Office: 383M

Phone: 723-2226

E-mail: andras "at" math.stanford.edu

Office hours: M 2-3, W 10:30-11:30 and W 1:30-3, from week 2. During week 1, office hour on M 10:30-11:30 and Friday 2-3pm.

Course Assistant: Jun Gao

E-mail: Jun Gao jung2 "at" stanford.edu

Office: 380N

Office hours: T3-6pm, W5-6pm, F3:30-5:30pm.

Class location: MWF 9:30-10:20 pm, Room 380-380D.

Textbook: due to the availability of lecture notes, the following are all `recommended'.

- Strauss' `Partial Differential Equations: An introduction' covers most topics, but the course is at a higher level, especially regarding first order PDE's, which is the first major topic covered, as well as distributions and the Fourier transform.
- Evans' `Partial Differential Equations' is a more advanced text, and it covers course topics not dealt with in Strauss' book.
- Pinchover and Rubinstein's `An introduction to partial differential equations'. This is a fairly good match for the level of difficulty of the course, though does not necessarily cover the same topics/the same way.
- All of these, as well as John's `Partial Differential Equations' will be on reserve at the Math library.

The running syllabus will change somewhat, corresponding to the extra lectures stated above, but should give an indication of the scope and speed of the course.

This course is similar to Math 220, but designed for undergraduate math majors.

It is the continuation of the honors analysis course 171, emphasizing rigorous (i.e. logically careful) proofs, in the spirit of 171.

The knowledge of measure theory, as presented in 172, is NOT a prerequisite of the class, though L^p spaces will be discussed as completions. The Fourier analysis part of 172 will be covered in this class.

Grading policy: The grade will be based on the weekly homework (25%), on the in-class (expected in the usual classroom, roughly at the usual class time, but with an extra half an hour, so perhaps at 9am) midterm exam (35%) and on the take home final exam (40%). The take home exam will be due at the end of the period of the regular exam for this time slot (11:30am on Tuesday, March 15), and will be available about a week before the deadline.

The homework will be due either in class or by 9am in the instructor's mailbox on the designated day, usually Thursdays. You are allowed to discuss the homework with others in the class, but you must write up your homework solution by yourself. Thus, you should understand the solution, and be able to reproduce it yourself. This ensures that, apart from satisfying a requirement for this class, you can solve the similar problems that are likely to arise on the exams.

- Introduction to PDE.
- Where do PDE come from?
- First order scalar semilinear equations.
- First order scalar quasilinear equations.
- Distributions and weak derivatives.
- Second order constant coefficient PDE.
- Properties of solutions of second order PDE.
- The Fourier transform -- basic properties and the inversion formula.
- The Fourier transform -- tempered distributions.
- PDEs and boundaries.
- Duhamel's principle.
- Separation of variables.
- Inner product spaces, symmetric operators, orthogonality.
- Convergence of the Fourier series.
- Solving PDEs.

- Problem Set 1, due Thursday, January 14, 9am. Solutions.
- Problem Set 2, due Thursday, January
21, 9am.
#### Postponed to Friday, January 22, 10am (in class).

Solutions. - Problem Set 3, due Thursday, January 28, 2pm. Solutions.
- Problem Set 4, due Monday, February 8, 10am. Solutions.
- Problem Set 5, due Friday, February
12, 10am.
#### Problem 6 delayed until Problem Set 6!

Solutions. - Problem Set 6, due Friday, February
19, 10am.
#### The constant in Problem 6, part ii, iv, in front of the exponential should be pi to the power -n/4; this is updated in the currently posted version.

Solutions. - Problem Set 7, due Friday, February
26, 10am.
#### Due to instructor unavailability, the problem set is postponed to Monday, February 29, 10am. Problem Set 8 will still be due on Friday, March 4.

Solutions. - Problem Set 8, due Friday, March 4, 10am. Solutions.
- Problem Set 9, due Wednesday, March 9, 10am. Solutions.