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AuthorTaylor, Michael E. author
TitlePseudodifferential Operators and Nonlinear PDE [electronic resource] / by Michael E. Taylor
ImprintBoston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 1991
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Descript IV, 216 p. online resource


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. One goal has been to build a bridge between two approaches which have been used in a number of papers written in the last decade, one being the theory of paradifferential operators, pioneered by Bony and Meyer, the other the study of pseudodifferential operators whose symbols have limited regularity. The latter approach is a natural successor to classical devices of deriving estimates for linear PDE whose coefficients have limited regularity in order to obtain results in nonlinear PDE. After developing the requisite tools, we proceed to demonstrate their effectiveness on a range of basic topics in nonlinear PDE. For example, for hyperbolic systems, known sufficient conditions for persistence of solutions are both sharpened and extended in scope. In the treatment of parabolic equations and elliptic boundary problems, it is shown that the results obtained here interface particularly easily with the DeGiorgi-Nash-Moser theory, when that theory applies. To make the work reasonable self-contained, there are appendices treating background topics in harmonic analysis and the DeGiorgi-Nash-Moser theory, as well as an introductory chapter on pseudodifferential operators as developed for linear 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


Introduction. -- 0. Pseudodifferential operators and linear PDE. -- ยง0.1 The Fourier integral representation and symbol classes -- ยง0.2 Schwartz kernels of pseudodifferential operators -- ยง0.3 Adjoints and products -- ยง0.4 Elliptic operators and parametrices -- ยง0.5 L2 estimates -- ยง0.6 Gรฅrdingโ{128}{153}s inequality -- ยง0.7 The sharp Gรฅrding inequality -- ยง0.8 Hyperbolic evolution equations -- ยง0.9 Egorovโ{128}{153}s theorem -- ยง0.10 Microlocal regularity -- ยง0.11 Lp estimates -- ยง0.12 Operators on manifolds -- 1. Symbols with limited smoothness. -- ยง1.1 Symbol classes -- ยง1.2 Some simple elliptic regularity theorems -- ยง1.3 Symbol smoothing -- 2. Operator estimates and elliptic regularity. -- ยง2.1 Bounds for operators with nonregular symbols -- ยง2.2 Further elliptic regularity theorems -- ยง2.3 Adjoints -- ยง2.4 Sharp Gรฅrding inequality -- 3. Paradifferential operators. -- ยง3.1 Composition and paraproducts -- ยง3.2 Various forms of paraproduct -- ยง3.3 Nonlinear PDE and paradifferential operators -- ยง3.4 Operator algebra -- ยง3.5 Product estimates -- ยง3.6 Commutator estimates -- 4. Calculus for OPC1Sclm. -- ยง4.1 Commutator estimates -- ยง4.2 Operator algebra -- ยง4.3 Gรฅrding inequality -- ยง4.4 C1-paradifferential calculus -- 5. Nonlinear hyperbolic systems. -- ยง5.1 Quasilinear symmetric hyperbolic systems -- ยง5.2 Symmetrizable hyperbolic systems -- ยง5.3 Higher order hyperbolic equations -- ยง5.4 Completely nonlinear hyperbolic systems -- 6. Propagation of singularities. -- ยง6.1 Propagation of singularities -- ยง6.2 Nonlinear formation of singularities -- ยง6.3 Egorovโ{128}{153}s theorem -- 7. Nonlinear parabolic systems. -- ยง7.1 Strongly parabolic quasilinear systems -- ยง7.2 Petrowski parabolic quasilinear systems -- ยง7.3 Sharper estimates -- ยง7.4 Semilinear parabolic systems -- 8. Nonlinear elliptic boundary problems. -- ยง8.1 Second order elliptic equations -- ยง8.2 Quasilinear elliptic equations -- ยง8.3 Interface with DeGiorgi-Nash-Moser theory -- 9. Extension of the Schauder estimates. -- ยง9.1 Nirenbergโ{128}{153}s refinement -- ยง9.2 Elliptic boundary problems -- A. Function spaces. -- ยงA.1 Hรถlder spaces, Zygmund spaces, and Sobolev spaces -- ยงA.2 Morrey spaces -- ยงA.3 BMO -- B. Sup norm estimates. -- C. DeGiorgi-Nash-Moser estimates. -- ยงC.2 Hรถlder continuity -- ยงC.3 Inhomogeneous equations -- ยงC.4 Boundary regularity -- D. Paraproduct estimates. -- Index of notation. -- References

Mathematics Mathematical analysis Analysis (Mathematics) Operator theory Partial differential equations Mathematics Analysis Partial Differential Equations Operator Theory


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