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.