* Partial Differential Equations: Outline *

ICE-EM / AMSI Summer School 2008

**Faculty**: Jerry L. Kazdan

email: `kazdan@math.upenn.edu`
Partial Differential Equations (PDEs) arise in many applications to
physics, geometry, and more recently the world of finance. This will
be a basic course.

In real life one can find explicit solutions of very few PDEs -- and
many of these are infinite series whose secrets are complicated to
extract. For more than a century the goal is to understand the
solutions -- even though there may not be a formula for the solution.

The historic heart of the subject (and of this course) are the three
fundamental linear equations: *wave equation, heat equation,
and Laplace equation * along with a few nonlinear equations
such as the minimal surface equation and others that arise from
problems in the calculus of variations.

*We seek insight and understanding rather than complicated formulas.*

**Prerequisites:** Linear algebra, calculus of several variables,
and basic ordinary differential equations. In particular I'll assume
some experience with the Stokes' and divergence theorems and a bit of
Fourier analysis. Previous acquaintantance with normed linear spaces
will also be assumed. Some of these topics will be reviewed as
needed.

** References:** For this course, the most important among the
following are the standard general texts by Strauss and Evans.

**A First Course:**

John, Fritz. *Partial Differential Equations*, 4th ed., Series:
Applied Mathematical Sciences, New York, NY: Springer-Verlag.

Strauss, Walter A., *Partial Differential Equations: An
Introduction*, New York, NY: Wiley, 1992.
**More Advanced**

Axler, S., Bourdin, P., and Ramey, W., *Harmonic Function Theory*,
accessible at
http://www.axler.net/HFT.pdf

Courant, Richard, and Hilbert, David, *Methods of Mathematical
Physics*, vol II. Wiley-Interscience, New York, 1962.

Evans, L.C., *Partial Differential Equations*, American
Mathematical Society, Providence, 1998.

Gilbarg, D., and Trudinger, N. S., *Elliptic Partial
Differential Equations of Second Order*, 2^{nd} Edition,
Springer-Verlag, 1983.

Jost, J., *Partial Differential Equations*, Series: Graduate
Texts in Mathematics, Vol. 214 . 2nd ed., 2007, 356 pp.

Kazdan, Jerry, Lecture Notes:
Applications of Partial Differential Equations to Some Problems in
Differential Geometry.