Pseudodifferential Operators and Nonlinear PDE

This book is number 100 in the Birkhauser Series, Progress in Mathematics. It deals with the use of pseudodifferential operators as a tool in nonlinear PDE.

One goal has been to build a bridge between two approaches that have been used in a number of works, one being the theory of paradifferential operators, introduced by J.-M. Bony, the other the study of pseudodifferential operators whose symbols have limited regularity.

To some degree the monograph Tools for PDE is a companion to this volume.

Contents

  • Chapter 0. Pseudodifferential operators and linear PDE-
    1. The Fourier integral representation and symbol classes
    2. Schwartz kernels of pseudodifferential operators
    3. Adjoints and products
    4. Elliptic operators and parametrices
    5. L2 estimates
    6. Garding’s inequality
    7. The sharp Garding inequality
    8. Hyperbolic evolution equations
    9. Egorov’s theorem
    10. Microlocal regularity
    11. Lp estimates
    12. Operators on manifolds
  • Chapter 1. Symbols with limited smoothness
    1. Symbol classes
    2. Some simple elliptic regularity theorems
    3. Symbol smoothing
  • Chapter 2. Operator estimates and elliptic regularity
    1. Bounds on operators with nonregular symbols
    2. Further elliptic regularity theorems
    3. Adjoints
    4. Sharp Garding inequality
  • Chapter 3. Paradifferential operators
    1. Composition and paraproducts
    2. Various forms of paraproduct
    3. Nonlinear PDE and paradifferential operators
    4. Operator algebra
    5. Product estimates
    6. Commutator estimates
  • Chapter 4. Calculus for OPC1Smcl
    1. Commutator estimates
    2. Operator algebra
    3. Garding inequality
    4. C1-paradifferential calculus
  • Chapter 5. Nonlinear hyperbolic systems
    1. Quasilinear symmetric hyperbolic systems
    2. Symmetrizable hyperbolic systems
    3. Higher order hyperbolic equations
    4. Completely nonlinear hyperbolic systems
  • Chapter 6. Propagation of singularities
    1. Propagation of singularities
    2. Nonlinear formation of singularities
    3. Egorov’s theorem
  • Chapter 7. Nonlinear parabolic systems
    1. Strongly parabolic quasilinear systems
    2. Petrowski parabolic quasilinear systems
    3. Sharper estimates
    4. Semilinear parabolic systems
  • Chapter 8. Nonlinear elliptic boundary problems
    1. Second order elliptic equations
    2. Quasilinear elliptic equations
    3. Interface with DeGiorgi-Nash-Moser theory
  • Chapter 9. Extension of the Schauder estimates
    1. Nirenberg’s refinement
    2. Elliptic boundary problems
  • Appendix A. Function spaces
    1. Holder spaces, Zygmund spaces, and Sobolev spaces
    2. Morrey spaces
    3. BMO
  • Appendix B. Sup norm estimates
    1. Loo estimates on pseudodifferential operators
    2. The spaces Cr#
  • Appendix C. DeGiorgi-Nash-Moser estimates
    1. Moser iteration and Loo estimates
    2. Holder continuity
    3. Inhomogeneous equations
    4. Boundary regularity
  • Paraproduct estimates