Lecture Notes

The lecture notes were prepared by two former students in the class. Zuoqin Wang prepared lecture notes 0 through 11 in LaTeX, and Yanir Rubinstein prepared lectures 12 through 24 in TeX. They used Professor Viaclovsky's handwritten notes in producing them.

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
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
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 (PDF)
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