Lectures: MWF 2:00-2:50 a.m. in APM 5829
Instructor: Kate Okikiolu   email: okikiolu@math.ucsd.edu
Office Hours: Will take the form of weekly problem sessions, Monday 5-6pm place TBA.   For other questions, use email or ask after class on Fridays.
Text : Partial Differential Equations by Lawrence C. Evans, Graduate Studies in Mathematics 19, American Mathematical Society
231A Syllabus:Chapters 2 and 3: Solution of classical Laplace, heat and wave equations, and non-linear first order equations.
Grading:Homework and/or Class Presentations

Homework 1, due 10/03    Homework 2, due 10/10    Selected Solutions 2,    Homework 3, due 10/17    Homework 4, due 10/24    Homework 5, due 10/31 (There will be no problem session on 10/31)    Homework 6, due 11/07    Homework 7, due 11/14   Homework for 11/21: This homework is review. Please go through your solutions and email me with those problems which you still have a problem with. If any problem gets more than one vote, we will go over it in class. I will post solutions to some of the problems with one vote on the web. There will be no problem session on 11/21.    Some Solutions

Partial differential equations (PDE) are used to model a vast array of "real life" phenomena. They are fundamental in many areas of physics, for example they describe all kinds of fluid flow, heat flow and wave phenomena. They are used in engineering, for example to describe vibrational and acoustic phenomena, and in bioengineering. They are used in economics and on Wall Street. Sometimes the emphasis in applications is to solve PDE numerically, but even in very practical investigations, some knowledge of the theory is important.

Questions about PDE motivated many of the developments in real and functional analysis, and PDE also occur as important tools or objects of study in other fields of pure mathematics, such as geometry, topology, complex analysis, dynamical systems and probability theory.

Given the huge range of subjects in which PDE occur, you might expect the field itself to be rather large and to employ techniques from many different areas, and indeed this is the case. This course, however, is intended as a standard introduction to the general theory of PDE. In the first quarter we will solve some classical second order linear equations: the Laplace, heat and wave equations. These are the simple examples of elliptic, parabolic and hyperbolic equations respectively. Many equations which arise in applications fall into these general categories and exhibit behavior somewhat similar to the classical examples. We will also solve general non-linear first order equations. In the second quarter we will go further into the theory of general second order linear equations. This theory is very well developed, and although the cutting edge research now mostly involves classes of non-linear equations which are important for applications, one often solves a non-linear equation by approximation around a linear equation, so the linear techniques are still fundamental.

In the third quarter we will either study some non-linear elliptic PDE in geometry, or study scattering theory, depending on the interests of the participants.

Lecture Schedule: Most of the time lecture notes will be posted the day after the lecture.

wk	date	Monday	Wednesday	Friday
0	09/19	-	-	Fourier series solution for the vibrating string
1	09/26	Classifying equations	Linear Transport	ODE
2	10/03	Laplacian	Fundamental Solution	Green's function
3	10/10	Poisson Kernel	Mean Values	Uniqueness
4	10/17	Fourier Transform	Fourier Inversion	Heat Equation
5	10/24	Duhamel's Principle	Maximum Principle	Energy Methods
6	10/31	Smoothness of solutions	Wave Equation	3 Dimensions
7	11/07	Method of Descent	Odd Dimensions	Holiday
8	11/14	Existence	Nonhomogeneous Wave Equation	Energy for Uniqueness
9	11/21	Cauchy-Kowalewski	No Class	Holiday
10	11/28	Cauchy-Kowalevskaya	Cauchy-Kowalevskaya	Cauchy-Kowalevskaya (by Michael Hansen)

Math Department Syllabus for 231A-B-C: Existence and uniqueness theorems. Cauchy-Kowalewski theorem, first order systems. Hamilton- Jacobi theory, initial value problems for hyperbolic and parabolic systems, boundary value problems for elliptic systems. Green's function, eigenvalue problems, perturbation theory.

Other References: