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
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.