| 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 (Monge-Ampere 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 | |