Teaching / Course archive

Math 580: Fall 2013

PDE 1

Course Record

Institution: McGill
Term: Fall 2013
Level: graduate
Topics: PDE

Overview

PDE I course centered on second order linear equations, with harmonic functions, Dirichlet problems, Sobolev spaces, weak solutions, regularity, spectral theory, and heat equation examples.

Schedule and Topics

Date	Topic
W 9/4	Green's identities and fundamental solutions.
M 9/9	Green's formula, mean value property, Harnack inequality, and maximum principles.
W 9/25	Harnack's inequality, Heine-Borel property, and the Dirichlet problem.
M 10/7	Strong derivatives, Sobolev spaces, and weak solutions.
W 10/16	Mollifiers, Du Bois-Reymond lemma, Leibniz rule, and Weyl's lemma.
W 10/23	Trace maps, Poisson equations, and variational method.
W 10/30	Moduli of continuity and interior regularity.
W 11/20	Rellich's lemma and Hilbert-Schmidt theory.
W 11/27	Courant's nodal domain theorem, minimax principle, and comparison theorems.
M 12/2	Weyl's law and functional calculus.

Topic List

Green's identities, harmonic functions, Harnack inequality, and fundamental solutions
Green's function, Poisson formula, and the Dirichlet problem
Sobolev spaces, weak and strong derivatives, and variational methods
Elliptic regularity, Sobolev embedding, and Rellich compactness
Laplace eigenvalues, eigenfunctions, Weyl's law, and heat equation examples

Books and Related Pages

Books

Lawrence C. Evans, Partial differential equations. AMS, 1998.
Fritz John, Partial differential equations. Springer, 1982.
Jurgen Jost, Partial differential equations. Springer, 2007.
Qing Han, A basic course in partial differential equations. AMS, 2011.

Assignments and Projects

Assignments and Exams

Assignments 1-3.
Take-home midterms 1-2.
Final exam.

Local Materials

Notes

Dirichlet pdf
Harmonic pdf
Poisson pdf

Course Files