Two general questions regarding partial differential equations are
explored in detail in this volume of the Encyclopaedia. The first
is the Cauchy problem, and its attendant question of well-posedness
(or correctness). The authors address this question in the context
of PDEs with constant coefficients and more general convolution
equations in the first two chapters. The third chapter extends a
number of these results to equations with variable coefficients.
The second topic is the qualitative theory of second order linear
PDEs, in particular, elliptic and parabolic equations. Thus, the
second part of the book is primarily a look at the behavior of
solutions of these equations. There are versions of the maximum
principle, the Phragmen-Lindel]f theorem and Harnack's inequality
discussed for both elliptic and parabolic equations. The book is
intended for readers who are already familiar with the basic
material in the theory of partial differential equations.
General
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