This volume is devoted to nonlinear PDE. There are treatments of equations arising in classical continuum mechanics, such as vibrating strings and membranes, and fluid flows. We also treat equations arising in differential geometry, nonlinear diffusion, and general relativity. Analytical tools introduced for these studies include Lp-Sobolev spaces, Morrey spaces, Hardy spaces, the Calderon-Zygmund theory, and paradifferential operator calculus, as well as more classical techniques such as energy estimates and maximum principles.

## Contents

- 13. Function Space and Operator Theory for Nonlinear Analysis
- L
^{p}-Sobolev spaces - Sobolev imbedding theorems
- Gagliardo-Nirenberg-Moser estimates
- Trudinger’s inequalities
- Singular integral operators on L
^{p} - The spaces H
^{s,p} - L
^{p}-spectral theory of the Laplace operator - Holder spaces and Zygmund spaces
- Pseudodifferential operators with nonregular symbols
- Paradifferential operators
- Young measures and fuzzy functions
- Hardy spaces

- L
- 14. Nonlinear Elliptic Equations
- A class of semilinear equations
- Surfaces with negative curvature
- Local solvability of nonlinear elliptic equations
- Elliptic regularity I (interior estimates)
- Isometric imbedding of Riemannian manifolds
- Minimal surfaces
- The minimal surface equation
- Elliptic regularity II (boundary estimates)
- Elliptic regularity III (DeGiorgi-Nash-Moser theory)
- The Dirichlet problem for quasi-linear elliptic equations
- Direct methods in the calculus of variations
- Quasi-linear elliptic systems
- Elliptic regularity IV (Krylov-Safonov estimates)
- Regularity for a class of completely nonlinear equations
- Monge-Ampere equations
- Elliptic equations in two variables
- Overdetermined elliptic systems

- 15. Nonlinear Parabolic Equations
- Semilinear parabolic equations
- Applications to harmonic maps
- Semilinear equations on regions with boundary
- Reaction-diffusion equations
- A nonlinear Trotter product formula
- The Stefan problem
- Quasi-linear parabolic problems I
- Quasi-linear parabolic problems II (sharper estimates)
- Quasi-linear parabolic problems III (Nash-Moser estimates)

- 16. Nonlinear Hyperbolic Equations
- Quasi-linear, symmetric hyperbolic systems
- Symmetrizable hyperbolic systems
- Second-order and higher-order hyperbolic systems
- Equations in the complex domain and the Cauchy-Kowalewsky theorem
- Compressible fluid motion
- Weak solutions to scalar conservation laws; the viscosity method
- Systems of conservation laws in one space variable; Riemann problems
- Entropy-flux pairs and Riemann invariants
- Global weak solutions of some 2 by 2 systems
- Vibrating strings revisited

- 17. Euler and Navier-Stokes Equations for Incompressible Fluids
- Euler’s equations for ideal incompressible fluid flow
- Existence of solutions to the Euler equation
- Euler flows on bounded regions
- Euler flows on a rotating surface
- Navier-Stokes equations
- Viscous flows on bounded regions
- Vanishing viscosity limits
- From velocity field convergence to flow convergence

- 18. Einstein’s Equations
- The gravitational field equations
- Spherically symmetric spacetimes and the Schwarzschild solution
- Stationary and static spacetimes
- Orbits in Schwarzschild spacetime
- Coupled Maxwell-Einstein equations
- Relativistic fluids
- Gravitational collapse
- The initial-value problem
- Geometry of initial surfaces
- Time slices and their evolution