An introduction to partial differential equations:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2004
|
| Edition: | 2. ed. |
| Series: | Texts in applied mathematics
13 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 434 S. graph. Darst. |
| ISBN: | 0387004440 9780387004440 9781441918208 |
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MARC
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| 100 | 1 | |a Renardy, Michael |d 1955- |e Verfasser |0 (DE-588)142008257 |4 aut | |
| 245 | 1 | 0 | |a An introduction to partial differential equations |c Michael Renardy ; Robert C. Rogers |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 2004 | |
| 300 | |a XIV, 434 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Texts in applied mathematics |v 13 | |
| 650 | 7 | |a Análise numérica |2 larpcal | |
| 650 | 7 | |a Equações diferenciais parciais |2 larpcal | |
| 650 | 7 | |a Partiële differentiaalvergelijkingen |2 gtt | |
| 650 | 4 | |a Differential equations, Partial | |
| 650 | 0 | 7 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |2 gnd |9 rswk-swf |
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Record in the Search Index
| _version_ | 1805072797097000960 |
|---|---|
| adam_text |
Contents
Series
Preface v
Preface
vii
1
Introduction
1
1.1
Basic
Mathematical
Questions
. 2
1.1.1
Existence
. 2
1.1.2
Multiplicity
. 4
1.1.3
Stability.
6
1.1.4
Linear
Systems
of ODEs and Asymptotic Stability
7
1.1.5
Well-Posed Problems
. 8
1.1.6
Representations
.
g
1.1.7
Estimation
.
Ю
1.1.8
Smoothness
. 12
1.2
Elementary Partial Differential Equations
. 14
1.2.1
Laplace's Equation
. 15
1.2.2
The Heat Equation
. 24
1.2.3
The Wave Equation
. 30
2
Characteristics 3g
2.1
Classification and Characteristics
. 3ß
2.1.1
The Symbol of a Differential Expression
. 37
2.1.2
Scalar Equations of Second Order
.
З8
2.1.3
Higher-Order Equations and Systems
. 41
χ
Contents
2.1.4
Nonlinear Equations
. 44
2.2
The Cauchy-Kovalevskaya Theorem
. 46
2.2.1
Real Analytic Functions
. 46
2.2.2
Majorization
. 50
2.2.3
Statement and Proof of the Theorem
. 51
2.2.4
Reduction of General Systems
. 53
2.2.5
A PDE without Solutions
. 57
2.3
Holmgren's Uniqueness Theorem
. 61
2.3.1
An Outline of the Main Idea
. 61
2.3.2
Statement and Proof of the Theorem
. 62
2.3.3
The
Weierstraß
Approximation Theorem
. 64
3
Conservation Laws and Shocks
67
3.1
Systems in One Space Dimension
. 68
3.2
Basic Definitions and Hypotheses
. 70
3.3
Blowup of Smooth Solutions
. 73
3.3.1
Single Conservation Laws
. 73
3.3.2
The
ρ
System
. 76
3.4
Weak Solutions
. 77
3.4.1
The Rankine-Hugoniot Condition
. 79
3.4.2
Multiplicity
. 81
3.4.3
The Lax Shock Condition
. 83
3.5
Riemann Problems
. 84
3.5.1
Single Equations
. 85
3.5.2
Systems
. 86
3.6
Other Selection Criteria
. 94
3.6.1
The Entropy Condition
. 94
3.6.2
Viscosity Solutions
. 97
3.6.3
Uniqueness
. 99
4
Maximum Principles
101
4.1
Maximum Principles of Elliptic Problems
. 102
4.1.1
The Weak Maximum Principle
. 102
4.1.2
The Strong Maximum Principle
. 103
4.1.3
A Priori Bounds
. 105
4.2
An Existence Proof for the Dirichlet Problem
. 107
4.2.1
The Dirichlet Problem on a Ball
. 108
4.2.2
Subharmonic Functions
. 109
4.2.3
The Arzela-Ascoli Theorem
. 110
4.2.4
Proof of Theorem
4.13. 112
4.3
Radial Symmetry
. 114
4.3.1
Two Auxiliary Lemmas
. 114
4.3.2
Proof of the Theorem
. 115
4.4
Maximum Principles for Parabolic Equations
. 117
4.4.1
The Weak Maximum Principle
. 117
Contents xi
4.4.2
The Strong Maximum Principle
. 118
Distributions
122
5.1
Test Functions and Distributions
. 122
5.1.1
Motivation
. 122
5.1.2
Test Functions
. 124
5.1.3
Distributions
. 126
5.1.4
Localization and Regularization
. 129
5.1.5
Convergence of Distributions
. 130
5.1.6
Tempered Distributions
. 132
5.2
Derivatives and Integrals
. 135
5.2.1
Basic Definitions
. 135
5.2.2
Examples
. 136
5.2.3
Primitives and Ordinary Differential Equations
. 140
5.3
Convolutions and Fundamental Solutions
. 143
5.3.1
The Direct Product of Distributions
. 143
5.3.2
Convolution of Distributions
. 145
5.3.3
Fundamental Solutions
. 147
5.4
The Fourier Transform
. 151
5.4.1
Fourier Transforms of Test Functions
. 151
5.4.2
Fourier Transforms of Tempered Distributions
. . 153
5.4.3
The Fundamental Solution for the Wave Equation
156
5.4.4
Fourier Transform of Convolutions
. 158
5.4.5
Laplace Transforms
. 159
5.5
Green's Functions
. 163
5.5.1
Boundary-Value Problems and their
Adjoints
. . 163
5.5.2
Green's Functions for Boundary-Value Problems
. 167
5.5.3
Boundary Integral Methods
. 170
Function Spaces
174
6.1
Banach Spaces and Hubert Spaces
. 174
6.1.1
Banach Spaces
. 174
6.1.2
Examples of Banach Spaces
. 177
6.1.3
Hubert Spaces
. 180
6.2
Bases in Hubert Spaces
. 184
6.2.1
The Existence of a Basis
. 184
6.2.2
Fourier Series
. 188
6.2.3
Orthogonal Polynomials
. 190
6.3
Duality and Weak Convergence
. 194
6.3.1
Bounded Linear Mappings
. 194
6.3.2
Examples of Dual Spaces
. 195
6.3.3
The Hahn-Banach Theorem
. 197
6.3.4
The Uniform Boundedness Theorem
. 198
6.3.5
Weak Convergence
. 199
xii Contents
7 Sobolev Spaces 203
7.1 Basic
Definitions
. 204
7.2
Characterizations of
Sobolev Spaces. 207
7.2.1
Some Comments on the Domain
Ω
. 207
7.2.2
Sobolev Spaces and Fourier Transform
. 208
7.2.3
The Sobolev Imbedding Theorem
. 209
7.2.4
Compactness Properties
. 210
7.2.5
The Trace Theorem
. 214
7-3
Negative Sobolev Spaces and Duality
. 218
7.4
Technical Results
. 220
7.4.1
Density Theorems
. 220
7.4.2
Coordinate Transformations and Sobolev Spaces on
Manifolds
. 221
7.4.3
Extension Theorems
. 223
7.4.4
Problems
. 225
8
Operator Theory
228
8.1
Basic Definitions and Examples
. 229
8.1.1
Operators
. 229
8.1.2
Inverse Operators
. 230
8.1.3
Bounded Operators, Extensions
. 230
8.1.4
Examples of Operators
. 232
8.1.5
Closed Operators
. 237
8.2
The Open Mapping Theorem
. 241
8.3
Spectrum and Resolvent
. 244
8.3.1
The Spectra of Bounded Operators
. 246
8.4
Symmetry and Self-adjointness
. 251
8.4.1
The Adjoint Operator
. 251
8.4.2
The Hubert Adjoint Operator
. 253
8.4.3
Adjoint Operators and Spectral Theory
. 256
8.4.4
Proof of the Bounded Inverse Theorem for Hubert
Spaces
. 257
8.5
Compact Operators
. 259
8.5.1
The Spectrum of a Compact Operator
. 265
8.6
Sturm-Liouville Boundary-Value Problems
. 271
8.7
The
Fredholm
Index
. 279
9
Linear Elliptic Equations
283
9.1
Definitions
. 283
9.2
Existence and Uniqueness of Solutions of the Dirichlet
Problem
. 287
9.2.1
The Dirichlet Problem—Types of Solutions
. 287
9.2.2
The Lax-Milgram Lemma
. 290
9.2.3
Garding's Inequality
. 292
9.2.4
Existence of Weak Solutions
. 298
Contents xiii
9.3
Eigenfunction Expansions
. 300
9.3.1
Fredholm
Theory
. 300
9.3.2
Eigenfunction Expansions
. 302
9.4
General Linear Elliptic Problems
. 303
9.4.1
The Neumann Problem
. 304
9.4.2
The Complementing Condition for Elliptic Systems
306
9.4.3
The Adjoint Boundary-Value Problem
. 311
9.4.4
Agmon's Condition and Coercive Problems
. 315
9.5
Interior Regularity
. 318
9.5.1
Difference Quotients
. 321
9.5.2
Second-Order Scalar Equations
. 323
9.6
Boundary Regularity
. 324
10
Nonlinear Elliptic Equations
335
10.1
Perturbation Results
. 335
10.1.1
The Banach Contraction Principle and the Implicit
Function Theorem
. 336
10.1.2
Applications to Elliptic PDEs
. 339
10.2
Nonlinear Variational Problems
. 342
10.2.1
Convex problems
. 342
10.2.2
Nonconvex Problems
. 355
10.3
Nonlinear Operator Theory Methods
. 359
10.3.1
Mappings on Finite-Dimensional Spaces
. 359
10.3.2
Monotone Mappings on Banach Spaces
. 363
10.3.3
Applications of Monotone Operators to Nonlinear
PDEs
. 366
10.3.4
Nemytskii Operators
. 370
10.3.5
Pseudo-monotone Operators
. 371
10.3.6
Application to PDEs
. 374
11
Energy Methods for Evolution Problems
380
11.1
Parabolic Equations
. 380
11.1.1
Banach Space Valued Functions and Distributions
380
11.1.2
Abstract Parabolic Initial-Value Problems
. 382
11.1.3
Applications
. 385
11.1.4
Regularity of Solutions
. 386
11.2
Hyperbolic Evolution Problems
. 388
11.2.1
Abstract Second-Order Evolution Problems
. 388
11.2.2
Existence of a Solution
. 389
11.2.3
Uniqueness of the Solution
. 391
11.2.4
Continuity of the Solution
. 392
12
Semigroup Methods
395
12.1
Semigroups and Infinitesimal Generators
. 397
12.1.1
Strongly Continuous Semigroups
. 397
xiv Contents
12.1.2 The Infinitesimal Generator. 399
12.1.3
Abstract ODEs
. 401
12.2 The Hille-Yosida Theorem. 403
12.2.1 The Hille-Yosida Theorem. 403
12.2.2 The Lumer-Phillips Theorem . 406
12.3 Applications
to PDEs
. 408
12.3.1 Symmetrie
Hyperbolic
Systems. 408
12.3.2
The Wave Equation.
410
12.3.3 The Schrödinger
Equation
. 411
12.4
Analytic Semigroups
. 413
12.4.1
Analytic Semigroups and Their Generators
. 413
12.4.2
Fractional Powers
. 416
12.4.3
Perturbations of Analytic Semigroups
. 419
12.4.4
Regularity of Mild Solutions
. 422
A References
426
A.I Elementary Texts
. 426
A.2 Basic Graduate Texts
. 427
A.3 Specialized or Advanced Texts
. 427
A.4
Multivolume
or Encyclopedic Works
. 429
A.5 Other References
. 429
Index
431 |
| any_adam_object | 1 |
| author | Renardy, Michael 1955- Rogers, Robert C. |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| language | English |
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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 2. ed. New York [u.a.] Springer 2004 XIV, 434 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Texts in applied mathematics 13 Análise numérica larpcal Equações diferenciais parciais larpcal Partiële differentiaalvergelijkingen gtt Differential equations, Partial 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 (DE-588)128614854 aut Texts in applied mathematics 13 (DE-604)BV002476038 13 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010250160&sequence=000002&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 Análise numérica larpcal Equações diferenciais parciais larpcal Partiële differentiaalvergelijkingen gtt Differential equations, Partial 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 | Análise numérica larpcal Equações diferenciais parciais larpcal Partiële differentiaalvergelijkingen gtt Differential equations, Partial Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Análise numérica Equações diferenciais parciais Partiële differentiaalvergelijkingen Differential equations, Partial Partielle Differentialgleichung Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010250160&sequence=000002&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 |