Partial Differential Equations III (eBook)

Nonlinear Equations

Michael E. Taylor (Autor)

eBook Download: PDF

2010 | 2nd ed. 2011
XXII, 715 Seiten
Springer New York (Verlag)
978-1-4419-7049-7 (ISBN)

Infos

The third of three volumes on partial differential equations, this is devoted to nonlinear PDE. It treats a number of equations of classical continuum mechanics, including relativistic versions, as well as various equations arising in differential geometry, such as in the study of minimal surfaces, isometric imbedding, conformal deformation, harmonic maps, and prescribed Gauss curvature. In addition, some nonlinear diffusion problems are studied. It also introduces such analytical tools as the theory of L Sobolev spaces, H lder spaces, Hardy spaces, and Morrey spaces, and also a development of Calderon-Zygmund theory and paradifferential operator calculus. The book is aimed at graduate students in mathematics, and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis and complex analysis

Michael E. Taylor is a Professor at University of North Carolina in the Department of Mathematics.

Inhalt

Contents 8
Contents of Volumes I and II 12
Preface 14
13 Function Space and Operator Theory for Nonlinear Analysis 24
1 Lp-Sobolev spaces 25
2 Sobolev imbedding theorems 27
3 Gagliardo–Nirenberg–Moser estimates 31
4 Trudinger's inequalities 37
5 Singular integral operators on Lp 40
6 The spaces Hs,p 47
7 Lp-spectral theory of the Laplace operator 54
8 Hölder spaces and Zygmund spaces 63
9 Pseudodifferential operators with nonregular symbols 73
10 Paradifferential operators 83
11 Young measures and fuzzy functions 97
12 Hardy spaces 109
A Variations on complex interpolation 119
References 125
14 Nonlinear Elliptic Equations 128
1 A class of semilinear equations 130
2 Surfaces with negative curvature 142
3 Local solvability of nonlinear elliptic equations 150
4 Elliptic regularity I (interior estimates) 158
5 Isometric imbedding of Riemannian manifolds 170
6 Minimal surfaces 175
6B Second variation of area 191
7 The minimal surface equation 199
8 Elliptic regularity II (boundary estimates) 208
9 Elliptic regularity III (DeGiorgi–Nash–Moser theory) 219
10 The Dirichlet problem for quasi-linear elliptic equations 231
11 Direct methods in the calculus of variations 245
12 Quasi-linear elliptic systems 252
12B Further results on quasi-linear systems 267
13 Elliptic regularity IV (Krylov–Safonov estimates) 281
14 Regularity for a class of completely nonlinear equations 296
15 Monge–Ampere equations 305
16 Elliptic equations in two variables 317
A Morrey spaces 322
B Leray–Schauder fixed-point theorems 325
References 327
15 Nonlinear Parabolic Equations 335
1 Semilinear parabolic equations 336
2 Applications to harmonic maps 347
3 Semilinear equations on regions with boundary 354
4 Reaction-diffusion equations 357
5 A nonlinear Trotter product formula 375
6 The Stefan problem 384
7 Quasi-linear parabolic equations I 398
8 Quasi-linear parabolic equations II (sharper estimates) 409
9 Quasi-linear parabolic equations III (Nash–Moser estimates) 418
References 429
16 Nonlinear Hyperbolic Equations 434
1 Quasi-linear, symmetric hyperbolic systems 435
2 Symmetrizable hyperbolic systems 446
3 Second-order and higher-order hyperbolic systems 453
4 Equations in the complex domain and the Cauchy–Kowalewsky theorem 466
5 Compressible fluid motion 469
6 Weak solutions to scalar conservation laws the viscosity method
7 Systems of conservation laws in one space variable Riemann problems
8 Entropy-flux pairs and Riemann invariants 519
9 Global weak solutions of some 2x2 systems 530
10 Vibrating strings revisited 538
References 545
17 Euler and Navier–Stokes Equations for Incompressible Fluids 551
1 Euler's equations for ideal incompressible fluid flow 552
2 Existence of solutions to the Euler equations 562
3 Euler flows on bounded regions 573
4 Navier–Stokes equations 581
5 Viscous flows on bounded regions 595
6 Vanishing viscosity limits 606
7 From velocity field convergence to flow convergence 619
A Regularity for the Stokes system on bounded domains 625
References 630
18 Einstein's Equations 635
1 The gravitational field equations 636
2 Spherically symmetric spacetimes and the Schwarzschild solution 646
3 Stationary and static spacetimes 659
4 Orbits in Schwarzschild spacetime 669
5 Coupled Maxwell–Einstein equations 676
6 Relativistic fluids 679
7 Gravitational collapse 690
8 The initial-value problem 697
9 Geometry of initial surfaces 707
10 Time slices and their evolution 719
References 725
Index 730

Produkt-Details

• Erscheint lt. Verlag: 2.11.2010
• Reihe/Serie: Applied Mathematical Sciences
• Reihe/Serie: Applied Mathematical Sciences
• Zusatzinfo: XXII, 715 p.
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• Schlagworte: Einstein's equations • navier-stokes equations • Nonlinear elliptic equations • Nonlinear hyperbolic equations • Partial differential equations
• ISBN-10: 1-4419-7049-5 / 1441970495
• ISBN-13: 978-1-4419-7049-7 / 9781441970497