Chapter I
Theory of Distributions
Introduction to Chapters I, II, III
Distributions and their Fourier transforms are the basis of the theory of
linear partial differential equations. We present the main elements of the
theory of distributions in the first twelve sections accompanied by many ex-
amples. The Sobolev spaces in
Rn
are studied in §13 using the theory of
distributions. In §14 we introduce the notion of wave front sets of distribu-
tions and give some of its applications. In §16 we study the Cauchy problem
for the heat, Schr¨ odinger, and wave equations. We also consider the Dirich-
let problem for the Laplace and Helmholtz (the reduced wave) equations.
The fundamental solutions for all these equations are constructed. In §17 we
demonstrate the power of distribution theory by constructing a fundamen-
tal solution for an arbitrary linear partial differential equation with constant
coefficients. In §18 we describe the class of hypoelliptic equations, i.e., linear
partial differential equations whose distribution solutions are
C∞
functions.
In §19 the existence and uniqueness of the solution of the nonhomogeneous
Helmholtz’s equation is proved in the class of solutions satisfying the radi-
ation conditions. For this purpose we prove and use in §19 the stationary
phase lemma.
The last §20 is devoted to the study of simple and double layer potentials
in bounded domains. These potentials will be used in Chapter V (scattering
theory).
Sections 6, 15 and 21 are the problem sets. We use some problems
to supplement the content of the book. For example, we always construct
fundamental solutions using the Fourier transform because this approach
can be applied to more general equations. However, for the wave and the
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http://dx.doi.org/10.1090/gsm/123/01
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