Chapter 1

INTRODUCTION

1.1 Partial diﬀerential equations

1.2 Examples

1.3 Strategies for studying PDE

1.4 Overview

1.5 Problems

1.6 References

This chapter surveys the principal theoretical issues concerning the solv-

ing of partial diﬀerential equations.

To follow the subsequent discussion, the reader should ﬁrst of all turn

to Appendix A and look over the notation presented there, particularly the

multiindex notation for partial derivatives.

1.1. PARTIAL DIFFERENTIAL EQUATIONS

A partial diﬀerential equation (PDE) is an equation involving an unknown

function of two or more variables and certain of its partial derivatives.

Using the notation explained in Appendix A, we can write out symbol-

ically a typical PDE, as follows. Fix an integer k ≥ 1 and let U denote an

open subset of

Rn.

DEFINITION. An expression of the form

(1) F

(Dku(x), Dk−1u(x),

. . . , Du(x), u(x), x) = 0 (x ∈ U)

is called a kth-order partial diﬀerential equation, where

F :

Rnk

×

Rnk−1

× · · · ×

Rn

× R × U → R

1