Taylor Partial Differential Equations III

Nonlinear Equations
2. Auflage 2011
ISBN: 978-1-4419-7049-7
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 715 Seiten

Reihe: Mathematics and Statistics

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 Sobolev spaces, H lder spaces, Hardy spaces, and Morrey spaces, and also a development of Calderon-Zygmund theory and paradifferential operator calculus. The book is aimed 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

Zielgruppe

Research

Autoren/Hrsg.

Taylor, Michael E.

Inhaltsverzeichnis

1;Contents;8
2;Contents of Volumes I and II;12
3;Preface;14
4;13 Function Space and Operator Theory for Nonlinear Analysis;24
4.1;1 Lp-Sobolev spaces;25
4.2;2 Sobolev imbedding theorems;27
4.3;3 Gagliardo–Nirenberg–Moser estimates;31
4.4;4 Trudinger's inequalities;37
4.5;5 Singular integral operators on Lp;40
4.6;6 The spaces Hs,p;47
4.7;7 Lp-spectral theory of the Laplace operator;54
4.8;8 Hölder spaces and Zygmund spaces;63
4.9;9 Pseudodifferential operators with nonregular symbols;73
4.10;10 Paradifferential operators;83
4.11;11 Young measures and fuzzy functions;97
4.12;12 Hardy spaces;109
4.13;A Variations on complex interpolation;119
4.14; References;125
5;14 Nonlinear Elliptic Equations;128
5.1;1 A class of semilinear equations;130
5.2;2 Surfaces with negative curvature;142
5.3;3 Local solvability of nonlinear elliptic equations;150
5.4;4 Elliptic regularity I (interior estimates);158
5.5;5 Isometric imbedding of Riemannian manifolds;170
5.6;6 Minimal surfaces;175
5.7;6B Second variation of area;191
5.8;7 The minimal surface equation;199
5.9;8 Elliptic regularity II (boundary estimates);208
5.10;9 Elliptic regularity III (DeGiorgi–Nash–Moser theory);219
5.11;10 The Dirichlet problem for quasi-linear elliptic equations;231
5.12;11 Direct methods in the calculus of variations;245
5.13;12 Quasi-linear elliptic systems;252
5.14;12B Further results on quasi-linear systems;267
5.15;13 Elliptic regularity IV (Krylov–Safonov estimates);281
5.16;14 Regularity for a class of completely nonlinear equations;296
5.17;15 Monge–Ampere equations;305
5.18;16 Elliptic equations in two variables;317
5.19;A Morrey spaces;322
5.20;B Leray–Schauder fixed-point theorems;325
5.21; References;327
6;15 Nonlinear Parabolic Equations;335
6.1;1 Semilinear parabolic equations;336
6.2;2 Applications to harmonic maps;347
6.3;3 Semilinear equations on regions with boundary;354
6.4;4 Reaction-diffusion equations;357
6.5;5 A nonlinear Trotter product formula;375
6.6;6 The Stefan problem;384
6.7;7 Quasi-linear parabolic equations I;398
6.8;8 Quasi-linear parabolic equations II (sharper estimates);409
6.9;9 Quasi-linear parabolic equations III (Nash–Moser estimates);418
6.10; References;429
7;16 Nonlinear Hyperbolic Equations;434
7.1;1 Quasi-linear, symmetric hyperbolic systems;435
7.2;2 Symmetrizable hyperbolic systems;446
7.3;3 Second-order and higher-order hyperbolic systems;453
7.4;4 Equations in the complex domain and the Cauchy–Kowalewsky theorem;466
7.5;5 Compressible fluid motion;469
7.6;6 Weak solutions to scalar conservation laws; the viscosity method;478
7.7;7 Systems of conservation laws in one space variable; Riemann problems;493
7.8;8 Entropy-flux pairs and Riemann invariants;519
7.9;9 Global weak solutions of some 2x2 systems;530
7.10;10 Vibrating strings revisited;538
7.11; References;545
8;17 Euler and Navier–Stokes Equations for Incompressible Fluids;551
8.1;1 Euler's equations for ideal incompressible fluid flow;552
8.2;2 Existence of solutions to the Euler equations;562
8.3;3 Euler flows on bounded regions;573
8.4;4 Navier–Stokes equations;581
8.5;5 Viscous flows on bounded regions;595
8.6;6 Vanishing viscosity limits;606
8.7;7 From velocity field convergence to flow convergence;619
8.8;A Regularity for the Stokes system on bounded domains;625
8.9; References;630
9;18 Einstein's Equations;635
9.1;1 The gravitational field equations;636
9.2;2 Spherically symmetric spacetimes and the Schwarzschild solution;646
9.3;3 Stationary and static spacetimes;659
9.4;4 Orbits in Schwarzschild spacetime;669
9.5;5 Coupled Maxwell–Einstein equations;676
9.6;6 Relativistic fluids;679
9.7;7 Gravitational collapse;690
9.8;8 The initial-value problem;697
9.9;9 Geometry of initial surfaces;707
9.10;10 Time slices and their evolution;719
9.11; References;725
10;Index;730

Über Autor(innen)

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