Preface

This book is based on the lectures on partial differential equations that I

have given for many years at UCLA. It does not assume any knowledge

of partial differential equations and can be considered as a first graduate

course in linear PDE. However, some basic knowledge of the Fourier trans-

form, Lebesgue integrals and elementary functional analysis is required. It

is organized as lecture notes with emphasis on clarity and accessibility.

We shall briefly describe the content of the book. The first three chap-

ters are the elementary theory of distributions and Fourier transforms of

distributions with applications to partial differential equations with con-

stant coeﬃcients. It is similar to the first chapters of the books by Gelfand

and Shilov [GSh] and Shilov [Sh]. Additional material includes the wave

front sets of distributions, Sobolev spaces, the stationary phase lemma, the

radiation conditions, and potential theory.

In Chapter IV the Dirichlet and the Neumann boundary value problems

are considered for second order elliptic equations in a smooth bounded do-

main. The existence, uniqueness, and regularity of solutions are proven. A

nontraditional topic of this chapter is the proof of the existence and unique-

ness of the solutions of the Neumann and Dirichlet problems for homoge-

neous equations in Sobolev spaces of negative order on the boundary.

Chapter V is devoted to scattering theory including inverse scattering,

inverse boundary value problem, and the obstacle problem.

Chapter VI starts with the theory of pseudodifferential operators with

classical symbols. It is followed by the theory of parabolic Cauchy problems

based on pseudodifferential operators with symbols analytic in the half-plane

and heat kernel asymptotics.

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