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.
| LEC # | TOPICS | LECTURE NOTES |
|---|---|---|
| 0 |
Course Overview
Examples of Harmonic Functions Fundamental Solutions for Laplacian and Heat Operator |
(PDF) |
| 1 |
Harmonic Functions and Mean Value Theorem
Maximum Principle and Uniqueness Harnack Inequality Derivative Estimates for Harmonic Functions Green’s Representation Formula |
(PDF) |
| 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 |
(PDF) |
| 3 |
Weak Solutions
Further Properties of Green’s Functions Weyl’s Lemma: Regularity of Weakly Harmonic Functions |
(PDF) |
| 4 |
A Removable Singularity Theorem
Laplacian in General Coordinate Systems Asymptotic Expansions |
(PDF) |
| 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 |
(PDF) |
| 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 |
(PDF) |
| 7 |
Quasilinear Equations (Minimal Surface Equation)
Fully Nonlinear Equations (Monge-Ampere Equation) Comparison Principle for Nonlinear Equations |
(PDF) |
| 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) |
(PDF) |
| 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) |
(PDF) |
| 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 |
(PDF) |
| 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 |
(PDF) |
| 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 |
(PDF) |
| 13 | Interior Schauder Estimate | (PDF) |
| 14 |
Global Schauder Estimate
Banach Spaces and Contraction Mapping Principle |
(PDF) |
| 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 |
(PDF) |
| 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 |
(PDF) |
| 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 |
(PDF) |
| 18 |
Sobolev Imbedding Theorem p < n
Morrey’s Inequality |
(PDF) |
| 19 |
Sobolev Imbedding for p > n, Hölder Continuity
Kondrachov Compactness Theorem Characterization of W^{1,p} in Terms of Difference Quotients |
(PDF) |
| 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 |
(PDF) |
| 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} |
(PDF) |
| 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 |
(PDF) |
| 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. |
(PDF) |
| 24 |
W^{2,p} Estimate for N.P., 1 < p < infty
W^{2,p} Estimate for Operators L with Continuous Leading Order Coefficients |
(PDF) |
