# MATH 424: Partial Differential Equations, Spring 2007

## Description:

A partial differential equation is an equation involving an unknown function of several variables and finitely many of its partial derivatives. The study of these began with the invention of calculus by Newton and Leibniz and developed to the present along with all the sciences, engineering, economics, etc. which have continually provided more equations and more important questions concerning the nature and accessibility of solutions. In this course we will focus on the mathematical theory of many important classes of  Nonlinear Partial Differential Equations.

In particular we will explore many equations coming from the calculus of variations where one considers functions, fields, surfaces,

etc. which are critical ( e.g. minima or maxima ) for some integral functionals. This property leads to an associated partial differential equation, called the Euler-Lagrange equation. We will consider some problems on existence including minimizers and mountain-pass type solutions. We also study several interesting examples of constrained problems arising in applications.

## Text:

Lawrence C. Evans , Partial Differential Equations, Methods and Applications,

American Math. Soc. Graduate studies vol.19.
This resource gives an excellent introduction to practically all major mathematical aspects of modern PDE, except numerical methods. In Math 424 we plan to cover chapter 8 and many parts of chapters 9 and 10.

## Prerequisites:

Here one should have some previous exposure to at least 400 level PDE.  For example either fall semester courses

Math 423 (D. Damanik) http://www.ruf.rice.edu/~dtd3/Ma423_Fall06

or CAAM 436 (W.Symes) http://www.caam.rice.edu/~caam436/

or CAAM 552 (D. Leykekhman) http://www.caam.rice.edu/~dmitriy/caam552/index.html would be sufficient.

Note that the topics we cover will be essentially distinct from the other Spring PDE course CAAM 437 (L. Borcea) so that students may usefully take both (and both courses will use the Evans book)

## Meets:

14:00-14:50 MWF in HB 427

## Instructor:

Robert Hardt

Office: HB 423; Office hours: 1-2 MTuW (and others by appt.)
Email: hardt@rice.edu
Telephone: ext 3280

## Homework:

There will be a homework assignment each week when there is not an hour exam scheduled. All homework is usually due each Wednesday, one week after it is assigned. The homework is not pledged. You are encouraged to discuss the homework, and to work together on the problems. However each student is responsible for the final preparation of his or her own homework papers.

I will be adding hints on this homepage to problems even after they are due. If you don't get a problem the first time, check the hint and give it another try. Then hand it in for some partial credit.

Current Homework (click pdf files)

Notes on Existence of a Minimizer