Partial Differential Equations III - Nonlinear Equations

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^p Sobolev spaces, Holder spaces, Hardy spaces, and Morrey spaces, and also a development of Calderon-Zygmund theory and paradifferential operator calculus. The book is targeted 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. The third edition further expands the material by incorporating new theorems and applications throughout the book, and by deepening connections and relating concepts across chapters. It includes new sections on rigid body motion, on probabilistic results related to random walks, on aspects of operator theory related to quantum mechanics, on overdetermined systems, and on the Euler equation for incompressible fluids. The appendices have also been updated with additional results, ranging from weak convergence of measures to the curvature of Kahler manifolds. Michael E. Taylor is a Professor of Mathematics at the University of North Carolina, Chapel Hill, NC. Review of first edition: “These volumes will be read by several generations of readers eager to learn the modern theory of partial differential equations of mathematical physics and the analysis in which this theory is rooted.” (Peter Lax, SIAM review, June 1998) Contents of Volumes I and II 7 Preface 8 Acknowledgments 16 Introduction to the Second Edition 17 Introduction to the Third Edition 17 Contents 19 13 Function Space and Operator Theory for Nonlinear Analysis 22 1 Lp-Sobolev spaces 23 2 Sobolev imbedding theorems 25 3 Gagliardo–Nirenberg–Moser estimates 29 4 Trudinger's inequalities 35 5 Singular integral operators on Lp 38 6 The spaces Hs,p 45 7 Lp-spectral theory of the Laplace operator 52 8 Hölder spaces and Zygmund spaces 61 9 Pseudodifferential operators with nonregular symbols 71 10 Paradifferential operators 81 11 Young measures and fuzzy functions 96 12 Hardy spaces 109 A Variations on complex interpolation 118 References 124 14 Nonlinear Elliptic Equations 128 1 A class of semilinear equations 131 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 171 6 Minimal surfaces 176 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 232 11 Direct methods in the calculus of variations 246 12 Quasi-linear elliptic systems 253 12B Further results on quasi-linear systems 268 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 318 17 Overdetermined elliptic systems 322 A Morrey spaces 336 B Leray–Schauder fixed-point theorems 339 C The Weyl tensor 341 References 347 15 Nonlinear Parabolic Equations 356 1 Semilinear parabolic equations 357 2 Applications to harmonic maps 368 3 Semilinear equations on regions with boundary 375 4 Reaction-diffusion equations 377 5 A nonlinear Trotter product formula 396 6 The Stefan problem 406 7 Quasi-linear parabolic equations I 419 8 Quasi-linear parabolic equations II (sharper estimates) 430 9 Quasi-linear parabolic equations III (Nash–Moser estimates) 439 References 450 16 Nonlinear Hyperbolic Equations 455 1 Quasi-linear, symmetric hyperbolic systems 456 2 Symmetrizable hyperbolic systems 467 3 Second-order and higher-order hyperbolic systems 474 4 Equations in the complex domain and the Cauchy–Kowalewsky theorem 487 5 Compressible fluid motion 490 6 Weak solutions to scalar conservation laws; the viscosity method 499 7 Systems of conservation laws in one space variable; Riemann problems 514 8 Entropy-flux pairs and Riemann invariants 540 9 Global weak solutions of some 22 systems 551 10 Vibrating strings revisited 559 References 566 17 Euler and Navier–Stokes Equations for Incompressible Fluids 573 1 Euler's equations for ideal incompressible fluid flow 574 2 Existence of solutions to the Euler equations 584 3 Euler flows on bounded regions 595 4 Euler equations on a rotating surface 603 5 Navier–Stokes equations 620 6 Viscous flows on bounded regions 634 7 Vanishing viscosity limits 645 8 From velocity field convergence to flow convergence 658 A Regularity for the Stokes system on bounded domains 664 References 669 18 Einstein's Equations 675 1 The gravitational field equations 676 2 Spherically symmetric spacetimes and the Schwarzschild solution 686 3 Stationary and static spacetimes 699 4 Orbits in Schwarzschild spacetime 708 5 Coupled Maxwell–Einstein equations 716 6 Relativistic fluids 719 7 Gravitational collapse 729 8 The initial-value problem 737 9 Geometry of initial surfaces 747 10 Time slices and their evolution 759 References 764 Index 770 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^p Sobolev spaces, Holder spaces, Hardy spaces, and Morrey spaces, and also a development of Calderon-Zygmund theory and paradifferential operator calculus. The book is targeted 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. The third edition further expands the material by incorporating new theorems and applications throughout the book, and by deepening connections and relating concepts across chapters. It includes new sections on rigid body motion, on probabilistic results related to random walks, on aspects of operator theory related to quantum mechanics, on overdetermined systems, and on the Euler equation for incompressible fluids. The appendices have also been updated with additional results, ranging from weak convergence of measures to the curvature of Kähler manifolds.