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Tools for PDE: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials
Michael E. Taylor, University of North Carolina, Chapel Hill, NC
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Mathematical Surveys and Monographs
2000; 257 pp; softcover
Volume: 81
ISBN-10: 0-8218-4378-8
ISBN-13: 978-0-8218-4378-9
List Price: US$65
Member Price: US$52
Order Code: SURV/81.S
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See also:

Morse Theoretic Aspects of \(p\)-Laplacian Type Operators - Kanishka Perera, Ravi P Agarwal and Donal O'Regan

Introduction to Differential Equations - Michael E Taylor

This book develops three related tools that are useful in the analysis of partial differential equations (PDEs), arising from the classical study of singular integral operators: pseudodifferential operators, paradifferential operators, and layer potentials.

A theme running throughout the work is the treatment of PDE in the presence of relatively little regularity. The first chapter studies classes of pseudodifferential operators whose symbols have a limited degree of regularity; the second chapter shows how paradifferential operators yield sharp estimates on the action of various nonlinear operators on function spaces. The third chapter applies this material to an assortment of results in PDE, including regularity results for elliptic PDE with rough coefficients, planar fluid flows on rough domains, estimates on Riemannian manifolds given weak bounds on Ricci tensor, div-curl estimates, and results on propagation of singularities for wave equations with rough coefficients. The last chapter studies the method of layer potentials on Lipschitz domains, concentrating on applications to boundary problems for elliptic PDE with variable coefficients.

Readership

Graduate students and research mathematicians interested in partial differential equations.

Table of Contents

  • Pseudodifferential operators with mildly regular symbols
  • Paradifferential operators and nonlinear estimates
  • Applications to PDE
  • Layer potentials on Lipschitz surfaces
  • Bibliography
  • List of symbols
  • Index
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