Date 
Material covered 
Remarks 
Aug 24

Introduction


Aug 29 
Notation and generalities. Harmonic and sub super harmonic
functions. Mean value formulas, maximum principle, and Harnack's
inequality. 

Sept 5 
Fundamental solution and Green's function, representation
formulas, Poisson integral. C^0 subharmonic functions and its
properties. 

Sept 7 
Solution to the Dirichlet problem for Laplace's equation via the
method of subharmonic functions. 

Sept 12 
The wave equation. D'Alembert's formula, Kirchhoff's formula,
and finite speed of propagation. 

Sept 19 
More on the wave equation. The heat equation. 

Sept 21 
Weak derivatives. Sobolev spaces. 

Sept 26 
Basic properties of Sobolev spaces. 

Oct 3 
More properties of Sobolev spaces. Density and trace theorems. 

Oct 5 
More properties of Sobolev spaces. Extensions. Holder spaces. 

Oct 10 
Sobolev inequalities. 

Oct 17 
More on Sobolev inequalities and compact embeddings. 

Oct 27 
Linear elliptic equations. Weak solutions. LaxMilgram. 

Oct 31 
The Fredholm alternative in Banach spaces. 

Nov 2 
The Fredholm alternative in Hilbert spaces. 

Nov 7 
Existence and uniqueness of weak solutions to linear elliptic
equations. The weak maximum principle. 

Nov 9 
Elliptic regularity. 

Nov 14 
Elliptic regularity up to the boundary. 

Nov 16 
Basic calculus in Frechet and topological vector spaces. 

Nov 28 
Inverse and implicit function theorems. 

Nov 30 
Picard iteration and Newton's method in Banach spaces. 

Dec 5 
The Nash implicit function theorem. 

Dec 7 
The Nash isometric embedding theorem. 
