偏微分方程(一)

The goal of this course is to introduce to graduate students the theory of elliptic partial differential equations, as well as that of the parabolic partial differential equations. If time permits, we will also briefly sketch the surface of the theory of wave equations. Our discussion will start with a detailed study of the classical solutions to the Laplace's equation. The study of Laplace's equation is important in that Laplace's equation is a classical example which illustrates many of the basic properties as shared by more general elliptic PDE's. Once being armed with a solid understanding of solutions to the Laplace's equation, we will proceed to discuss solutions to more general elliptic PDE's. However, in our study of more general elliptic PDE's, we will find that it could be very helpful for us to talk about solutions in the weak sense instead of the more traditional sense of classical smoothness. Since the notion of weak solutions to PDE's currently plays a dominant role in the research and study of PDE's, the focus of our disscussion will be switched to the notion of weak solutions to an elliptic PDE. Next, we will study the heat equation, which is a classical example of more general parabolic type PDE's. We will use the heat equation to illustrate important properties of solutions to parabolic type PDE's, such as the maximum principle, and the gradient estimate to a solution of a parabolic PDE. In this course, we will not have time to discuss first order PDE's such as the Burger's equation and the theory of Hamilton-Jacobi equations. The study of Hamilton-Jacobi equations is difficult since it originated from the development of the framework of classical mechanics since the time of Newton, Euler, Lagrange, Hamilton and Jacobi. Another difficulty in the learning of the subject of Hamilton-Jacobi equations comes from the fact that the geometric and physical-mechanical aspect of a Hamilton-Jacobi equation is now described in terms of the modern language of differential geometry. So, students who are interested in the study of first order PDE's should consider to take graduate level PDE course in the spring semester.

A student taking this course should have a solid understanding and working knowledge of undergraduate level analysis, as well as that of linear algebra. Moreover, a student taking this course should have some basic knowledge of multi-variable calculus. Techniques from analysis, linear algebra, and multi-variable caluclus will be invoked during class lectures whenever the need arises.

The grading policy will be based on the following: Homework assignments will contribute 20% towards the final grade. A take home final exam counts 10% towards the final grade. Attendance record counts 40% towards the final grade. A final written report on a selected topic and a presentation based on the written report will count 30% towards the final grade.

There is no required textbook for the course. However, the following classical books are good references. 1. Partial Differential Equations, by Evans, published by AMS. 2. Elliptic Partial Differential Equations by Gilbarg and Trudinger.