An Introduction to Partial Differential Equations

✍ Scribed by Michael Renardy, Robert C. Rogers

• Publisher: Springer
• Year: 2010
• Tongue: English
• Leaves: 449
• Series: Texts in Applied Mathematics
• Edition: 2nd ed.
• Category: Library

✦ Synopsis

Partial differential equations are fundamental to the modeling of natural phenomena. The desire to understand the solutions of these equations has always had a prominent place in the efforts of mathematicians and has inspired such diverse fields as complex function theory, functional analysis, and algebraic topology. This book, meant for a beginning graduate audience, provides a thorough introduction to partial differential equations.

✦ Table of Contents

Cover......Page 1
Preface......Page 8
Notes on the second edition......Page 9
Contents......Page 10
1 Introduction......Page 16
1.1 Basic Mathematical Questions......Page 17
1.2 Elementary Partial Differential Equations......Page 29
2.1 Classification and Characteristics......Page 51
2.2 The Cauchy-Kovalevskaya Theorem......Page 61
2.3 Holmgren's Uniqueness Theorem......Page 76
3 Conservation Laws and Shocks......Page 82
3.1 Systems in One Space Dimension......Page 83
3.2 Basic Definitions and Hypotheses......Page 85
3.3 Blowup of Smooth Solutions......Page 88
3.4 Weak Solutions......Page 92
3.5 Riemann Problems......Page 99
3.6 Other Selection Criteria......Page 109
4 Maximum Principles......Page 116
4.1 Maximum Principles of Elliptic Problems......Page 117
4.2 An Existence Proof for the Dirichlet Problem......Page 122
4.3 Radial Symmetry......Page 129
4.4 Maximum Principles for Parabolic Equations......Page 132
5.1 Test Functions and Distributions......Page 137
5.2 Derivatives and Integrals......Page 150
5.3 Convolutions and Fundamental Solutions......Page 158
5.4 The Fourier Transform......Page 166
5.5 Green's Functions......Page 178
6.1 Banach Spaces and Hilbert Spaces......Page 189
6.2 Bases in Hilbert Spaces......Page 199
6.3 Duality and Weak Convergence......Page 209
7 Sobolev Spaces......Page 218
7.1 Basic Definitions......Page 219
7.2 Characterizations of Sobolev Spaces......Page 222
7.3 Negative Sobolev Spaces and Duality......Page 233
7.4 Technical Results......Page 235
8 Operator Theory......Page 243
8.1 Basic Definitions and Examples......Page 244
8.2 The Open Mapping Theorem......Page 256
8.3 Spectrum and Resolvent......Page 259
8.4 Symmetry and Self-adjointness......Page 266
8.5 Compact Operators......Page 274
8.6 Sturm-Liouville Boundary-Value Problems......Page 286
8.7 The Fredholm Index......Page 294
9.1 Definitions......Page 298
9.2 Existence and Uniqueness of Solutions of the Dirichlet Problem......Page 302
9.3 Eigenfunction Expansions......Page 315
9.4 General Linear Elliptic Problems......Page 318
9.5 Interior Regularity......Page 333
9.6 Boundary Regularity......Page 339
10.1 Perturbation Results......Page 350
10.2 Nonlinear Variational Problems......Page 357
10.3 Nonlinear Operator Theory Methods......Page 374
11.1 Parabolic Equations......Page 395
11.2 Hyperbolic Evolution Problems......Page 403
12 Semigroup Methods......Page 410
12.1 Semigroups and Infinitesimal Generators......Page 412
12.2 The Hille-Yosida Theorem......Page 418
12.3 Applications to PDEs......Page 423
12.4 Analytic Semigroups......Page 428
A. 1 Elementary Texts......Page 441
A.3 Specialized or Advanced Texts......Page 442
A.5 Other References......Page 444
Index......Page 446