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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