Welcome to the MATH 836 website for this semester
This is what the guide says we will do...
Sobolev spaces, potential theory, variational methods for elliptic equations, inverse problems. PREREQ: MATH806.
This is where and when...
MWF
10:10AM - 11:00AM at Drake Hall 074
The book
Sandro Salsa. Partial Differential Equations in Action. From Modelling to Theory. Chapters 3, 6, 7 and 8.
Newsbox
| (05/06) Final problemsheet.
We will work on these problems in the final
lectures. |
Table with the detailed schedule
Click on the Problems cells to see or download the problem collections. Sections in the book are given as help to read a possibly different point of view. Students are supposed to be able to solve all the recommended problems.
| Week |
Lecture |
Section |
Description |
Problems |
|
| 1 |
02/04 |
M |
7.1-7.2 |
I. DISTRIBUTIONS 1. Test functions |
Problem sheet #1 |
| 02/06 |
W |
7.3 |
2. Distributions |
||
| 02/08 |
F |
7.4 |
3. Convergence and
differentiation |
||
| 2 |
02/11 |
M |
7.4 |
4. Vanishing gradients and fundamental
solutions |
|
| 02/13 |
W |
7.7.2 |
II. THE
HOMOGENEOUS DIRICHLET PROBLEM 1. The Sobolev space H^1 |
||
| 02/15 |
F |
7.2 |
2. Cut-off, mollification and density |
Problem sheet
#2 |
|
| 3 |
02/18 |
M |
7.7.3 |
3. H^1_0 and the
Poincare-Friedrichs inequality |
|
| 02/20 |
W |
4. Three forms of the
Dirichlet problem |
|||
| 02/22 |
F |
6.5 |
5. The Riesz-Frechet
representation as an existence theorem |
EXAM
#1 DUE |
|
| 4 |
02/25 |
M |
III.
NON-HOMOGENEOUS DIRICHLET B.C. 1. Lipschitz transformations and Lipschitz domains |
||
| 02/27 |
W |
7.8 |
2. Localization and pull-back |
||
| 03/01 |
F |
7.8 |
3. The extension theorem | ||
| 5 |
03/04 |
M |
7.9 |
4. The trace theorem | Problem sheet #3 |
| 03/06 |
W |
5. The kernel and image of the trace operator | |||
| 03/08 |
F |
8.4.1 |
6. The non-homogeneous Dirichlet problem | ||
| 6 |
03/11 |
M |
IN CLASS TEST |
EXAM #2 |
|
| 03/13 |
W |
6.6 |
IV.
NON-SYMMETRIC AND COMPLEX PROBLEMS 1. The Lax-Milgram lemma |
||
| 03/15 |
F |
8.5.2 |
2. Convection-diffusion problems |
||
| 7 |
03/18 |
M |
3. Complex spaces and
complexified spaces |
Problem sheet #4 |
|
| 03/20 |
W |
4. Resolvent equations |
|||
| 03/22 |
F |
INTERMISSION. Linear elasticity |
|||
| 8 |
Spring break |
||||
| 9 |
04/01 |
M |
V. THE NEUMANN PROBLEM 1. H(div) and the normal trace |
||
| 04/03 |
W |
2. Easy problems with
Neumann conditions |
|||
| 04/05 |
F |
3. The Deny-Lions theorem
and a flavor of compactness |
|||
| 10 |
04/08 |
M |
4. Neumann problems for elliptic equations |
EXAM #3 DUE | |
| 04/10 |
W |
5. The Neumann problem for the Poisson equation | |||
| 04/12 |
F |
VI. THE
FREDHOLM ALTERNATIVE AND HELMHOLTZ EQUATIONS 1. The Rellich-Kondrachov theorem |
|||
| 11 |
04/15 |
M |
2. The Fredholm alternative in the
self-adjoint case |
Problem sheet #5 |
|
| 04/17 |
W |
3. Helmholtz equation and
Neumann revisited |
|||
| 04/19 |
F |
4. Impedance conditions
and the bilaplacian |
|||
| 12 |
04/22 |
M |
5. General Fredholm theorems | Problem sheet
#6 |
|
| 04.24 |
W |
6. Convection-diffusion problems |
|||
| 04/26 |
F |
VII.
EIGENVALUES OF ELLIPTIC OPERATORS 1. Eigenvalues of elliptic operators |
|||
| 13 |
04/29 |
M |
IN CLASS TEST (five
problems, closed book) |
EXAM #4 |
|
| 05/01 |
W |
2. The Hilbert-Schmidt
theorem |
|||
| 05/03 |
F |
3. Proofs and applications |
|||
| 14 |
05/06 |
M |
4. Series characterizations of Sobolev
spaces |
||
| 05/08 |
W |
5. More eigenvalue problems |
Problem sheet #7 | ||
| 05/10 |
F |
||||
| 15 |
05/13 |
M |
|||
| 05/17 |
F |
EXAM
#5 due |
|||
| And
we are done |
|||||
