Pseudodifferential operators and nonlinear PDE

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Bibliographic Information

Pseudodifferential operators and nonlinear PDE

Michael E. Taylor

（Progress in mathematics, v. 100）

Birkhäuser, c1991

  • : us
  • : gw

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Note

Bibliographical reference: p. [209]-213

Description and Table of Contents

Volume

: us ISBN 9780817635954

Description

For the past 25 years the theory of pseudodifferential operators has played an important role in many exciting and deep investigations into linear PDE. Over the past decade, this tool has also begun to yield interesting results in nonlinear PDE. This book is devoted to a summary and reconsideration of some used of pseudodifferential operator techniques in nonlinear PDE. The book should be of interest to graduate students, instructors, and researchers interested in partial differential equations, nonlinear analysis in classical mathematical physics and differential geometry, and in harmonic analysis.

Table of Contents

Introduction..- 0. Pseudodifferential operators and linear PDE..- 0.1 The Fourier integral representation and symbol classes.- 0.2 Schwartz kernels of pseudodifferential operators.- 0.3 Adjoints and products.- 0.4 Elliptic operators and parametrices.- 0.5 L2 estimates.- 0.6 Garding's inequality.- 0.7 The sharp Garding inequality.- 0.8 Hyperbolic evolution equations.- 0.9 Egorov's theorem.- 0.10 Microlocal regularity.- 0.11 Lp estimates.- 0.12 Operators on manifolds.- 1. Symbols with limited smoothness..- 1.1 Symbol classes.- 1.2 Some simple elliptic regularity theorems.- 1.3 Symbol smoothing.- 2. Operator estimates and elliptic regularity..- 2.1 Bounds for operators with nonregular symbols.- 2.2 Further elliptic regularity theorems.- 2.3 Adjoints.- 2.4 Sharp Garding inequality.- 3. Paradifferential operators..- 3.1 Composition and paraproducts.- 3.2 Various forms of paraproduct.- 3.3 Nonlinear PDE and paradifferential operators.- 3.4 Operator algebra.- 3.5 Product estimates.- 3.6 Commutator estimates.- 4. Calculus for OPC1Sclm..- 4.1 Commutator estimates.- 4.2 Operator algebra.- 4.3 Garding inequality.- 4.4 C1-paradifferential calculus.- 5. Nonlinear hyperbolic systems..- 5.1 Quasilinear symmetric hyperbolic systems.- 5.2 Symmetrizable hyperbolic systems.- 5.3 Higher order hyperbolic equations.- 5.4 Completely nonlinear hyperbolic systems.- 6. Propagation of singularities..- 6.1 Propagation of singularities.- 6.2 Nonlinear formation of singularities.- 6.3 Egorov's theorem.- 7. Nonlinear parabolic systems..- 7.1 Strongly parabolic quasilinear systems.- 7.2 Petrowski parabolic quasilinear systems.- 7.3 Sharper estimates.- 7.4 Semilinear parabolic systems.- 8. Nonlinear elliptic boundary problems..- 8.1 Second order elliptic equations.- 8.2 Quasilinear elliptic equations.- 8.3 Interface with DeGiorgi-Nash-Moser theory.- 9. Extension of the Schauder estimates..- 9.1 Nirenberg's refinement.- 9.2 Elliptic boundary problems.- A. Function spaces..- A.1 Hoelder spaces, Zygmund spaces, and Sobolev spaces.- A.2 Morrey spaces.- A.3 BMO.- B. Sup norm estimates..- C. DeGiorgi-Nash-Moser estimates..- C.2 Hoelder continuity.- C.3 Inhomogeneous equations.- C.4 Boundary regularity.- D. Paraproduct estimates..- Index of notation..- References..
Volume

: gw ISBN 9783764335953

Description

The theory of pseudodifferential operators has played an important role in many investigations into linear PDE. This book is devoted to a summary and reconsideration of some uses of pseudodifferential operator techniques in nonlinear PDE.

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