Partial Differential Equations: Outline
ICE-EM / AMSI Summer School 2008

Faculty: Jerry L. Kazdan

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
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, 2nd 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.