Class meets: MTh 10:20-11:40am, Hill-425.
Office Hours: TTh9-10am, or by appointment.
Email: zchan at math dot rutgers dot edu
Canvas: The course will use Canvas for all material during the semester. All enrolled students should have automatic access to the site after logging in to Canvas. Current information about syllabus, homework assignments and exams will be found there.

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 some methods and techniques through studying some prototype equations ---to build up experience and intuition--- than to learn the most general theories. This first semester course will be more basic, emphasizing the study of constructions and properties of solutions using relatively elementary tools; but the approaches that we will take will motivate and naturally lead to more sophisticated generalizations in more advanced treatment.

In order to cover enough topics to provide the students with a relatively wide perspective, we will not have time to do thorough 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: I will mostly follow my lecture notes prepared and uploaded to the canvas site. It is important that a graduate student does not limit his or her study from a single source: there are often different approaches to the same problem, and different approaches may be adapted more easily in different contexts. I will put the following books on reserve in the math library. Some of the Springer books have free electronic versions via the Rutgers Springer Mathematics E-books package---Rutgers affiliated users can also order each print copy for $24.95 (Shipping and handling are included):

• 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).
• 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.
• Lectures on elliptic and parabolic equations in Hölder spaces, by N. V. Krylov, Providence, R.I. : American Mathematical Society, 1996.
• Partial Differential Equations: An Introduction, by W. Strauss, John Wiley & sons, 2008.
• Partial differential equations by Jeffrey Rauch, Springer-Verlag, 1991.
• A Course on Partial Differential Equations by Walter Craig, AMS, 2018.
• Partial Differential Equations: An Accessible Route Through Theory and Applications by Andras Easy, AMS, 2015.

Organization of the Material: The course can be thought of as developing in several stages. At 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---easily adapted to more general situations, such as the maximum principle and the energy methods, will also be introduced in this simple setting.

At the next stage, we will begin to adapt some of our earlier approaches to more general settings. 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: 60%.
• Midterm and Final Exam: 20% each.

Collaboration on homework is encouraged, but students must write up and turn in their solutions individually. If you work with another student on a problem, please acknowledge him or her in your solution. You are also welcome to stop by my office to report/discuss ideas for solving the problems. Written work should be neat and labelled clearly, and are due strictly by the announced deadline, unless a late submission date has been approved by me in advance. If there is demand, I am willing to hold occasional problem sessions.

The midterm assignment and the the take-home final exam have the format of the regularly assigned problems (perhaps slightly expanded and more comprehensive), and serve to provide some kind of closure on certain parts of our course development.