偏微分方程(第3卷)(第2版)

书籍目录

Contents of Volumes I and H

Preface

13  Function Space and Operator Theory for Nonlinear Analysis

1  LP-Sobolev spaces

2  Sobolev imbedding theorems

3  Gagliardo-Nirenberg-Moser estimates

4  Trudinger's inequalities

5  Singular integral operators on Lp

6  The spaces Hs'p

7  LP-spectral theory of the Laplace operator

8  Holder spaces and Zygmund spaces

9  Pseudodifferential operators with nonregular symbols

10  Paradifferential operators

11  Young measures and fuzzy functions

12  Hardy spaces

A  Variations on complex interpolation

References

14  Nonlinear Elliptic Equations

1  A class of semilinear equations

2  Surfaces with negative curvature

3  Local solvability of nonlinear elliptic equations

4  Elliptic regularity I (interior estimates)

5  Isometric imbedding of Riemannian manifolds

6  Minimal surfaces

6B  Second variation of area

7  The minimal surface equation

8  Elliptic regularity II (boundary estimates)

9  Elliptic regularity III (DeGiorgi-Nash-Moser theory)

10  The Dirichlet problem for quasi-linear elliptic equations

11  Direct methods in the calculus of variations

12  Quasi-linear elliptic systems

12B  Further results on quasi-linear systems

13  Elliptic regularity IV (Krylov-Safonov estimates)

14  Regularity for a class of completely nonlinear equations

15  Monge-Ampereequations

16  Elliptic equations in two variables

A  Morrey spaces

B  Leray-Schauder fixed-point theorems

References

15  Nonlinear Parabolic Equations

1  Semitinear parabolic equations

2  Applications to harmonic maps

3  Semilinear equations on regions with boundary

4  Reaction-diffusion equations

5  A nonlinear Trotter product formula

6  The Stefan problem

7  Quasi-linear parabolic equations I

8  Quasi-linear parabolic equations II (sharper estimates)

9  Quasi-linear parabolic equations III (Nash-Moser estimates)

References

16  Nonlinear Hyperbolic Equations

1  Quasi-linear, symmetric hyperbolic systems

2  Symmetrizable hyperbolic systems

3  Second-orderand higher-orderhyperbolic systems

4  Equations in the complex domain and the Cauchy- Kowalewsky theorem

5  Compressible fluid motion

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

9  Global weak solutions of some 2 x 2 systems

10  Vibrating strings revisited

References

17  Euler and Navier-Stokes Equations for Incompressible Fluids

1  Euler's equations for ideal incompressible fluid flow

2  Existence of solutions to the Euler equations

3  Euler flows on bounded regions

4  Navier-Stokes equations

5  VisCous flows on bounded regions

6  Vanishing viscosity limits

7  From velocity field convergence to flow convergence

A  Regularity for the Stokes system on bounded domains

References

18  Einstein's Equations

1  The gravitational field equations

2  Spherically symmetric spacetimes and the Schwarzschild solution

3  Stationary and static spacetimes

4  Orbits in Schwarzschild spacetime

5  Coupled Maxwell-Einstein equations

6  Relativistic fluids

7  Gravitational collapse

8  The initial-value problem

9  Geometry of initial surfaces

10  Time slices and their evolution

References

Index

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