Preface to the first
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.
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