xx Preface to the ﬁrst edition

c. Understanding generalized solutions is fundamental. Many of the

partial diﬀerential equations we study, especially nonlinear ﬁrst-order equa-

tions, do not in general possess smooth solutions. It is therefore essential to

devise some kind of proper notion of generalized or weak solution. This is

an important but subtle undertaking, and much of the hardest material in

this book concerns the uniqueness of appropriately deﬁned weak solutions.

d. PDE theory is not a branch of functional analysis. Whereas

certain classes of equations can proﬁtably be viewed as generating abstract

operators between Banach spaces, the insistence on an overly abstract view-

point, and consequent ignoring of deep calculus and measure theoretic esti-

mates, is ultimately limiting.

e. Notation is a nightmare. I have really tried to introduce consistent

notation, which works for all the important classes of equations studied.

This attempt is sometimes at variance with notational conventions within a

given subarea.

f. Good theory is (almost) as useful as exact formulas. I incorporate

this principle into the overall organization of the text, which is subdivided

into three parts, roughly mimicking the historical development of PDE the-

ory itself. Part I concerns the search for explicit formulas for solutions, and

Part II the abandoning of this quest in favor of general theory asserting

the existence and other properties of solutions for linear equations. Part III

is the mostly modern endeavor of fashioning general theory for important

classes of nonlinear PDE.

Let me also explicitly comment here that I intend the development

within each section to be rigorous and complete (exceptions being the frankly

heuristic treatment of asymptotics in §4.5 and an occasional reference to a

research paper). This means that even locally within each chapter the topics

do not necessarily progress logically from “easy” to “hard” concepts. There

are many diﬃcult proofs and computations early on, but as compensation

many easier ideas later. The student should certainly omit on ﬁrst reading

some of the more arcane proofs.

I wish next to emphasize that this is a textbook, and not a reference

book. I have tried everywhere to present the essential ideas in the clearest

possible settings, and therefore have almost never established sharp versions

of any of the theorems. Research articles and advanced monographs, many

of them listed in the Bibliography, provide such precision and generality.

My goal has rather been to explain, as best I can, the many fundamental

ideas of the subject within fairly simple contexts.