Search Index
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CONTENTS
SOME REMARKS ON THE HISTORY AND OBJECTIVES
O
F
THE CALCULUS O
F
VARIATIONS 1
1. DIRECT METHODS O
F
THE CALCULUS O
F
VARIATIONS 15
1.1 THE FUNDAMENTAL THEOREM OF THE CALCULUS OF VARIATIONS .
.
.
. 15
1.2 APPLYING THE FUNDAMENTAL THEOREM IN BANACH SPACES 20
1.2.1 SEQUENTIALLY LOWER SEMICONTINUOUS FUNCTIONALS 22
1.3 MINIMISING SPECIAL CLASSES OF FUNCTIONS 25
1.3.1 QUADRATIC FUNCTIONALS 28
1.4 SOME REMARKS ON LINEAR OPTIMISATION 30
1.5 RITZ'S APPROXIMATION METHOD 31
2. DIFFERENTIAL CALCULUS IN BANACH SPACES 35
2.1 GENERAL REMARKS 35
2.2 THE FRECHET DERIVATIVE 36
2.2.1 HIGHER DERIVATIVES 43
2.2.2 SOME PROPERTIES OF FRECHET DERIVATIVES 44
2.3 THE GATEAUX DERIVATIVE 46
2.4 NTH VARIATION 49
2.5 THE ASSUMPTIONS OF THE FUNDAMENTAL THEOREM
OF VARIATIONAL CALCULUS 51
2.6 CONVEXITY OF / AND MONOTONICITY OF /
' 52
3. EXTREMA O
F
DIFFERENTIABLE FUNCTIONS 54
3.1 EXTREMA AND CRITICAL VALUES 54
3.2 NECESSARY CONDITIONS FOR AN EXTREMUM 55
3.3 SUFFICIENT CONDITIONS FOR AN EXTREMUM 60
4. CONSTRAINED MINIMISATION PROBLEMS
(METHOD O
F
LAGRANGE MULTIPLIERS) 63
4.1 GEOMETRICAL INTERPRETATION
OF CONSTRAINED MINIMISATION PROBLEMS 63
4.2 LJUSTERNIK'S THEOREMS 66
HTTP://D-NB.INFO/920209963
X CONTENTS
4.3 NECESSARY AND SUFFICIENT CONDITIONS
FOR EXTREMA SUBJECT TO CONSTRAINTS 72
4.4 A SPECIAL CASE 75
5. CLASSICAL VARIATIONAL PROBLEMS 77
5.1 GENERAL REMARKS 77
5.2 HAMILTON'S PRINCIPLE IN CLASSICAL MECHANICS 80
5.2.1 SYSTEMS WITH ONE DEGREE OF FREEDOM 81
5.2.2 SYSTEMS WITH SEVERAL DEGREES OF FREEDOM 95
5.2.3 AN EXAMPLE FROM CLASSICAL MECHANICS 105
5.3 SYMMETRIES AND CONSERVATION LAWS IN CLASSICAL MECHANICS . 107
5.3.1 HAMILTONIAN FORMULATION OF CLASSICAL MECHANICS 107
5.3.2 COORDINATE TRANSFORMATIONS AND INTEGRALS OF MOTION 109
5.4 THE BRACHYSTOCHRONE PROBLEM 113
5.5 SYSTEMS WITH INFINITELY MANY DEGREES OF FREEDOM: FIELD THEORY 116
5.5.1 HAMILTON'S PRINCIPLE IN LOCAL FIELD THEORY 117
5.5.2 EXAMPLES OF LOCAL CLASSICAL FIELD THEORIES 122
5.6 NOETHER'S THEOREM IN CLASSICAL FIELD THEORY 124
5.7 THE PRINCIPLE OF SYMMETRIC CRITICALITY 130
6. T
H
E VARIATIONAL APPROACH TO LINEAR BOUNDARY
AND EIGENVALUE PROBLEMS 142
6.1 THE SPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS.
COURANT'S CLASSICAL MINIMAX PRINCIPLE. PROJECTION THEOREM . 142
6.2 DIFFERENTIAL OPERATORS AND FORMS 148
6.3 THE THEOREM OF LAX-MILGRAM AND SOME GENERALISATIONS 152
6.4 THE SPECTRUM OF ELLIPTIC DIFFERENTIAL OPERATORS IN A BOUNDED
DOMAIN. SOME PROBLEMS FROM CLASSICAL POTENTIAL THEORY 156
6.5 VARIATIONAL SOLUTION OF PARABOLIC DIFFERENTIAL EQUATIONS.
THE HEAT CONDUCTION EQUATION. THE STOKES EQUATIONS 159
6.5.1 A GENERAL FRAMEWORK FOR THE VARIATIONAL SOLUTION
OF PARABOLIC PROBLEMS 161
6.5.2 THE HEAT CONDUCTION EQUATION 166
6.5.3 THE STOKES EQUATIONS IN HYDRODYNAMICS 167
7. NONLINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS
AND MONOTONIC OPERATORS 171
7.1 FORMS AND OPERATORS - BOUNDARY VALUE PROBLEMS 171
7.2 SURJECTIVITY OF COERCIVE MONOTONIC OPERATORS.
THEOREMS OF BROWDER AND MINTY 173
7.3 NONLINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS.
A VARIATIONAL SOLUTION 178
CONTENTS XI
8. NONLINEAR ELLIPTIC EIGENVALUE PROBLEMS 192
8.1 INTRODUCTION 192
8.2 DETERMINATION OF THE GROUND STATE
IN NONLINEAR ELLIPTIC EIGENVALUE PROBLEMS 195
8.2.1 ABSTRACT VERSIONS OF SOME EXISTENCE THEOREMS 195
8.2.2 DETERMINING THE GROUND STATE SOLUTION
FOR NONLINEAR ELLIPTIC EIGENVALUE PROBLEMS 203
8.3 LJUSTERNIK-SCHNIRELMAN THEORY FOR COMPACT MANIFOLDS 205
8.3.1 THE TOPOLOGICAL BASIS OF THE GENERALISED MINIMAX PRINCIPLE . 205
8.3.2 THE DEFORMATION THEOREM 207
8.3.3 THE LJUSTERNIK-SCHNIRELMAN CATEGORY AND THE GENUS OF A SET 210
8.3.4 MINIMAX CHARACTERISATION OF CRITICAL VALUES
OF LJUSTERNIK-SCHNIRELMAN 215
8.4 THE EXISTENCE OF INFINITELY MANY SOLUTIONS
OF NONLINEAR ELLIPTIC EIGENVALUE PROBLEMS 217
8.4.1 SPHERE-LIKE CONSTRAINTS 217
8.4.2 GALERKIN APPROXIMATION FOR NONLINEAR EIGENVALUE PROBLEMS
IN SEPARABLE BANACH SPACES 220
8.4.3 THE EXISTENCE OF INFINITELY MANY CRITICAL POINTS AS SOLUTIONS
OF ABSTRACT EIGENVALUE PROBLEMS IN SEPARABLE BANACH SPACES 225
8.4.4 THE EXISTENCE OF INFINITELY MANY SOLUTIONS
OF NONLINEAR EIGENVALUE PROBLEMS 228
9. SEMILINEAR ELLIPTIC DIFFERENTIAL EQUATIONS.
SOME RECENT RESULTS O
N GLOBAL SOLUTIONS 241
9.1 INTRODUCTION 241
9.2 TECHNICAL PRELIMINARIES 247
9.2.1 SOME FUNCTION SPACES AND THEIR PROPERTIES 247
9.2.2 SOME CONTINUITY RESULTS FOR NIEMYTSKI OPERATORS 252
9.2.3 SOME RESULTS ON CONCENTRATION OF FUNCTION SEQUENCES 256
9.2.4. A ONE-DIMENSIONAL VARIATIONAL PROBLEM 262
9.3 SOME PROPERTIES OF WEAK SOLUTIONS
OF SEMILINEAR ELLIPTIC EQUATIONS 266
9.3.1 REGULARITY OF WEAK SOLUTIONS 266
9.3.2 POHOZAEV'S IDENTITIES 278
9.4 BEST CONSTANT IN SOBOLEV INEQUALITY 283
9.5 THE LOCAL CASE WITH CRITICAL SOBOLEV EXPONENT 287
9.6 THE CONSTRAINED MINIMISATION METHOD UNDER SCALE COVARIANCE 294
9.7 EXISTENCE OF A MINIMISER I: SOME GENERAL RESULTS 302
9.7.1 SYMMETRIES 302
9.7.2. NECESSARY AND SUFFICIENT CONDITIONS 304
9.7.3 THE CONCENTRATION CONDITION 305
9.7.4 MINIMISING SUBSETS 308
XII CONTENTS
9.7.5 GROWTH RESTRICTIONS ON THE POTENTIAL 310
9.8 EXISTENCE OF A MINIMISER II: SOME EXAMPLES 312
9.8.1 SOME NON-TRANSLATION-INVARIANT CASES 313
9.8.2 SPHERICALLY SYMMETRIC CASES 316
9.8.3 THE TRANSLATION-INVARIANT CASE WITHOUT SPHERICAL SYMMETRY 319
9.9 NONLINEAR FIELD EQUATIONS IN TWO DIMENSIONS 322
9.9.1 SOME PROPERTIES OF NIEMYTSKI OPERATORS ON E
Q
323
9.9.2 SOLUTION OF SOME TWO-DIMENSIONAL VECTOR FIELD EQUATIONS . 326
9.10 CONCLUSION AND COMMENTS 332
9.10.1 CONCLUSION 332
9.10.2 GENERALISATIONS 334
9.10.3 COMMENTS 335
9.11 COMPLEMENTARY REMARKS 337
10. THOMAS-FERMI THEORY 340
10.1 GENERAL REMARKS 340
10.2 SOME RESULTS FROM THE THEORY OF L
P
SPACES (1 P OO) .
.
.
. 342
10.3 MINIMISATION OF THE THOMAS-FERMI ENERGY FUNCTIONAL 344
10.4 THOMAS-FERMI EQUATIONS AND THE MINIMISATION PROBLEM
FOR THE T
F
FUNCTIONAL 351
10.5 SOLUTION OF T
F
EQUATIONS FOR POTENTIALS
OF THE FORM Y(X) =
3
5
7
10.6 REMARKS ON RECENT DEVELOPMENTS IN THOMAS-FERMI
AND RELATED THEORIES 361
APPENDIX A
. BANACH SPACES 363
APPENDIX B
. CONTINUITY AND SEMICONTINUITY 371
APPENDIX C. COMPACTNESS IN BANACH SPACES 373
APPENDIX D
. THE SOBOLEV SPACES
W
M,P
(N)
380
D.L DEFINITION AND PROPERTIES 380
D.2 POINCARE'S INEQUALITY 385
D.3 CONTINUOUS EMBEDDINGS OF SOBOLEV SPACES 386
D.4 COMPACT EMBEDDINGS OF SOBOLEV SPACES 388
APPENDIX E 391
E.L BESSEL POTENTIALS 391
E.2 SOME PROPERTIES OF WEAKLY DIFFERENTIABLE FUNCTIONS 392
E.3 PROOF OF THEOREM 9.2.3 393
REFERENCES 395
INDEX O
F
NAMES 405
SUBJECT INDEX 407 |
| any_adam_object | 1 |
| author | Blanchard, Philippe Brüning, Erwin |
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| discipline | Mathematik |
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| indexdate | 2024-12-02T09:06:04Z |
| institution | BVB |
| isbn | 3540161902 0387161902 |
| language | English |
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| publishDate | 1992 |
| publishDateSearch | 1992 |
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| publisher | Springer |
| record_format | marc |
| series2 | Texts and monographs in physics |
| spelling | Blanchard, Philippe Verfasser aut Variational methods in mathematical physics a unified approach Philippe Blanchard ; Erwin Brüning Berlin ; Heidelberg ; New York ; London ; Paris ; Tokyo ; Hong Kong ; Barcelona ; Budapest Springer 1992 txt rdacontent n rdamedia nc rdacarrier Texts and monographs in physics Dt. Ausg. u.d.T.: Blanchard, Philippe: Direkte Methoden der Variationsrechnung Thomas-Fermi-Methode (DE-588)4185320-9 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Direkte Methode (DE-588)4705893-6 gnd rswk-swf Semilineare elliptische Differentialgleichung (DE-588)4225683-5 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 s Variationsrechnung (DE-588)4062355-5 s DE-604 Semilineare elliptische Differentialgleichung (DE-588)4225683-5 s Thomas-Fermi-Methode (DE-588)4185320-9 s Direkte Methode (DE-588)4705893-6 s Brüning, Erwin Verfasser (DE-588)124673198 aut DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019107277&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Blanchard, Philippe Brüning, Erwin Variational methods in mathematical physics a unified approach Thomas-Fermi-Methode (DE-588)4185320-9 gnd Mathematische Physik (DE-588)4037952-8 gnd Direkte Methode (DE-588)4705893-6 gnd Semilineare elliptische Differentialgleichung (DE-588)4225683-5 gnd Variationsrechnung (DE-588)4062355-5 gnd |
| subject_GND | (DE-588)4185320-9 (DE-588)4037952-8 (DE-588)4705893-6 (DE-588)4225683-5 (DE-588)4062355-5 |
| title | Variational methods in mathematical physics a unified approach |
| title_auth | Variational methods in mathematical physics a unified approach |
| title_exact_search | Variational methods in mathematical physics a unified approach |
| title_full | Variational methods in mathematical physics a unified approach Philippe Blanchard ; Erwin Brüning |
| title_fullStr | Variational methods in mathematical physics a unified approach Philippe Blanchard ; Erwin Brüning |
| title_full_unstemmed | Variational methods in mathematical physics a unified approach Philippe Blanchard ; Erwin Brüning |
| title_short | Variational methods in mathematical physics |
| title_sort | variational methods in mathematical physics a unified approach |
| title_sub | a unified approach |
| topic | Thomas-Fermi-Methode (DE-588)4185320-9 gnd Mathematische Physik (DE-588)4037952-8 gnd Direkte Methode (DE-588)4705893-6 gnd Semilineare elliptische Differentialgleichung (DE-588)4225683-5 gnd Variationsrechnung (DE-588)4062355-5 gnd |
| topic_facet | Thomas-Fermi-Methode Mathematische Physik Direkte Methode Semilineare elliptische Differentialgleichung Variationsrechnung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019107277&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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