Math 580 Fall 2011 | Tsogtgerel Gantumur

Course Record

Institution: McGill
Term: Fall 2011
Level: graduate
Topics: PDE

Overview

PDE I course on classification and well-posedness, energy methods, Dirichlet principle, distributions, weak derivatives, harmonic functions, heat equations, wave equations, and scalar conservation laws.

Schedule and Topics

Date	Topic
Sep 1	Examples of PDE.
Sep 13	Cauchy-Kovalevskaya theorem.
Sep 20	First order semilinear equations and method of characteristics.
Sep 27	Weak solutions, Riemann problem, shock waves, Rankine-Hugoniot condition, and entropy conditions.
Oct 4	Green's formula, mean value property, maximum principles, and comparison principle.
Oct 13	Removable singularity theorem, Harnack inequalities, and Harnack convergence theorems.
Oct 25	Poisson equation and Newtonian potential.
Nov 8	Global Schauder estimates and semilinear examples.
Nov 24	Nonlinear reaction-diffusion equations and maximum principles.
Dec 1	Tychonov's example, Kirchhoff's formula, and wave energy.

Topic List

Cauchy-Kovalevskaya theorem and analytic functions
Quasilinear first order equations, conservation laws, shocks, and entropy criteria
Harmonic functions, Green's identities, maximum principles, Harnack inequalities, and Poisson formula
Dirichlet problem, Schauder estimates, method of continuity, and nonlinear elliptic examples
Heat equation, Duhamel's principle, maximum principle, and nonlinear reaction-diffusion equations
Wave equation, Huygens principle, energy method, and symmetric hyperbolic systems

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 1-6.
Take-home midterm exam.
Final project.

Math 580: Fall 2011