MAT257 Analysis II, University of Toronto 2005-06

# MAT 1060 Partial Differential Equations I (Fall 2009)

## Almut Burchard, Instructor

How to reach me: Almut Burchard, 215 Huron # 1024, 6-4174.
almut @math , www.math.utoronto.ca/almut/
Lectures MWF 11:10-12noon, BA 6183
Office hours: Mon 1:30-3pm
Course description: This is a basic introduction to partial differential equations as they arise in physics, geometry and optimization. It is meant to be accessible to beginners with little or no prior knowledge of the field. It is also meant to introduce beautiful ideas and techniques which are part of most analysts' basic bag of tools. A key theme will be the development of techniques for studying non-smooth solutions to these equations.
Text:   "Partial Differential Equations", by Lawrence C. Evans,
AMS Graduate Studies in Mathematics, Vol. 19, ISBN 0-8218-0772-2.
Chapters 1-3 and 5; selected topics from Chapters 4, 8 and 9.

Evaluation:
50% : 6 hand-in homework sets
50% : Final Exam

## Tentative schedule:

Week 1 (September 9-11)
Chapter 1, Appendix A, Section 2.1.
W: What is a PDE? Notation, examples, well-posedness, classical solutions.
F: Transport equation. Initial-value problem.
Week 2 (September 14-18)
Appendix C and Section 2.2.
M: Laplace's and Poisson's equations. Fundamental solution.
W: Mean value property.
F: Maximum principle. Harmonic functions.
Week 3 (September 21-25)
Section 2.2.4, 8.8.1-3, and 2.3.1-2
M: Liouville-s theorem. Harnack inequality.
W: Green's function. Poisson's problem on the half-space.
Assignment 1 due.
F: Poisson's problem on the ball.
Week 4 (September 28-October 2)
Sections 2.3 and 9.2.1.
M: Brief excursion into the Calculus of Variations.
W: Heat equation. Fundamental solution.
F: Duhamel's formula. The parabolic maximum principle.
Week 5 (October 5-9)
Section 2.4.
M: The parabolic maximum principle on R^n. Uniqueness for the Cauchy problem.
W: One-dimensional wave equation. D'Alembert's formula.
Assignment 2 due.
F: Wave equation in higher dimensions. Spherical means. Kirchhoff's formula in three dimensions.
Week 6 (October 12-16)
Section 3.2
M: Thanksgiving.
W: Hadamard's method of descent. Solution in two dimensions. Loss of regularity.
F: Energy methods. Domain of dependence, domain of influence.
Week 7 (October 19-23)
Section 3.3
M: Nonhomogeneous wave equation.
W: Nonlinear first-order PDE. Characteristics. Boundary conditions.
Assignment 3 due.
F: Local solutions of nonlinear first-order PDE. Applications.
Week 8 (October 26-30)
Sections 3.3
M: Hamilton-Jacobi equations.
W: Hamiltonian and Lagrangian. Legendre transform.
F: Hopf-Lax formula.
Week 9 (November 2-6)
Section 3.4
M: Weak solutions, uniqueness.
W: Conservation laws.
Assignment 4 due.
F: Shocks and the Rankine-Hugoniot condition.
Week 10 (November 9-13)
Section 3.4
M: Nonuniqueness. Rarefaction waves. Entropy condition.
W: Weak solutions. Lax-Oleinik formula. Long-time behavior.
F: Fall break.
Week 11 (November 16-20)
Sections 5.1 and 5.2.
M: No lecture (to be rescheduled)
W: Hölder spaces and Sobolev spaces.
Assignment 5 due.
F: Weak derivatives. Smooth approximation.
Week 12 (November 23-27)
Sections 5.3. and 5.4
M: Extensions of Sobolev functions.
W: Traces of The Sobolev inequality for W^{1,p}.
F: Sobolev inequalities. Morrey's inequality.
Week 13 (November 30-December 4)
Sections 5.6 and 5.7.
M: Rellich-Kondrachev theorem and compact embeddings.
W: Poincaré inequalities.
F: Other spaces of functions.
Assignment 6 due.
Exam Week (December 7-11)
Sections 4.1-4.3
F: 2-5pm Final Exam, BA 6183