This self-contained text offers an elementary introduction to partial
differential equations (pdes), primarily focusing on linear equations,
but also providing some perspective on nonlinear equations. The classical
treatment is mathematically rigorous with a generally theoretical layout,
though indications to some of the physical origins of pdes are made
throughout in references to potential theory, similarity solutions for
the porous medium equation, generalized Riemann problems, and others.

The material begins with a focus on the Cauchy.Kowalewski theorem, dis-
cussing the notion of characteristic surfaces to classify pdes. Next, the
Laplace equation and connected elliptic theory are treated, as well as
integral equations and solutions to eigenvalue problems. The heat equation
and related parabolic theory are then presented, followed by the wave
equation in its basic aspects. An introduction to conservation laws, the
uniqueness theorem, viscosity solutions, ill-posed problems, and nonlinear
equations of first order round out the key subject matter.

Large parts of this revised second edition have been streamlined and
rewritten to incorporate years of classroom feedback, correct errors, and
improve clarity. Most of the necessary background material has been
incorporated into the complements and certain nonessential topics have been
given reduced attention (noticeably, numerical methods) to improve the flow
of presentation.

The exposition is replete with examples, problems and solutions that
complement the material to enhance understanding and solidify comprehension.
The only prerequisites are advanced differential calculus and some basic Lp
theory. The work can serve as a text for advanced undergraduates and graduate
students in mathematics, physics, engineering, and the natural sciences, as
well as an excellent reference for applied mathematicians and mathematical
physicists.

From Reviews of the First Edition:

"The author's intent is to present an elementary introduction to pdes...
In contrast to other elementary textbooks on pdes...much more material is
presented on the three basic equations: Laplace's equation, the heat and wave
equations...The presentation is clear and well organized...The text is
complemented by numerous exercises and hints to proofs."

http://www.amazon.com/Differential-Equations-Cornerstones-Emmanuele-Dibenedetto/dp/0817645519

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