An introduction to partial differential equations:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York u.a.
Springer
1993
|
| Series: | Texts in applied mathematics
13 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturangaben |
| Physical Description: | XIII, 428 S. graph. Darst. |
| ISBN: | 0387979522 3540979522 |
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| 245 | 1 | 0 | |a An introduction to partial differential equations |c Michael Renardy ; Robert C. Rogers |
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Record in the Search Index
| _version_ | 1823026259278430208 |
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| adam_text |
MICHAEL RENARDY ROBERT C. ROGERS
AN INTRODUCTION TO
PARTIAL
DIFFERENTIAL
EQUATIONS
WITH 21 ILLUSTRATIONS
SPRINGER-VERLAG
NEW
YORK BERLIN HEIDELBERG LONDON PARIS
TOKYO
HONG KONG BARCELONA BUDAPEST
CONTENTS
SERIES PREFACE V
PREFACE VII
1 INTRODUCTION 1
1.1 BASIC MATHEMATICAL QUESTIONS 1
1.1.1 EXISTENCE 1
1.1.2 MULTIPLICITY 4
1.1.3 STABILITY 5
1.1.4 LINEAR SYSTEMS OF ODES AND ASYMPTOTIC STABILITY . 7
1.1.5
WELL-POSED PROBLEMS 8
1.1.6 REPRESENTATION 9
1.1.7 ESTIMATION YY 10
1.1.8 SMOOTHNESS 12
1.2 ELEMENTARY PARTIAL DIFFERENTIAL EQUATIONS 14
1.2.1 LAPLACE'S EQUATION 14
1.2.2 THE HEAT EQUATION 23
1.2.3 THE WAVE EQUATION 29
2
CHARACTERISTICS
37
2.1 CLASSIFICATION AND CHARACTERISTICS 37
2.1.1 THE SYMBOL OF A DIFFERENTIAL EXPRESSION 38
2.1.2 SCALAR EQUATIONS OF SECOND ORDER 39
2.1.3 HIGHER-ORDER EQUATIONS AND SYSTEMS 42
2.1.4 NONLINEAR EQUATIONS 45
2.2 THE CAUCHY-KOVALEVSKAYA THEOREM 46
2.2.1 REAL ANALYTIC FUNCTIONS 47
2.2.2 MAJORIZATION 51
2.2.3 STATEMENT AND PROOF OF THE THEOREM 51
2.2.4 REDUCTION OF GENERAL SYSTEMS 54
2.2.5 A PDE WITHOUT SOLUTIONS 58
2.3 HOLMGREN'S UNIQUENESS THEOREM 61
2.3.1 AN OUTLINE OF THE MAIN IDEA 62
2.3.2 STATEMENT AND PROOF OF THE THEOREM 63
2.3.3 THE WEIERSTRASS APPROXIMATION THEOREM 65
X CONTENTS
3 CONSERVATION LAWS AND SHOCKS 69
3.1 SYSTEMS IN ONE SPACE DIMENSION 69
3.2 BASIC DEFMITIONS AND HYPOTHESES 72
3.3 BLOWUP OF SMOOTH SOLUTIONS 75
3.3.1 SINGLE CONSERVATION LAWS 75
3.3.2 THE
P
SYSTEM 78
3.4 WEAK SOLUTIONS 80
3.4.1 THE RANKINE-HUGONIOT CONDITION 81
3.4.2 MULTIPLICITY 83
3.4.3 THE LAX SHOCK CONDITION 85
3.5 RIEMANN PROBLEMS 87
3.5.1 SINGLE EQUATIONS 88
3.5.2 SYSTEMS 89
3.6 OTHER SELECTION CRITERIA 96
3.6.1 THE ENTROPY CONDITION 98
3.6.2 VISCOSITY SOLUTIONS 100
3.6.3 UNIQUENESS 102
4 MAXIMUM PRINCIPLES 105
4.1 MAXIMUM PRINCIPLES OF EUIPTIC PROBLEMS 105
4.1.1 THE WEAK MAXIMUM PRINCIPLE 105
4.1.2 THE STRONG MAXIMUM PRINCIPLE 107
4.1.3 A PRIORI BOUNDS 109
4.2 AN EXISTENCE PROOF FOR THE DIRICHLET PROBLEM 111
4.2.1 THE DIRICHLET PROBLEM ON A BALL 112
4.2.2 SUBHARMONIC FUNCTIONS 113
4.2.3 THE ARZELA-ASCOLI THEOREM 114
4.2.4 PROOF OF THEOREM 4.13 116
4.3 RADIAL SYMMETRY 118
4.3.1 TWO AUXILIARY LEMMAS 118
4.3.2 PROOF OF THE THEOREM 119
4.4 MAXIMUM PRINCIPLES FOR PARABOLIC EQUATIONS 121
4.4.1 THE WEAK MAXIMUM PRINCIPLE 121
4.4.2 THE STRONG MAXIMUM PRINCIPLE 122
5 DISTRIBUTIONS 127
5.1 TEST FUNCTIONS AND DISTRIBUTIONS 127
5.1.1 MOTIVATION 127
5.1.2 TEST FUNCTIONS 128
5.1.3 DISTRIBUTIONS 131
5.1.4 LOCALIZATION 133
5.1.5 CONVERGENCE OF DISTRIBUTIONS . 133
5.1.6 TEMPERED DISTRIBUTIONS 136
5.2 DERIVATIVES AND INTEGRALS 138
5.2.1 BASIC DEFMITIONS 138
CONTENTS XI
5.2.2 EXAMPLES 139
5.2.3 PRIMITIVES AND ORDINARY DIFFERENTIAL EQUATIONS . . . 142
5.3
CONVOLUTIONS AND FUNDAMENTAL SOLUTIONS 145
5.3.1 THE DIRECT PRODUCT OF DISTRIBUTIONS 145
5.3.2 CONVOLUTION OF DISTRIBUTIONS 147
5.3.3 FUNDAMENTAL SOLUTIONS 149
5.4 THE FOURIER TRANSFORM 154
5.4.1 FOURIER TRANSFORMS OF TEST FUNCTIONS 154
5.4.2 FOURIER TRANSFORMS OF TEMPERED DISTRIBUTIONS . . . 155
5.4.3
THE FUNDAMENTAL SOLUTION FOR THE WAVE EQUATION . 158
5.4.4
FOURIER TRANSFORM OF CONVOLUTIONS 160
5.4.5 LAPLACE TRANSFORMS 161
5.5 GREEN'S FUNCTIONS 165
5.5.1 BOUNDARY-VALUE PROBLEMS AND THEIR ADJOINTS .
.
.
. 165
5.5.2
GREEN'S FUNCTIONS FOR BOUNDARY-VALUE PROBLEMS . . 169
5.5.3
BOUNDARY INTEGRAL METHODS 172
6 FUNCTION SPACES 177
6.1 BANACH SPACES AND HUBERT SPACES 177
6.1.1 BANACH SPACES 177
6.1.2 EXAMPLES OF BANACH SPACES 179
6.1.3 HUBERT SPACES 183
6.2 BASES IN HUBERT SPACES 187
6.2.1 THE EXISTENCE OF A BASIS 187
6.2.2 FOURIER SERIES 191
6.2.3 ORTHOGONAL POLYNOMIALS 193
6.3 DUALITY AND WEAK CONVERGENCE 196
6.3.1 BOUNDED LINEAR MAPPINGS 196
6.3.2 EXAMPLES OF DUAL SPACES 198
6.3.3 THE HAHN-BANACH THEOREM 200
6.3.4 THE UNIFORM BOUNDEDNESS THEOREM 200
6.3.5 WEAK CONVERGENCE 202
6.4 SOBOLEV SPACES 205
6.4.1
BASIC DEFINITIONS 205
6.4.2 SOBOLEV SPACES AND FOURIER TRANSFORM 208
6.4.3 DENSITY THEOREMS 209
6.4.4 COORDINATE TRANSFORMATIONS AND SOBOLEV SPACES ON
MANIFOLDS
210
6.4.5 EXTENSION THEOREMS 212
6.4.6 THE SOBOLEV IMBEDDING THEOREM 214
6.4.7 COMPACTNESS PROPERTIES 215
6.4.8 THE TRACE THEOREM 218
6.4.9 NEGATIVE SOBOLEV SPACES AND DUALITY 222
XII CONTENTS
7 OPERATOR THEORY 227
7.1 BASIC DEFINITIONS AND EXAMPLES 228
7.1.1 OPERATORS 228
7.1.2 INVERSE OPERATORS 229
7.1.3 BOUNDED OPERATORS, EXTENSIONS 229
7.1.4 EXAMPLES OF OPERATORS 231
7.1.5 CLOSED OPERATORS 236
7.2 THE OPEN MAPPING THEOREM 240
7.3 SPECTRUM AND RESOLVENT 243
7.3.1 THE SPECTRA OF BOUNDED OPERATORS 245
7.4 SYMMETRY AND SELF-ADJOINTNESS YYYY YY 250
7.4.1 THE ADJOINT OPERATOR 250
7.4.2 THE HUBERT ADJOINT OPERATOR 251
7.4.3 ADJOINT OPERATORS AND SPECTRAL THEORY 255
7.4.4 PROOF OF THE BOUNDED INVERSE THEOREM FOR HUBERT
SPACES
256
7.5 COMPACT OPERATORS 259
7.5.1 THE SPECTRUM OF A COMPACT OPERATOR 264
7.6 STURM-LIOUVILLE BOUNDARY-VALUE PROBLEMS 270
7.7 THE FREDHOLM INDEX 279
8 LINEAR EUIPTIC EQUATIONS 283
8.1 DEFINITIONS 283
8.2 EXISTENCE AND UNIQUENOESS OF THE SOLUTIONS
OF
THE DIRICHLET PROBLEM , 287
8.2.1 THE DIRICHLET PROBLEM-TYPES OF SOLUTIONS 287
8.2.2 THE LAX-MILGRAM LEMMA 290
8.2.3 GAERDING'S INEQUALITY 292
8.2.4 EXISTENCE OF WEAK SOLUTIONS 298
8.3 EIGENFUNCTION EXPANSIONS 299
8.3.1 FREDHOLM THEORY 299
8.3.2 EIGENFUNCTION EXPANSIONS 302
8.4 GENERAL LINEAR EUIPTIC PROBLEMS 303
8.4.1 THE NEUMANN PROBLEM 303
8.4.2 THE COMPLEMENTING CONDITION FOR EUIPTIC SYSTEMS . 306
8.4.3
THE ADJOINT BOUNDARY VALUE PROBLEM 310
8.4.4 AGMON'S CONDITION AND COERCIVE PROBLEMS 315
8.5 INTERIOR REGULARITY 318
8.5.1 DIFFERENCE QUOTIENTS 320
8.5.2 SECOND-ORDER SCALAR EQUATIONS 322
8.6 BOUNDARY REGULARITY 324
CONTENTS XIII
9 NONLINEAR ELLIPTIC EQUATIONS 335
9.1 PERTURBATION RESULTS 335
9.1.1 THE BANACH CONTRACTION PRINCIPLE AND THE IMPLICIT
FUNCTION
THEOREM 335
9.1.2 APPLICATIONS TO ELLIPTIC PDES 339
9.2 NONLINEAR VARIATIONAL PROBLEMS 341
9.3 NONLINEAR OPERATOR THEORY METHODS 355
9.3.1 MAPPINGS ON FINITE-DIMENSIONAL SPACES 356
9.3.2 MONOTONE MAPPINGS ON BANACH SPACES 359
9.3.3 APPLICATIONS OF MONOTONE OPERATORS TO NONLINEAR
PDES
362
9.3.4 NEMYTSKII OPERATORS 366
9.3.5 PSEUDO-MONOTONE OPERATORS 367
9.3.6 APPLICATION TO PDES 370
10 ENERGY METHODS FOR EVOLUTION PROBLEMS 377
10.1 PARABOLIC EQUATIONS 377
10.1.1 BANACH SPACE VALUED FUNCTIONS AND DISTRIBUTIONS . 377
10.1.2
ABSTRACT PARABOLIC INITIAL-VALUE PROBLEMS 379
10.1.3 APPLICATIONS 382
10.1.4 REGULARITY OF SOLUTIONS 383
10.2 HYPERBOLIC EVOLUTION PROBLEMS 385
10.2.1 ABSTRACT SECOND ORDER EVOLUTION PROBLEMS 385
10.2.2
EXISTENCE OF A SOLUTION 386
10.2.3 UNIQUENESS OF THE SOLUTION 388
10.2.4 CONTINUITY OF THE SOLUTION 389
11 SEMIGROUP METHODS 393
11.1 SEMIGROUPS AND INFINITESIMAL GENERATORS 395
11.1.1 STRONGLY CONTINUOUS SEMIGROUPS 395
11.1.2 THE INFINITESIMAL GENERATOR 397
11.1.3 ABSTRACT ODES 399
11.2 THE HILLE-YOSIDA THEOREM 401
11.2.1 THE HILLE-YOSIDA THEOREM 401
11.2.2 THE LUMER-PHILLIPS THEOREM 404
11.3 APPLICATIONS TO PDES 407
11.3.1 SYMMETRIE HYPERBOLIC SYSTEMS 407
11.3.2 THE WAVE EQUATION 409
11.3.3 THE SCHROEDINGER EQUATION 410
11.4 ANALYTIC SEMIGROUPS 411
11.4.1 ANALYTIC SEMIGROUPS AND THEIR GENERATORS 411
11.4.2 FRACTIONAL POWERS 415
11.4.3 PERTURBATIONS OF ANALYTIC SEMIGROUPS 418
11.4.4 REGULARITY OF MILD SOLUTIONS 420
INDEX
425 |
| any_adam_object | 1 |
| author | Renardy, Michael 1955- Rogers, Robert C. |
| author_GND | (DE-588)142008257 |
| author_facet | Renardy, Michael 1955- Rogers, Robert C. |
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| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.353 |
| dewey-search | 515/.353 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| series2 | Texts in applied mathematics |
| spelling | Renardy, Michael 1955- Verfasser (DE-588)142008257 aut An introduction to partial differential equations Michael Renardy ; Robert C. Rogers New York u.a. Springer 1993 XIII, 428 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Texts in applied mathematics 13 Literaturangaben Distribution lc Equations aux dérivées partielles - Premier ordre Espace Sobolev lc Principe maximum lc Théorème Holmgren lc Équations aux dérivées partielles ram Équations différentielles ram Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Partielle Differentialgleichung (DE-588)4044779-0 s DE-604 Rogers, Robert C. Verfasser aut Texts in applied mathematics 13 (DE-604)BV002476038 13 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005401180&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Renardy, Michael 1955- Rogers, Robert C. An introduction to partial differential equations Texts in applied mathematics Distribution lc Equations aux dérivées partielles - Premier ordre Espace Sobolev lc Principe maximum lc Théorème Holmgren lc Équations aux dérivées partielles ram Équations différentielles ram Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4123623-3 |
| title | An introduction to partial differential equations |
| title_auth | An introduction to partial differential equations |
| title_exact_search | An introduction to partial differential equations |
| title_full | An introduction to partial differential equations Michael Renardy ; Robert C. Rogers |
| title_fullStr | An introduction to partial differential equations Michael Renardy ; Robert C. Rogers |
| title_full_unstemmed | An introduction to partial differential equations Michael Renardy ; Robert C. Rogers |
| title_short | An introduction to partial differential equations |
| title_sort | an introduction to partial differential equations |
| topic | Distribution lc Equations aux dérivées partielles - Premier ordre Espace Sobolev lc Principe maximum lc Théorème Holmgren lc Équations aux dérivées partielles ram Équations différentielles ram Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Distribution Equations aux dérivées partielles - Premier ordre Espace Sobolev Principe maximum Théorème Holmgren Équations aux dérivées partielles Équations différentielles Partielle Differentialgleichung Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005401180&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV002476038 |
| work_keys_str_mv | AT renardymichael anintroductiontopartialdifferentialequations AT rogersrobertc anintroductiontopartialdifferentialequations |