| Contents | 8 |
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| Contents of Volumes I and II | 12 |
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| Preface | 14 |
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| 13 Function Space and Operator Theory for Nonlinear Analysis | 24 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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