Preface to the ﬁrst

edition

I present in this book a wide-ranging survey of many important topics in

the theory of partial diﬀerential equations (PDE), with particular emphasis

on various modern approaches. I have made a huge number of editorial

decisions about what to keep and what to toss out, and can only claim

that this selection seems to me about right. I of course include the usual

formulas for solutions of the usual linear PDE, but also devote large amounts

of exposition to energy methods within Sobolev spaces, to the calculus of

variations, to conservation laws, etc.

My general working principles in the writing have been these:

a. PDE theory is (mostly) not restricted to two independent vari-

ables. Many texts describe PDE as if functions of the two variables (x, y)

or (x, t) were all that matter. This emphasis seems to me misleading, as

modern discoveries concerning many types of equations, both linear and

nonlinear, have allowed for the rigorous treatment of these in any number

of dimensions. I also ﬁnd it unsatisfactory to “classify” partial diﬀerential

equations: this is possible in two variables, but creates the false impression

that there is some kind of general and useful classiﬁcation scheme available

in general.

b. Many interesting equations are nonlinear. My view is that overall

we know too much about linear PDE and too little about nonlinear PDE. I

have accordingly introduced nonlinear concepts early in the text and have

tried hard to emphasize everywhere nonlinear analogues of the linear theory.

xix