## Introduction to PDE, Fall 1999 - Spring 2000.

A graduate course in Mathematics at the Department of Mathematics, National Tsing Hua University.

Instructor: Professor Wei-Cheng Wang,

Meeting: General III building, room 707, T5R5R6.

First Class: September 16, 1999

Grading: The grade will be based on weekly homework assignments. For those who take this course, you are encouraged to work on the homework assignment together, but everyone has to to write down his/her own version independently in the end.

Textbook and References: Textbook: L. C. Evans: Partial Differential Equations ; F. John: Partial Differential Equations. (Evans' will be the prime one)

Course description:

Part I : Difinition of PDE; Notation; Linear PDE's; Four impotatant linear PDE's: linear transport equation, Wave equation, Laplace equation and heat equation.

Part II : General techniques for solving linear PDE's: exact formula, reduction to ODE (method of characteristics), Fourier transform and Laplace transform. Separation of variables (Eigenfunction expansions), ...

Part III : Theory of Linear PDE: Generalized functions (distributions); Weak dereivatives, Weak solutions (distribution-valued solutions), Well posedness, Initial value problem, boundary value problem, Initial boundary value problem, Hilbert space method, Maximum principle, Energy method, Smooth approximations, Fourier analysis revisited, ...

Part V : Nonlinear PDE's: (Details later).

Lectures:

Week 1: Lecture 1.

Week 2: Lecture 2, Lecture 3.

Week 3: Lecture 4, Lecture 5.

Week 4: Lecture 6, Lecture 7.

Week 5: Lecture 8, Lecture 9.

Week 6: Lecture 10, Lecture 11. The uniqueness/nonuniqueness theorems for the heat equation, maximum principle for the heat equation. Local property of the fundamental solution, scaling argument. Stochastic interpretation of the Poisson and the heat equation.

Week 7 and on (tentative): Energy methods revisited. Finite speed of propagation for wave equation and nearly finite speed of propagation for transport diffusion equations. Alterntive proof for maximum principle. From examples to general theory: characteristic manifolds and the Cauchy problem. More equations (linear and nonlinear). Theory of distributions. Fourier analysis revisited. Weak derivatives and Sobolev spaces. Modern methods.

Homework assignments

Assignment 1, in plain text format.

Assignment 2, in plain text format.

Assignment 3, in plain text format.

Assignment 4, in plain text format.

Assignment 5, given 10/28/99 in plain text format.

Assignment 6, given 11/11/99 in plain text format.