Why do people study partial differential equations? A compelling reason is for the applications. Indeed there are a vast range of processes in the real world which are described by partial differential equations. The basic theory of the subject begins with ideas from calculus and analysis, but very soon it is clear that geometry is needed. Conversely the theory of partial differential equations is a powerful tool in understanding geometry. Much of the modern theory of partial differential equations is driven by concrete applications to the real world and to geometry.

This quarter we cover some basic existence, uniqueness and regularity theory for linear elliptic partial differential equations. Roughly speaking the theory goes like this. Obviously if L is a second order linear partial differential operator and f is a function with partial derivatives up to order k, then the function Lf has partial derivatives up to order k-2. The special feature of elliptic operators is that they can in general be inverted, so that if g is a function with partial derivatives up to order k-2, then we can find a function f with partial derivatives up to order k so that Lf=g. There are several ways to make this statement precise - we will use Sobolev spaces. If you know the simple definition of an elliptic operator you can appreciate the power of the statement. It is saying that by knowing about some of the order k partial derivatives of a function, you can deduce the behavior of the rest of the order k partial derivatives.

This elliptic theory has a lot of applications. The Schrödinger operator is an elliptic operator, and the steady state solutions of the wave equation and the heat equation also satisfy elliptic equations. We will give some applications of elliptic theory to geometry. In fact we will begin the course with a famous non-linear elliptic problem which can be solved using this theory - the classical uniformization theorem.

Instructor:   Kate Okikiolu     okikiolu@math.ucsd.edu

Lectures: MWF 2-2:50 pm in APM 7421

Office Hours: F 4-5pm in AP&M 7426

Homework: There will be two homework sets due the last lecture.   Homework 1    Homework 2 

Text: "Partial Differential Equations" by Lawrence C. Evans published by the American Mathematical Society.

Other texts: "Elliptic Partial Differential Equations of Second Order" by D. Gilbarg and N.S. Trudinger.
"Partial differential Equations. Basic Theory" (="Partial Differential Equations. I.") by Michael E. Taylor.
"Partial differential Equations." by Jürgen Jost.

Syllabus: Sobolev spaces, Sobolev inequalities. Existence, uniqueness and regularity of solutions to linear elliptic systems. Boundary value problems. Elliptic equations on manifolds. Green's function. Non-linear geometric problem: The classical uniformization theorem for surfaces.

Prerequisites: MATH 20F, MATH 109, MATH 140. MATH 231A is not assumed.

Exams:


Grading:

Approximate Lecture Schedule (may be updated during the course): The day after each lecture, I might write a lecture summary which can be obtained by clicking the relevant section number.