Class meets:MTh2 10:20-11:40, in HLL 525.
Office Hours: Tuesday, 4:00--5:00pm, or by appointment.
Email: zchan at math dot rutgers dot edu

Do not forget to "reload" the assignments pages - if you visited them before, your browser may be showing you only the old cached page. At the bottom portion of this document you can find dated information on this course, including notes and homework assignment.

Some General Comments: This is the first half of the year-long introductory graduate course on PDE. PDE is an enormously vast field, and for the entering students, it is probably more important to learn the methods and techniques than to know the most general results. Also, in order to cover enough topics to provide the students with a relatively wide perspective, we will not have the time to do thorough, fine treatment of some of the topics. My intention is to explain the key ideas in the lectures, leaving some of the verifications/generalizations to exercises. I would like to emphasize that it is crucial that students do enough computations/proofs on their own, instead of just listening to the lectures or reading through proofs in the books.

Text and Additional Books on Reserve: Our text for the course will be Partial Differential Equations: Second Edition, by Lawrence C. Evans, Publisher: American Mathematical Society; 2 edition (March 3, 2010), (# ISBN-10: 0821849743 # ISBN-13: 978-0821849743). I will also put the following books on reserve in the math library:

• Partial Differential Equations: Second Edition, by Emmanuele DiBenedetto. Publisher: Birkhäuser Boston; 2nd ed. edition (November 12, 2009). (# ISBN-10: 0817645519 # ISBN-13: 978-0817645519).
• Introduction to Partial Differential Equations, by G.B. Folland, Princeton University Press, 1995.
• Partial Differential Equations, 4th ed., by F. John, Springer-Verlag, 1982.
• Partial Differential Equations: An Introduction, by W. Strauss, John Wiley & sons, 2008.
• Partial differential equations by Jeffrey Rauch, Springer-Verlag, 1991.

Organization of the Material: The course can be thought of as developing in several stages. In the first stage, we will study the prototype PDEs arising from mathematical physics and other contexts: the Laplace equation, the standard heat equation, and the D'Alembert wave equation. Explicit representation formulas will be established and used to prove fundamental properties for solutions of each class of the equations. Other more general methods, such as the maximum principle and the energy methods, will also be introduced in this simple setting. As we will see, for more general equations such as those with variable coefficients or nonlinear ones, it may be impossible to obtain explicit representation formulas for solutions. However, as long as we have means to establish similar properties(estimates) for the solutions---the maximum principle/energy method will do in many situations, the construction/existence of the solutions in the more general situation can often be carried out in a way that is not too much different from the approaches in dealing with these prototype equations.

Homework Assignments and Grading Policy: The grade for the course will be based on a combination of graded homework problems, a midterm assignment, a take-home final exam. Here is the break up:

• Graded homework: 50%.
• Midterm and Final Exam: 25% each.

Dated Course Material

• Here are some notes for the first couple lectures.
• Assignment 1: Exercise 2 and parts (1) and (2) of Exercise 3 in the notes. Due date: September 20.
• Assignment 2: Exercises 5, 6 and 8 in the notes. Due date: September 27.
• Here are some notes on harmonic functions.
• Assignment 3: Problems 5, 9, 10 in 2.5 of Evans' text. Due date: October 18.
• Here are some notes on maximum principles.
• Here is the problem set for the take-home midterm exam, due at 10am, Nov. 8.
• Here are some notes on the fundamental solutions to the heat equation.
• Here are some notes on the Holder estimates for heat equation.
• Here is Assignment 4, due at 10 am on Dec. 2.