MAT1061 PDE II, University of Toronto 2007-8
 

MAT 1061 Partial Differential Equations II (Spring 2008)

Robert Jerrard, Instructor

How to reach me: Robert Jerrard, 6236 Bahen Center, 8-5164.
rjerrard at math dot toronto dot edu.
Lectures: Mondays 3-4 in BA 1220, Tuesdays 12:40-2pm in BA 2135, and Fridays 11-12 in BA 3000
Office hours: Tuesday and Thursday 3-4pm.
Course description: This is a sequel to MAT 1061. Topics to be covered include linear and nonlinear equations of elliptic, parabolic, and hyperbolic type.
Text:   The first part of the course will cover most of Chapters 6-8 of "Partial Differential Equations", by Lawrence C. Evans.
AMS Graduate Studies in Mathematics, Vol. 19, ISBN 0-8218-0772-2.

The later part of the course will draw on material from a number of sources.

Evaluation: (tentative)
35% : 2 quizzes
20% : 4 hand-in homework sets
45% : Final Exam
The quiz questions will be drawn from a list of questions that will be updated as the semester progreses and that will be posted on the course web site. I will allow people to consult handwritten notes (in your own handwriting only!) while taking the first quiz. You are therefore encouraged to solve everything on the list of potential questions and to keep your solutions in a neat and well-organized format.

The first quiz will take place on Friday, February 8. List of possible questions for the 1st quiz

The second quiz will take place on Friday, April 11. List of possible questions for the 2nd quiz

Supplementary material:

Summary of existence results for 2nd order elliptic equations This summarizes some topics covered in Evans, Chapter 6.

notes on determinants This gives an alternative treatment of some topics covered in Evans, Section 8.1.4.

notes on nonlinear ground states. This considers a variational problem relevant to nonlinear Schrodinger equations.

The relevance of the Calculus of Variations to Evolution Equations . This discusses some ways in which the calculus of variations is relevant to evolutions equations. In particular, many parabolic equations can e seen as ``gradient flows'', and hence can be solved via discretization procedures that involve an iterative minimization procedure. And many wave and Schrodinger equations are formally the Euler-Lagrange equations for certain functionals. While it is not in general possible to use this fact to construct solutions by an variational argument, nonetheless ideas such as Noether's Principle, which establishes a general relationship between symmetries of an "action functional" and conserved quantities for the associated Euler-Lagrange equation, can provide useful information.

Parabolic interior regularity This elaborates on a remark made in Evans, that for parabolic equations one can prove interior regularity results analogous to those for elliptic equations.

Note on nonlinear wave equations (incomplete)