It is generally well known that the Fourier-Laplace transform converts a linear constant coefficient PDE P(D)u=f on Rn to an equation P(§)u-(§)=f-(§), for the transforms u-, f- of u and f,so that solving P(D)u=f just amounts to division by the polynomial P(§). The practical application was suspect, and ill understood, however, until theory of distributions provided a basis for a logically consistent theory. Thereafter it became the Fourier-Laplacemethod for solving initial-boundary problems for standard PDE. We recall these facts in some detail in sec's 1-4 of ch.0.
【目录】
Chapter 0. Introductory discussions
0.0. Some special notations, used in the book
0.1. The Fourier transform; elementary facts
0.2. Fourier analysis for temperate distributions on Rn
0.3. The Paley-Wiener theorem; Fourier transform for general u∈D''
0.4. The Fourier-Laplace method; examples
0.5. Abstract solutions and hypo-ellipticity
0.6. Exponentiating a first order linear differential operator
0.7. Solving a nonlinear first order partial differen-tial equation
0.8. Characteristics and bicharacteristics of a linear PDE
0.9. Lie groups and Lie algebras for classical analysts
Chapter 1. Calculus of pseudodifferential operators
1.0. Introduction
1.1. Definition of do''s
1.2. Elementary properties of do''s
1.3. Hoermander symbols; Weyl do''s; distribution kernels
1.4. The composition formulas of Beals
1.5. The Leibniz'' formulas with integral remainder
1.6. Calculus of do''s for symbols of Hoermander type
1.7. Strictly classical symbols; some lemmata for application
Chapter 2. Elliptic operators and parametrices in Rn
2.0. Introduction
2.1. Elliptic and md-elliptic do''s
2.2. Formally hypo-elliptic do''s
2.3. Local md-ellipticity and local md-hypo-ellipticity
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