Pseudodifferential operators and nonlinear PDE
This book is devoted to recent developments in the theory of pseudodifferential operators, namely the calculus of paradifferential operators and the study of pseudodifferential operators with symbols of limited smoothness, which have led to interesting results in nonlinear PDE over the past decade. The author succeeded in a quite optimal choice and a very readable presentation of the material, which is of interest to each graduate student and researcher who wishes to get acquainted with those new tools in the regularity theory of nonlinear PDE. The book begins with a preliminary chapter on the basics of the theory of pseudodifferential operators, including applications to linear elliptic and hyperbolic equations. Many results are proved in detail. Chapter 1 and 2 are devoted to the study of pseudodifferential operators with symbols \(p(x,\xi)\), having limited smoothness in \(x\). Here the basic technique of symbol smoothing is introduced, bounds for operators with nonregular symbols are established, and first regularity results for nonlinear elliptic PDE are derived. Chapter 3 develops the key tool of paradifferential operator calculus, including the commutator estimates of Coifman and Meyer and of Kato and Ponce. Chapter 4 deals with a sharp operator calculus for \(C^ 1\) symbols and discusses its relation to the \(C^ 1\) paradifferential calculus of Bony. The subsequent chapters are concerned with important applications in nonliner PDE. Chapter 5 is devoted to nonlinear hyperbolic equations and contains the sharpest results on regular solutions to symmetric and symmetrizable systems. In Chapter 6 a variant of Bony's theorem on the propagation of singularities is established. Nonlinear parabolic equations and elliptic boundary value problems are treated in Chapters 7 and 8. In Chapter 9, Nirenberg's refinement of the Schauder estimates for elliptic equations is discussed, using the paradifferential calculus. At the end there are four appendices on function spaces, sup norm estimates, De Giorgi-Nash- Moser theory and paraproduct estimates.
scientific article; zbMATH DE number 52678