Teaching / Course archive

Math 581: Winter 2014

PDE 2

Course Record

Institution: McGill
Term: Winter 2014
Level: graduate
Topics: PDE

Overview

PDE II course on heat kernels, Cauchy-Kovalevskaya theory, distributions, Fourier transform, constant coefficient operators, hypoellipticity, and pseudodifferential directions.

Schedule and Topics

Date	Topic
M 1/6	Euclidean heat kernel, parabolic maximum principles, and Tychonov uniqueness theorem.
M 1/27	Cauchy-Kovalevskaya theorem and Janet-Cartan isometric embedding theorem.
M 2/17	Test functions and distributions.
M 3/10	Complex-valued distributions and fundamental solutions.
W 3/19	Fourier transform and Paley-Wiener theorem.
W 3/26	Laurent expansion and Malgrange-Ehrenpreis theorem.
W 4/2	Hormander's theorem.
W 4/9	Petrowsky's theorem and fundamental solution of the Cauchy problem.

Topic List

Heat equation, maximum principles, heat kernel, and uniqueness theorems
Nirenberg-Nishida, Cauchy-Kovalevskaya, and Holmgren theorems
Distributions, convolution, and Fourier transform
Constant coefficient operators, fundamental solutions, and hypoellipticity
Malgrange-Ehrenpreis theorem and Hormander's characterization
Introduction to pseudodifferential operators, if time permits

Books and Related Pages

Books

Gerald B. Folland, Introduction to partial differential equations. Princeton, 1995.
Gregory I. Eskin, Lectures on linear partial differential equations. AMS, 2011.
Joseph Wloka, Partial differential equations. Cambridge, 1987.
Francois Treves, Basic linear partial differential equations. Dover, 2006.

Assignments and Projects

Assignments and Exams

Assignments 1-5.
Course project.

Projects

Y. Barsheshat: The Yamabe problem
S. Lu: Lin's local isometric embedding theorem
G. Martine-La Boissoniere: Approximations in quantum mechanics
M. Stevenson: The heat kernel for forms

Local Materials

Notes

Baire pdf
Distrib pdf

Course Files