Partial Differential Equations III (eBook)

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

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