Math 290J: Introduction to Elliptic PDE
Tuesdays 3:00--3:50 in MS6221.
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.