MAT257 Analysis II, University of Toronto 200506
MAT 1060 Partial Differential Equations I (Fall 2009)
Almut Burchard, Instructor
How to reach me: Almut Burchard, 215 Huron # 1024,
64174.
 almut @math ,
www.math.utoronto.ca/almut/
 Lectures MWF 11:1012noon, BA 6183
 Office hours: Mon 1:303pm
Teaching assistant: Ehsan Kamalinejad,
ehsan.kamainejad @gmail.com .
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 nonsmooth solutions to these equations.
Text:
"Partial Differential Equations", by Lawrence C. Evans,
AMS Graduate Studies in Mathematics, Vol. 19, ISBN 0821807722.
Chapters 13 and 5; selected topics from Chapters 4, 8 and 9.
Evaluation:
 50% : 6 handin homework sets
 50% : Final Exam
Tentative schedule:
Week 1 (September 911)
Chapter 1, Appendix A, Section 2.1.
 W: What is a PDE?
Notation, examples, wellposedness,
classical solutions.
 F: Transport equation. Initialvalue problem.
Week 2 (September 1418)
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 2125)
Section 2.2.4, 8.8.13, and 2.3.12
 M:
Liouvilles theorem. Harnack inequality.
 W: Green's function.
Poisson's problem on the halfspace.
Assignment 1 due.
 F: Poisson's problem on the ball.
Week 4 (September 28October 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 59)
Section 2.4.

M: The parabolic maximum principle on R^n.
Uniqueness for the Cauchy problem.
 W: Onedimensional 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 1216)
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 1923)
Section 3.3
 M:
Nonhomogeneous wave equation.
 W:
Nonlinear firstorder PDE.
Characteristics. Boundary conditions.
Assignment 3 due.
 F:
Local solutions of nonlinear firstorder PDE. Applications.
Week 8 (October 2630)
Sections 3.3
 M:
HamiltonJacobi equations.
 W:
Hamiltonian and Lagrangian. Legendre transform.
 F:
HopfLax formula.
Week 9 (November 26)
Section 3.4

M:
Weak solutions, uniqueness.
 W:
Conservation laws.
Assignment 4 due.

F:
Shocks and the RankineHugoniot condition.
Week 10 (November 913)
Section 3.4
 M:
Nonuniqueness. Rarefaction waves. Entropy condition.
 W: Weak solutions.
LaxOleinik formula. Longtime behavior.
 F: Fall break.
Week 11 (November 1620)
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 2327)
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 30December 4)
Sections 5.6 and 5.7.

M: RellichKondrachev theorem and compact embeddings.
 W: Poincaré inequalities.
 F: Other spaces of functions.
Assignment 6 due.
Exam Week (December 711)
Sections 4.14.3

 F: 25pm Final Exam, BA 6183