AM 223-224 (MA 237-238) is the core graduate sequence on partial
differential equations. In the first semester, the focus is
on the basic theory of linear equations and
quasilinear equations. This forms a basis for the study of nonlinear
equations in the second semester. The topics for the first semester
will include:
The classical theory of Laplace's equation, the heat equation
and the wave equation.
The theory of distributions and weak solutions.
First order equations.
The treatment is for the most part rigorous and self-contained. We
will begin with explicit solution formulas and work towards a general theory.
Prerequisites: Real analysis (AM 211, MA 221 or equivalent). A
good rigorous
introduction to advanced calculus will suffice for the most part, if
you are willing to take some things on faith. Strictly
speaking, no prior exposure to PDE is necessary, but of course it
would help.
Textbooks: There are several excellent texts on partial
differential equations, each with a different perspective. John's book
includes a lot of the material for the first semester, but not all. If
you plan to take both semesters, you should invest in Evans' book. The
subject is vast, and browsing is strongly encouraged.
F. John, Partial differential equations , Springer 1981.
L. C. Evans, Partial differential equations , AMS 1998.
J. Rauch, Partial differential equations ,
Springer, 1991.
D. Gilbarg and N. Trudinger, Elliptic partial differential
equations of second order , Springer 2001.
Lecture notes
I typed up lecture notes for this class until I ran out of steam. Here
is the current version of lectures on
Laplace's equation and the heat equation.
The remainder of the course is available
online thanks to Andreas
Kloeckner's real-time(!)
TeX transcription of lectures.
However, this has not been proofread, and I am not sure I will clean
it up until I teach the class again.