We will follow the book

*Elliptic PDE of Second Order* by David Gilbarg and Neil S. Trudinger

developing the linear theory from part I. We will not have much time to discuss Part II on the quasi-linear case.

Chapter 1 gives an outline of the topics to be covered in the book (and hence the course), while Chapter 2 is both a review of basic facts about Laplace's equation and a preview of the types of results we would like to prove for more general equations. For this reason, we will begin the presentations with Chapter 3. Here is a brief guide for the presentation of the topics:

**No meeting March 30**: Everyone should review the first few sections of Chapter 2.-
**Weak Maximum Principle:**Define*elliptic*and present the results of Section 3.1. Continue with Section 10.1 where some immediate consequences in the quasi-linear case are described. Perhaps note the applicability to the equation of prescibed (e.g. constant) mean curvature, or its cousin, the equation of capillarity. -
**Strong Maximum Principle:**Present the results of Section 3.2. -
**Deducing a priori bounds:**The main results here are Theorems 3.7 and 3.10. Present also Theorem 10.3, which is the application of the same technique in the quasi-linear case. -
**Interior regularity (constant coeff.):**Sections 4.2 and 4.3. This is a lot of material, the speaker should be careful not to get too bogged down in computations; Theorem 4.9 can be skipped. -
**Boundary regularity (constant coeff.):**Sections 4.4 and 4.5.

Chapter 5 is a review of basic functional analysis and will be skipped. -
**Interior regularity:**Chapter 6 to Theorem 6.2 and Lemma 6.4 (the proof of Lemma 6.32 can be skipped). -
**Boundary regularity:**Remainder of Section 6.2 and Theorem 6.8. -
**The Dirichlet Problem:**Corollary 6.9 to Theorem 6.14.