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Taylor M.E. - Partial Differential Equations 2

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Название: Partial Differential Equations 2
Автор: Taylor M.E.
Категория: Математика
Тип: Книга
Дата: 31.12.2008 00:36:42
Скачано: 73
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Описание: In this chapter we discuss the basic theory of pseudodifferential operators as it has been developed to treat problems in linear PDE. We define pseudodifferential operators with symbols in classes denoted S™s, introduced by L. Hormander. In §2 we derive some useful properties of their Schwartz kernels. In §3 we discuss adjoints and products of pseudodifferential operators. In §4 we show how the algebraic properties can be used to establish the regularity of solutions to elliptic PDE with smooth coefficients. In §5 we discuss mapping properties on L2 and on the Sobolev spaces Hs. In §6 we establish Garding's inequality. In §7 we apply some of the previous material to establish the existence of solutions to hyperbolic equations. In §8 we show that certain important classes of pseudodifferential operators are preserved under the action of conjugation by solution operators to (scalar) hyperbolic equations, a result of Y. Egorov. We introduce the notion of wave front set in §9 and discuss the microlocal regularity of solutions to elliptic equations. We also discuss how solution operators to a class of hyperbolic equations propagate wave front sets. In §10 there is a brief discussion of pseudodifferential operators on manifolds. We give some further applications of pseudodifferential operators in the next three sections. In §11 we discuss, from the perspective of the pseudodifferential operator calculus, the classical method of layer potentials, applied particularly to the Dirichlet and Neumann boundary problems for the Laplace operator. Historically, this sort of application was one of the earliest stimuli for the development of the theory of singular integral equations. One function of §11 is to provide a warm-up for the use of similar integral equations to tackle problems in scattering theory, in §7 of Chapter 9. Section 12 looks at general regular elliptic boundary problems and includes material complementary to that developed in §11 of Chapter 5. In §13 we construct a parametrix for the heat equation and apply this to obtain an asymptotic expansion of the trace of the solution operator. This expansion will be useful in studies of the spectrum in Chapter 8 and in index theory in Chapter 10. Finally, in §14 we introduce the Weyl calculus. This can provide a powerful alternative to the operator calculus developed in §§1-6, as can be seen in [Ho4]
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