Note: This schedule is subject to revision during the term.
Thursday, September 1
Examples of PDE.
Tuesday, September 6
Classical function spaces.
Thursday, September 8
Cauchy's method of majorants.
Analytic existence theorem for ODE.
Tuesday, September 13
Thursday, September 15 - (Assignment 1 due)
Basic classifications of PDE.
Tuesday, September 20
First order semilinear equations.
Method of characteristics.
Thursday, September 22
First order quasilinear equations.
Scalar conservation laws.
Tuesday, September 27
Thursday, September 29 - (Assignment 2 due)
Variable substitution in second order linear equations.
Fundamental solution of the Laplace operator.
Tuesday, October 4
Mean value property.
Thursday, October 6
Koebe's converse of the mean value theorem.
Tuesday, October 11
Analyticity of harmonic functions.
Thursday, October 13 - (Assignment 3 due)
Removable singularity theorem.
Harnack convergence theorems.
Tuesday, October 18
Poincare's method of sweeping out.
Thursday, October 20
Tuesday, October 25
Thursday, October 27 - (Assignment 4 due)
Second order elliptic equations.
Method of continuity.
Introduction to a priori estimates.
Tuesday, November 1
Hölder norm estimates for the Newtonian potential.
Local Schauder estimates for the Laplace operator.
Thursday, November 3 - (Midterm due)
Local Schauder estimates.
Tuesday, November 8
Global Schauder estimates. Overview of the Schauder theory for linear and nonlinear elliptic equations.
Thursday, November 10 - (2 lectures)
Cauchy problem for the heat equation.
The heat propagator.
Tuesday, November 15 - No lecture
Thursday, November 17 - No lecture
Tuesday, November 22 - (Assignment 5 due)
Thursday, November 24 - (2 lectures)
Nonlinear reaction-diffusion equations.
Tuesday, November 29
Examples of finite time blow-up.
Examples of global well posedness.
Thursday, December 1 - (Assignment 6 due)
Kirchhoff's (Poisson's) formula.