LEC #  TOPICS  KEY DATES 

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 