Partial Differential Equations 440/540

Time and Place

Place: Bryan 324

Time: MWF 11:10 - 12:00

Problem Sessions: CUE 318, MTuWTh 5:00-6:00

Extra Problem Session: CUE 418, Sunday 3:00-5:00

Temporary copy of first few pages of text until your textbooks arrive: pdf of pages 1-35

Introduction

I will be using Karl Gustafson's "Introduction to Partial Differential Equations and Hilbert Space Methods", published by Dover (publishers price = 18.95). I am also recommending a free book by Pivato that can be found here.

We will cover Karl Gustafson's book, pages 1-230 (up through section 2.7). We will cover about 6-7 pages in each lecture. What is expected of you: Revised

There will be two take home tests. I will grade the undergraduates separately from the graduate students.

You should each try to do all the assigned exercises. collaboration is encouraged, problem sessions are advised.

Read the text before coming to class.

Grading, etc

This is basically a graduate course that makes concessions and modifications for motivated undergraduates - this is definitely not an undergraduate course that is stepped up a bit for graduate students.

You will be expected to work very diligently! If you are looking for an easy A, this is not the class for you. But if you are willing to work hard, you will have a lot of fun and get a good grade. I will not be giving tricky tests or otherwise trying to reduce the number of people I give good grades to.

First take home test is here.

Final take home test is here.

Notes

8-27-2009: Notes on the divergence of the normal field

10-21-2009: a few pages from Chavel's book on isoperimetric inequalities -- see pages 8-11

11-2-2009: a short excerpt from Zeidler's Nonlinear Functional Analysis, volume 1 (pages 15-22) covering the Banach Fixed Point Theorem

11-4-2009: a short excerpt from Folland's book (pages 49-56) covering theorems on integration

References

Here is a list of references for the course. As there is interest I will point people at other references that cover various aspects of PDEs and their applications.

Our text: Karl Gustafson's "Introduction to Partial Differential Equations and Hilbert Space Methods", published by Dover (publishers price = 18.95).
Pivato's free book that can be found here.
Craig Evans' book, "Partial Differential Equations" published by the American Mathematical Society. If you join this society, you get a discount on the book. While joining to get the discount does not make sense, if you are mathematically inclined, this is a good society to belong to. This is a must have for anybody serious about PDEs. (The appendices are superb at singling out useful pieces of analysis and functional analysis.)
Gilbarg and Trudinger, "Elliptic Partial Differential Equations of Second Order", published by Springer Verlag. This is an excellent reference, very well written and pretty much self contained. (Actually I would always start on Evans' book and then read this one after the basics are covered.)

Extra References

For spectral theory of linear operators in finite dimensions (matrices), see "Differential Equations, Dynamicsl Systems, and Linear Algebra" by Morris W. Hirsch and Stephen Smale. Nicest reference I know on spectral theory in finite dimensions.
A good reference on real analysis is "Real Analysis: Modern Techniques and Applications" by Gerald B. Folland. I recommend this text very highly. I think it belongs in the library of anyone who is taking this class to really understand the subject. (See short excerpt above)
A great reference for facts on usable topology, function spaces and functional analysis can be found in the extensive appendix in "Nonlinear Functional Analysis Volume I: Fixed Point Theorems" By Eberhard Zeidler. Highly recommended. It is worth the considerable investment required to own this book. (See short excerpt above)

Schedule of lectures

This is a preliminary list of lectures, homework, notes, etc. This will be updated as we go. The reading and problems should be done before the date on which it is listed (at least that is the idea ;~).

M Aug 24, 2009
W Aug 26, 2009 Read pages 1-8; do exercises 1,2,3 on page 4 and exercises 1,2,3 on page 9.
F Aug 28, 2009 Read pages 9-13, do exercises 1,2,3 on page 13
M Aug 31, 2009 read pages 14-21; do exercises 1-3 on pages 17-18 and exercises 1-3 on pages 21-22.
W Sep 02, 2009 (no new assignment)
F Sep 04, 2009 read 22-38; do exercises 1-3 on page 28 and exercise 1 on page 33
M Sep 07, 2009 Labor day Vacation
W Sep 09, 2009 read 22-38 again; do exercises 2-3 on page 33 and exercises 1-3 on page 38.
F Sep 11, 2009 read 38-45 exercises 1-9
M Sep 14, 2009 read 38-45 exercises 1-9
W Sep 16, 2009 read 38-45 exercises 1-9
F Sep 18, 2009 read 38-45 exercises 1-9
M Sep 21, 2009 read 38-45 exercises 1-9
W Sep 23, 2009 read 38-45 exercises 1-9
F Sep 25, 2009 class cancelled
M Sep 28, 2009 read 45-57 p 58 ex. 1-3
W Sep 30, 2009 read 45-57 p 58 ex. 1-3
F Oct 02, 2009 read 58-69 p 69 ex. 1-3
M Oct 05, 2009 read 58-69 p 69 ex. 1-3
W Oct 07, 2009 read 69-76 p. 76 ex 1-3
F Oct 09, 2009 read 77-82 ex 1-9
M Oct 12, 2009 read 77-82 ex 1-9
W Oct 14, 2009 read 77-82 ex 1-9
F Oct 16, 2009 read 77-82 ex 1-9
M Oct 19, 2009 read 77-82 ex 1-9
W Oct 21, 2009 read 77-82 ex 1-9
F Oct 23, 2009 read 77-82 ex 1-9

For the rest of the semester, you will be expected to read pages 141-209 in the text, attempt the 27 problems and exercises found in those pages, and work on the last take home test.

M Oct 26, 2009 141-157 plus extra functional analysis
W Oct 28, 2009 141-157 plus extra functional analysis
F Oct 30, 2009 141-157 plus extra functional analysis, take home test is posted
M Nov 02, 2009
W Nov 04, 2009
F Nov 06, 2009
M Nov 09, 2009
W Nov 11, 2009 Veteran's Day Vacation
F Nov 13, 2009
M Nov 16, 2009
W Nov 18, 2009
F Nov 20, 2009
M Nov 23, 2009 Thanksgiving Vacation
W Nov 25, 2009 Thanksgiving Vacation
F Nov 27, 2009 Thanksgiving Vacation
M Nov 30, 2009
W Dec 02, 2009
F Dec 04, 2009
M Dec 07, 2009
W Dec 09, 2009
F Dec 11, 2009