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