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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.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract
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 paper is devoted to a summary and reconsideration of some uses of pseudodifferential operator techniques in nonlinear PDE.
Michael E. Taylor

Chapter 0. Pseudodifferential operators and linear PDE

Abstract
In this preliminary chapter we give an outline of the theory of pseudodifferential operators as it has been developed to treat problems in linear PDE, and which will provide a basis for further developments to be discussed in the following chapters. Many results will be proved in detail, but some proofs are only sketched, with references to more details in the literature. We define pseudodifferential operators with symbols in Hörmander’s classes S p,δ m , derive some useful properties of their Schwartz kernels, discuss their algebraic properties, then show how they can be used to establish regularity of solutions to elliptic PDE with smooth coefficients. We proceed to a discussion of mapping properties on L2 and on the Sobolev spaces H s , then discuss Gårding’s inequality, and some of its refinements, known as sharp Gårding inequalities. In §0.8 we apply some of the previous material to establish existence of solutions to hyperbolic equations. We introduce the notion of wave front set in §0.10 and discuss microlocal regularity of solutions to elliptic equations. We also discuss how solution operators to a class of hyperbolic equations propagate wave front sets. In §0.11 we discuss L p estimates, particularly some fundamental results of Calderon and Zygmund, and applications to Littlewood-Paley Theory, which will be an important technical tool for basic estimates established in Chapter 2.We end this introduction with a brief discussion of pseudodifferential operators on manifolds.
Michael E. Taylor

Chapter 1. Symbols with limited smoothness

Abstract
Here we establish some very general facts about symbols p(x,ξ) with limited smoothness in x. We prove some operator bounds on D) when p(x, D) when p(x,ξ) is homogeneous in ξ.
Michael E. Taylor

Chapter 2. Operator estimates and elliptic regularity

Abstract
In order to make use of the symbol smoothing of §1.3, we need operator estimates on p(x, D) when p(x, ξ) ∈CsS 1, δ m . We give a number of such results in §2.1, most of them following from work of Bourdaud [BG]. The main result, Theorem 2.1.A, treats the case δ = 1. This will be very useful for the treatment of paradifferential operators in Chapter 3.
Michael E. Taylor

Chapter 3. Paradifferential Operators

Abstract
The key tool of paradifferential operator calculus is developed in this chapter, beginning with Meyer’s ingenious formula for F(u) as M(x, D)u + R where F is smooth in its argument (s), u belongs to a Hölder or Sobolev space, M(x, D) is a pseudodifferential operator of type 1,1, and R is smooth. From there, one applies symbol smoothing to M(x, ξ) and makes use of results established in Chapter 2. The tool that arises is quite powerful in nonlinear analysis. The first glimpse we give of this is that it automatically encompasses some important Moser estimates. We re-derive elliptic regularity results established in Chapter 2, after establishing some microlocal regularity results. In §3.3 we do this using symbol smoothing with δ < 1; in §3.4 we present some results of Bony and Meyer dealing with the δ = 1 case, the case of genuine paradifferential operators.
Michael E. Taylor

Chapter 4. Calculus for OPC1S cl m

Abstract
In the last chapter, we developed an operator calculus and used it for several purposes, including obtaining commutator estimates in §3.6. Here we work in the opposite order. In §4.1 we recall the estimate (3.6.2) of Coifman-Meyer (generalizing results of Calderon) and show how it leads to further commutator estimates for operators with C1-regular symbols. Then we use these commutator estimates to establish an operator calculus for symbols in C1S cl m . For this, Calderon’s estimates suffice, and much of the material of §4.2 is contained in [Ca2], [Ca3]. In §4.3, we look at a Gårding inequality, more precise, though less general, than the Gårding inequality in Proposition 2.4.B.
Michael E. Taylor

Chapter 5. Nonlinear hyperbolic systems

Abstract
In this chapter we treat various types of hyperbolic equations, beginning in §5.1 with first order symmetric hyperbolic systems. In this case, little direct use of pseudodifferential operator techniques is made, mainly an appeal to the Kato-Ponce estimates. We use Friedrichs mollifiers to set up a modified Galerkin method for producing solutions, and some of their properties, such as (5.1.43), can be approached from a pseudodifferential operator perspective. The idea to use Moser type estimates and to aim for results on persistence of solutions as long as the C1-norms remain bounded was influenced by [Mj]. We provide a slight sharpening, demonstrating persistence of solutions as long as the C1*-norm is bounded. In §5.2 we study two types of symmetrizable systems, the latter type involving pseudodifferential operators in an essential way. Here and in subsequent sections, including a treatment of higher order hyperbolic equations, we make strong use of the C1S cl m -calculus developed in Chapter 4.
Michael E. Taylor

Chapter 6. Propagation of singularities

Abstract
We present a proof of Bony’s propagation of singularities result for solutions to nonlinear PDE. As mentioned in the Introduction, we emphasize how C r regularity of solutions rather than Hn/2+r regularity yields propagation of higher order microlocal regularity, giving in that sense a slightly more precise result than usual. Our proof also differs from most in using S 1, δ m calculus, with δ < 1. This simplifies the linear analysis to some degree, but because of this, in another sense our result is slightly weaker than that obtained using B r S 1,1 m calculus by Bony and Meyer; see also Hörmander’s treatment [H4] using \( \tilde{S}_{{1,1}}^m \)calculus. Material developed in §3.4 could be used to supplement the arguments of §6.1, yielding this more precise result. In common with other approaches, our argument is modeled on Hörmander's classic analysis of the linear case.
Michael E. Taylor

Chapter 7. Nonlinear parabolic systems

Abstract
We examine existence, uniqueness, and regularity of solutions to nonlinear parabolic systems. We begin with an approach to strongly parabolic quasilinear equations using techniques very similar to those applied to hyperbolic systems in Chapter 5, moving on to symmetrizable quasilinear parabolic systems in §7.2. It turns out that another approach, making stronger use of techniques of Chapter 3, yields sharper results. We explore this in §7.3, treating there completely nonlinear as well as quasilinear systems. For a class of scalar equations in divergence form, we make contact with the DeGiorgi-Nash-Moser theory and show how some global existence results follow.
Michael E. Taylor

Chapter 8. Nonlinear elliptic boundary problems

Abstract
We establish estimates and regularity for solutions to nonlinear elliptic boundary problems. In §8.1 we treat completely nonlinear second order equations, obtaining L2-Sobolev estimates for solutions assumed a priori to belong to \({C^{2 + r}}(\overline M )\), r > 0. The analysis here is done on a quite general level, and extends readily to higher order elliptic systems, by amalgamating the nonlinear analysis in §8.1 with the approach to linear elliptic boundary problems taken in Chapter 5 of [T2]. In §8.2 we make note of improved estimates for solutions to quasilinear second order equations. In §8.3 we show how such results, when supplemented by the DeGiorgi-Nash-Moser theory, apply to solvability of the Dirichlet problem for certain quasilinear elliptic PDE.
Michael E. Taylor

Chapter 9. Extension of the Schauder estimates

Abstract
We describe here, from the point of view of the paradifferential calculus as used in Chapters 3 and 8, a refinement of the Schauder estimates due to Nirenberg. We also describe an important application of this to Monge-Ampere equations, given in [CNS].
Michael E. Taylor

Backmatter

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