13 Function Space and Operator Theory for Nonlinear Analysis 1
1 Lp-Sobolev spaces 2
2 Sobolev imbedding theorems 4
3 Gagliardo-Nirenberg-Moser estimates 8
4 Trudinger's inequalities 14
5 Singular integral operators on Lp 17
6 The spaces Hs,p 24
7 Lp-spectral theory of the Laplace operator 31
8 H？lder spaces and Zygmund spaces 40
9 Pseudodifferential operators with nonregular symbols 50
10 Paradifierential operators 60
11 Young measures and fuzzy functions 74
12 Hardy spaces 86
A Variations on complex interpolation 96
References 102
14 Nonlinear Elliptic Equations 105
1 A class of semilinear equations 107
2 Surfaces with negative curvature 119
3 Local solvability of nonlinear elliptic equations 127
4 Elliptic regularity Ⅰ（interior estimates） 135
5 Isometric imbedding of Riemannian manifolds 147
6 Minimal surfaces 152
6B Second variation of area 168
7 The minimal surface equation 176
8 Elliptic regularity Ⅱ（boundary estimates） 185
9 Elliptic regularity Ⅲ（DeGiorgi-Nash-Moser theory） 196
10 The Dirichlet problem for quasi-linear elliptic equations 208
11 Direct methods in the calculus of variations 222
12 Quasi-linear elliptic systems 229
12B Further results on quasi-linear systems 244
13 Elliptic regularity Ⅳ（Krylov--Safonov estimates） 258
14 Regularity for a class of completely nonlinear equations 273
15 Monge-Ampere equations 282
16 Elliptic equations in two variables 294
A Morrey spaces 299
B Leray-Schauder fixed-point theorems 302
References 304
15 Nonlinear Parabolic Equations 313
1 Semilinear parabolic equations 314
2 Applications to harmonic maps 325
3 Semilinear equations on regions with boundary 332
4 Reaction-diffusion equations 335
5 A nonlinear Trotter product formula 353
6 The Stefan problem 362
7 Quasi-linear parabolic equationsⅠ 376
8 Quasi-linear parabolic equations Ⅱ（sharper estimates） 387
9 Quasi-linear parabolic equations Ⅲ（Nash-Moser estimates） 396
References 407
16 Nonlinear Hyperbolic Equations 413
1 Quasi-linear,symmetric hyperbolic systems 414
2 Symmetrizable hyperbolic systems 425
3 Second-order and higher-order hyperbolic systems 432
4 Equations in the complex domain and the Cauchy-Kowalewsky theorem 445
5 Compressible fluid motion 448
6 Weak solutions to scalar conservation laws;the viscosity method 457
7 Systems of conservation laws in one space variable;Riemann problems 472
8 Entropy-flux pairs and Riemann invariants 498
9 Global weak solutions of some 2×2 systems 509
10 Vibrating strings revisited 517
References 524
17 Euler and Navier-Stokes Equations for Incompressible Fluids 531
1 Euler's equations for ideal incompressible fluid flow 532
2 Existence of solutions to the Euler equations 542
3 Euler flows on bounded regions 553
4 Navier-Stokes equations 561
5 Viscous flows on bounded regions 575
6 Vanishing viscosity limits 586
7 From velocity field convergence to flow convergence 599
A Regularity for the Stokes system on bounded domains 605
References 610
18 Einstein's Equations 615
1 The gravitational field equations 616
2 Spherically symmetric spacetimes and the Schwarzschild solution 626
3 Stationary and static spacetimes 639
4 Orbits in Schwarzschild spacetime 649
5 Coupled Maxwell-Einstein equations 656
6 Relativistic fluids 659
7 Gravitational collapse 670
8 The initial-value problem 677
9 Geometry of initial surfaces 687
10 Time slices and their evolution 699
References 705
Index 711