Lecture Notes

The lecture notes for this course were prepared by Dale Winter, a student in the class, in collaboration with Prof. Colding.

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)