0 
Course Overview
Examples of Harmonic Functions
Fundamental Solutions for Laplacian and Heat Operator 

1 
Harmonic Functions and Mean Value Theorem
Maximum Principle and Uniqueness
Harnack Inequality
Derivative Estimates for Harmonic Functions
Green's Representation Formula 

2 
Definition of Green's Function for General Domains
Green's Function for a Ball
The Poisson Kernel and Poisson Integral
Solution of Dirichlet Problem in Balls for Continuous Boundary Data
Continuous + Mean Value Property <> Harmonic 

3 
Weak Solutions
Further Properties of Green's Functions
Weyl's Lemma: Regularity of Weakly Harmonic Functions 

4 
A Removable Singularity Theorem
Laplacian in General Coordinate Systems
Asymptotic Expansions 

5 
Kelvin Transform I: Direct Computation
Harmonicity at Infinity, and Decay Rates of Harmonic Functions
Kelvin II: Poission Integral Formula Proof
Kelvin III: Conformal Geometry Proof 

6 
Weak Maximum Princple for Linear Elliptic Operators
Uniqueness of Solutions to Dirichlet Problem
A Priori C^0 Estimates for Solutions to Lu = f, c leq 0
Strong Maximum Principle 
Homework 1 due 
7 
Quasilinear Equations (Minimal Surface Equation)
Fully Nonlinear Equations (MongeAmpere Equation)
Comparison Principle for Nonlinear Equations 

8 
If Delta u in L^{infty}, then u in C^{1,alpha}, any 0 < alpha < 1
If Delta u in L^{p}, p > n, then u in C^{1,alpha}, p = n/(1  alpha) 

9 
If Delta u in C^{alpha}, alpha > 0, then u in C^{2}
Moreover, if alpha < 1, then u in C^{2,alpha} (Proof to be completed next lecture) 

10 
Interior C^{2,alpha} Estimate for Newtonian Potential
Interior C^{2,alpha} Estimates for Poisson's Equation
Boundary Estimate on Newtonian Potential: C^{2,alpha} Estimate up to the Boundary for Domain with Flat Boundary Portion 

11 
Schwartz Reflection Reviewed
Green's Function for Upper Half Space Reviewed
C^{2,alpha} Boundary Estimate for Poisson's Equation for Flat Boundary Portion
Global C^{2,alpha} Estimate for Poisson's Equation in a Ball for Zero Boundary Data
C^{2,alpha} Regularity of Dirichlet Problem in a Ball for C^{2,alpha} Boundary Data 
Homework 2 due 
12 
Global C^{2,alpha} Solution of Poisson's Equation Delta u = f in C^{alpha}, for C^{2,alpha} Boundary Values in Balls
Constant Coefficient Operators
Interpolation between Hölder Norms 

13 
Interior Schauder Estimate 

14 
Global Schauder Estimate
Banach Spaces and Contraction Mapping Principle 

15 
Continuity Method
Can Solve Dirichlet Problem for General L Provided can Solve for Laplacian
Corollary: Solution of C^{2, alpha} Dirichlet Problem in Balls for General L
Solution of Dirichlet Problem in C^{2,alpha} for Continuous Boundary Values, in Balls 

16 
Elliptic Regularity: If f and Coefficients of L in C^{k,alpha}, Lu = f, then u in C^{k+2,alpha}
C^{2,alpha} Regularity up to the Boundary 

17 
C^{k,alpha} Regularity up to the Boundary
Hilbert Spaces and Riesz Representation Theorem
Weak Solution of Dirichlet Problem for Laplacian in W^{1,2}_0
Weak Derivatives
Sobolev Spaces 

18 
Sobolev Imbedding Theorem p < n
Morrey's Inequality 

19 
Sobolev Imbedding for p > n, Hölder Continuity
Kondrachov Compactness Theorem
Characterization of W^{1,p} in Terms of Difference Quotients 

20 
Characterization of W^{1,p} in Terms of Difference Quotients (cont.)
Interior W^{2,2} Estimates for W^{1,2}_0 Solutions of Lu = f in L^2 

21 
Interior W^{k+2,2} Estimates for Solutions of Lu = f in W^{k,2}
Global (up to the Boundary) W^{k+2,2} Estimates for Solutions of Lu = f in W^{k,2} 

22 
Weak L^2 Maximum Principle
Global a priori W^{k+2,2} Estimate for Lu = f, f in W^{k,2}, c(x) leq 0 

23 
Cube Decomposition
Marcinkiewicz Interpolation Theorem
L^p Estimate for the Newtonian Potential
W^{1,p} Estimate for N.P.
W^{2,2} Estimate for N.P. 

24 
W^{2,p} Estimate for N.P., 1 < p < infty
W^{2,p} Estimate for Operators L with Continuous Leading Order Coefficients 
