# Class schedule

*Note*: This schedule is subject to revision during the term.
**Thursday, September 1**

Examples of PDE.
**Tuesday, September 6**

Classical function spaces.
Analyticity.
**Thursday, September 8**

Cauchy's method of majorants.
Analytic existence theorem for ODE.
**Tuesday, September 13**

Cauchy-Kovalevskaya theorem.
**Thursday, September 15** - (*Assignment 1 due*)

Characteristic surfaces.
Well posedness.
Basic classifications of PDE.
**Tuesday, September 20**

First order semilinear equations.
Method of characteristics.
Local theory.
**Thursday, September 22**

First order quasilinear equations.
Burgers equation.
Scalar conservation laws.
Wave breaking.
**Tuesday, September 27**

Weak solutions.
Riemann problem.
Shock waves.
Rankine-Hugoniot condition.
Entropy conditions.
**Thursday, September 29** - (*Assignment 2 due*)

Variable substitution in second order linear equations.
Fundamental solution of the Laplace operator.
Green's identities.
Uniqueness theorems.
**Tuesday, October 4**

Green's formula.
Mean value property.
Maximum principles.
Comparison principle.
**Thursday, October 6**

Koebe's converse of the mean value theorem.
Derivative estimates.
Liouville's theorem.
**Tuesday, October 11**

Analyticity of harmonic functions.
Green's function.
Poisson's formula.
**Thursday, October 13** - (*Assignment 3 due*)

Removable singularity theorem.
Harnack inequalities.
Harnack convergence theorems.
**Tuesday, October 18**

Dirichlet problem.
Poincare's method of sweeping out.
**Thursday, October 20**

Dirichlet domains.
Barriers.
Boundary regularity.
**Tuesday, October 25**

Poisson equation.
Newtonian potential.
**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.
Semilinear examples.
**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*)

Duhamel's principle.
Uniqueness.
**Thursday, November 24** - (*2 lectures*)

Nonlinear reaction-diffusion equations.
Maximum principles.
**Tuesday, November 29**

Examples of finite time blow-up.
Examples of global well posedness.
Wave equation.
**Thursday, December 1** - (*Assignment 6 due*)

Tychonov's example.
Kirchhoff's (Poisson's) formula.
Wave energy.