The lecture notes for this course were prepared by Dale Winter, a student in the class, in collaboration with Prof. Colding.
Course notes.
| LEC # |
TOPICS |
| 1 |
Harmonic Functions and the Harnack Inequality (PDF) |
| 2 |
The Gradient Estimate (PDF) |
| 3 |
The Hopf Maximum Principle (PDF) |
| 4 |
The Poincare Inequalities (PDF) |
| 5 |
The Cacciopolli Inequality (PDF) |
| 6 |
More General Operators (PDF) |
| 7 |
Consequences of Cacciopolli (PDF) |
| 8 |
Maximum Principles and Gradient Estimates (PDF) |
| 9 |
Hopf and Harnack for L-harmonic Functions (PDF) |
| 10 |
An Improved Gradient Estimate for Harmonic Functions (PDF) |
| 11 |
More on Harmonic Functions on a Ball (PDF) |
| 12 |
Solving the Laplace Equation in R2: The Dirichlet Problem (PDF) |
| 13 |
The Heat Equation (PDF) |
| 14 |
A Gradient Estimate for the Heat Equation on a Ball (PDF) |
| 15 |
Campanato's Lemma and Morrey's Lemma (PDF) |
| 16 |
Five Inequalities for Harmonic Functions (PDF) |
| 17 |
Regularity of L-harmonic Functions Part I (PDF) |
| 18 |
Regularity of L-harmonic Functions Part II (PDF) |
| 19 |
Regularity of L-harmonic Functions Part III (PDF) |
| 20 |
Smoothness of L-harmonic Functions (PDF) |
| 21 |
The Mean Value Inequality Revisited Part I (PDF) |
| 22 |
The Mean Value Inequality Revisited Part II (PDF) |
| 23 |
Moser's Approach (PDF) |