Is it really necessary to classify partial differential equations (PDEs) and to
employ different methods to discuss different types of equations? Why is it
important to derive a priori estimates of solutions before even proving the
existence of solutions? These are only a few questions any students who
just start studying PDEs might ask. Students may find answers to these
questions only at the end of a one-semester course in basic PDEs, sometimes
after they have already lost interest in the subject. In this book, we attempt
to address these issues at the beginning. There are several notable features
in this book.
First, the importance of a priori estimates is addressed at the beginning
and emphasized throughout this book. This is well illustrated by the chapter
on first-order PDEs. Although first-order linear PDEs can be solved by
the method of characteristics, we provide a detailed analysis of a priori
estimates of solutions in sup-norms and in integral norms. To emphasize the
importance of these estimates, we demonstrate how to prove the existence
of weak solutions with the help of basic results from functional analysis.
The setting here is easy, since
are needed only. Meanwhile, all
important ideas are in full display. In this book, we do attempt to derive
explicit expressions for solutions whenever possible. However, these explicit
expressions of solutions of special equations usually serve mostly to suggest
the correct form of estimates for solutions of general equations.
The second feature is the illustration of the necessity to classify second-
order PDEs at the beginning. In the chapter on general second-order linear
PDEs, immediately after classifying second-order PDEs into elliptic, para-
bolic and hyperbolic type, we discuss various boundary-value problems and
initial/boundary-value problems for the Laplace equation, the heat equation
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