Tools for PDE : Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials

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. Cover......Page 1 Title: Tools for PDE......Page 3 QA377.T37 2000 515'.353 — dc2l......Page 5 Contents......Page 8 Preface......Page 10 Introduction......Page 12 1. Spaces of continuous functions......Page 14 2. Operator estimates on L^p, h^2 and bmo......Page 28 3. Symbol classes and symbol smoothing......Page 42 4. Operator estimates on Sobolev-like spaces......Page 48 5. Operator estimates on spaces C^(\lambda)......Page 55 6. Products......Page 65 7. Commutator estimates......Page 69 8. Operators with Sobolev coefficients......Page 72 9. Operators with double symbols......Page 74 10. The CRW commutator estimate......Page 86 12. Estimates on a class of Besov spaces......Page 93 13. Operators with coefficients in a function algebra......Page 97 14. Some BKM-type estimates......Page 99 15. Variations on an estimate of Thmanov......Page 103 16. Estimates on Morrey-type spaces......Page 105 Introduction......Page 112 1. A product estimate......Page 116 2. A commutator estimate......Page 117 3. Some handy estimates involving maximal functions......Page 119 4. A composition estimate......Page 121 5. More general composition estimate......Page 123 6. Continuity of u ---> f(u) on H^(1, p)......Page 124 7. Estimates on F(u) —> F(v)......Page 127 8. A pseudodiffereritial operator estimate......Page 129 9. Paradifferential operators on the spaces C^(\lambda)......Page 131 A. Paracomposition......Page 136 B. Alinhac's lemma......Page 143 Introduction......Page 146 1. Interior elliptic regularity......Page 148 2. Some natural first-order operators......Page 159 3. Estimates for the Dirichlet problem......Page 166 4. Layer potentials on C^1' surfaces......Page 170 5. Parametrix estimates and trace asymptotics......Page 184 6. Euler flows on rough planar domains......Page 189 7. Persistence of solutions to semilinear wave equations......Page 194 8. Div-curl estimates......Page 197 9. Harmonic coordinates......Page 205 10. Riemannian manifolds with bounded Ricci tensor......Page 213 11. Propagation of singularities......Page 216 Introduction......Page 228 1. Cauchy kerne!s on Lipschitz curves......Page 229 2. The method of rotations and extensions to higher dimensions......Page 239 3. The variable-coefficient case......Page 241 4. Boundary integral operators......Page 246 5. The Dirichiet problem on Lipschitz domains......Page 252 A. The Koebe-Bieberbach distortion theorem......Page 257 Bibliography......Page 260 List of Symbols......Page 266 Index......Page 268 Back Cover......Page 269 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. This work focuses on the treatment of PDE in the presence of relatively little regularity. This text develops three related tools that are useful in the analysis of partial differential equations, arising from the classical study of singular integral operators: pseudodifferential operators, paradifferential operators, and layer potentials.