Thursday, September 1Examples of PDE.
Tuesday, September 6Classical function spaces. Analyticity.
Thursday, September 8Cauchy's method of majorants. Analytic existence theorem for ODE.
Tuesday, September 13Cauchy-Kovalevskaya theorem.
Thursday, September 15 - (Assignment 1 due)Characteristic surfaces. Well posedness. Basic classifications of PDE.
Tuesday, September 20First order semilinear equations. Method of characteristics. Local theory.
Thursday, September 22First order quasilinear equations. Burgers equation. Scalar conservation laws. Wave breaking.
Tuesday, September 27Weak 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 4Green's formula. Mean value property. Maximum principles. Comparison principle.
Thursday, October 6Koebe's converse of the mean value theorem. Derivative estimates. Liouville's theorem.
Tuesday, October 11Analyticity 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 18Dirichlet problem. Poincare's method of sweeping out.
Thursday, October 20Dirichlet domains. Barriers. Boundary regularity.
Tuesday, October 25Poisson equation. Newtonian potential.
Thursday, October 27 - (Assignment 4 due)Second order elliptic equations. Method of continuity. Introduction to a priori estimates.
Tuesday, November 1Hölder norm estimates for the Newtonian potential. Local Schauder estimates for the Laplace operator.
Thursday, November 3 - (Midterm due)Local Schauder estimates.
Tuesday, November 8Global 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 29Examples 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.