Math 246 A,B,C, Syllabus


Partial Differential Equations

I.

Techniques and Notation

1.

Multi-indecies and properties of solutions to PDE's

2.

Smooth functions and convolutions

3.

The mean-value and implicit function theorems

4.

The Fourier transform

II.

Distribution Theory

1.

Basic properties of distributions

2.

Homogeneous distribution

3.

Distributions with compact support

4.

Convolutions of distributions

III.

The PDE's of Mathematical Physics

1.

Poisson's and Laplace's equations

2.

The heat equation

3.

The Schrödinger equation

4.

The wave equation

IV.

Sobolev Spaces

1.

Sobolev, L^p and Hölder spaces

2.

The Sobolev embedding theorem

3.

Interpolation and the Gagliardo-Nirenberg inequalities

V.

Semilinear Hyper- and Parabolic PDE's

1.

The contraction mapping principle

2.

The nonlinear wave and Schrödinger equations

3.

The maximum principle and the nonlinear heat equation

4.

Blow-up in finite-time