Welcome to the course on Differential operators and Function spaces,
Fall 2010, Block 2. Lecturer: Gerd
Grubb, room 04.2.03.

The book by Gerd Grubb: "Distributions and Operators",
Graduate Texts in Mathematics, Springer Verlag 2009, will be used.

In this course we shall read Chapters 1-3, 12, and 4-6 (in that order),
with some omissions, and with consultations of the appendices.

The exercise classes will be conducted by Heiko Gimperlein. Because of
travels, he will be replaced by Kim Petersen in the first week and Phan Thanh

Weakly scheduled hours:

Lecture 1 Monday from 15:00 to 17:00 in Auditorium 10

Lecture 2 Wednesday from 10:00 to 12:00 in Auditorium 9

Lecture 3 Wednesday from 15:00 to 16:00 in Auditorium 9

Exercise class: Wednesday from 13:00 to
15:00 in A109

It is customary at the university to start 15 minutes
into the hour. So, add 15 min.s to each of the starting times above.

The first lecture begins Monday November 15, 2010 at
15.15 in Aud. 10 at the HCØrsted Institute, Universitetsparken 5.

**Teaching period: **November 15 2010 – January 28 2011, in the weeks 46-50 of 2010 and 1-4
of 2011.

There is homework every week, and the
homework in weeks 49 of 2010 and 2 of 2011 will be obligatory (counting with
20% each in the final result). The dates for handing in these
two obligatory homeworks are so far planned to be: December 8 and January
12, 2011. (These dates may be modified if some outer circumstances make
it necessary.)

In week 4 2011, on the last day of the course January
26, there will be a written exam of 3 hours counting with 60%. There will be
given a grade and there will be an external censor.

You are welcome to collaborate with others in
solving the homework problems, but you must write your own formulation of the
answers (we do not accept copying). Write in Danish or English.

List of comments to the book: corrections

PLAN OF THE COURSE. Past lectures will be described in detail,
and tentative plans for future lectures will be posted here.
The exercises for each week will be posted here, on or before the
Wednesday of the preceding week. The underlined exercises are to be handed in as
written homework, at the exercise session on Wednesdays.

PAST LECTURES:

Week 46, Nov. 15 and 17: After
explaining the need for generalizations of the concept of a derivative, and
mentioning weak derivatives (Chapter 1), we considered the relevant function
spaces introduced in Chapter 2. First of all, we showed the existence of test
functions. Next, we recalled the Banach spaces of continuous or k times
continuously differentiable functions on closed intervals in one dimension,
closed boxes in n dimensions. Using material from Appendix B, leaving out most
proofs, we introduced Fréchet spaces and the important construction of
such spaces by use of countable families of seminorms. This was applied to
function spaces over open sets and with infinitely many derivatives. Finally,
the inductive limit topology on the union of an increasing family of Fréchet
spaces was introduced, and applied to the test function space. The lectures
covered Chapter 2 essentially up to page 15, and covered the material in
Appendix B superficially.

Week 47, Nov. 22 and 24: We went
through the rest of Sections 2.1-2.3 and began Chapter 3. Distributions were
defined, and illustrated by the important examples: locally integrable
functions, the delta-measure, and their derivatives. Some cases of conventions
for operation with distributions were shown. The lectures covered up to page
34, skipping Remark 3.3.

Week 48, Nov. 29 and Dec. 1: In Chapter 3, we covered the material until
page 46, with main topics being the various operations on distributions, the special
estimates for distributions with compact support, and the approximation
theorems. Lemma 3.6 and Theorem 3.16 were superficially explained, Theorem 3.20
was not mentioned. Section 3.5 is left as voluntary reading. In Chapter 12 on
unbounded operators, we covered up to and including Theorem 12.5.

Week 49, Dec. 6 and 8: We covered the rest of Section 12.2, and Sections
12.3-5, except for Theorems 12.11 and 12.12. The uniqueness results in
Corollaries 12.22 and 12.25 were superficially mentioned.

Week 50, Dec. 13 and 15: Most of Chapter 4 was covered. In Section 4.1
on realizations, Remark 4.4 was skipped. In Section 4.2 on Sobolev spaces, we
omitted the proofs of Theorem 4.9, Thm. 4.10 2^o and 3^o, and Theorem 4.12
(except for some brief indications). In Section 4.3 on the one-dimensional
situation, the emphasis was on the basic results for H^1. In Section 4.4 on
higher dimensions, we established boundary values of H^1-functions and
discussed the Friedrichs extension of the minimal realization of the

Week 1, 2011, Jan. 3 and 5: The Dirichlet and Neumann realizations in
Section 4.4 were deduced by use of the Lax-Milgram Theorem (Th. 12.18), and the
Poincaré inequality was proved. In Chapter 5, we did Sections 5.1-3 on Fourier
transformation, Schwartz functions and Schwartz distributions. (For lack of
time, we have to skip the subsequent results pertaining to the

Week 2, Jan. 10 and 12: We
continued in Chapter 6, covering Sobolev spaces on R^n of all orders, their
dualities, the Sobolev imbedding theorem and the Structure Theorem, and began the study of elliptic operators in
Section 6.4, including Theorem 6.22. In Lemma 6.17 2^o, the case of noninteger
s was skipped.

Week 3, Jan. 17 and 19: Because
of travels, the lectures were given by Heiko Gimperlein. They dealt with the rest of Section 6.4 on
elliptic operators, and Section 5.6 on the non-integrable function 1/t and
related subjects. Section 5.6 will not be part of the final test. The lectures took place Monday 15-17 and
Wednesday 10-12.

UPCOMING LECTURES:

Week
4, Jan. 24 and 26: Question hour (spørgetime) Monday 15-17. The question hour (15-17.30)
was used to do some exercises in full detail. Final written test Wednesday 9-12
in Mariendals-Hal B, Mariendalsvej 21 C,

**NB! The place for the final exam has
been changed from my previous messages! GG **

**NB!** Please bring
your corrected obligatory exercises along, and hand them in together with the
final test. If you enclose an envelope with your address, or just a sheet with
your address, I can send the exercises to you after the evaluation is over (and
some waiting time has passed). GG

**The result resultat**

Feb. 11, 2011. Thanks to everybody for having participated in the course
in good spirits. Good Luck! Gerd G.

EXERCISES:

Week 46: We use this first
session to go through Appendix A, refreshing some known background facts and getting
acquainted with others. Do the exercises A.2 (a) (and (b) if there is time) and
A.3 in class. Kim Petersen will conduct the exercises this week.

Week 47: 2.2, 2.3, 2.4, __2.5__, B.3, B.10, __B.11__, B.12.
Phan Thanh

Week 48: 2.6, __2.7__, 3.1, __3.2(a)__,
3.3, 3.4, 3.5, __3.8__, 3.9.

Week 49: 3.7, 3.10, 3.11, 3.12,
3.13 __3.17__, 6.9, __6.31__, __6.36__.
The underlined exercises are **obligatory
homework**, to be handed in on December 8. In Exercise 6.36, you can use the
results of Exercise 3.9, and you should skip the question on f^.

Week 50: 12.1, 12.3, 12.4, __12.8__,
12.10, __12.12__, 12.23, 12.35, 6.39 (this is posted in corrections).

Week 1, 2011: 4.2, 4.3, 4.4, 4.7,
4.10, 4.11, __4.14__, 4.23, __4.24__. In Exercise 4.14, you can use the
outcome of Remark 4.21 without further explanation.

Week 2: __4.13__, 4.25, 5.1,
5.3, 5.4(a), 5.6, 5.7, 5.10, __6.40__ (from the sheet of corrections).
Underlined exercises are **obligatory
homework**, to be handed in on January 12, 2011.

Week 3: The session was from 13h to 16h. Exercises
5.8, 6.1, 6.4, 6.5, 6.6(a,c), 6.7, 6.22, 6.28, 6.29 and possibly 6.10, 6.34,
6.35. (In 6.29(c), assume that Re{b} > -2.) Results were listed for the
exercises not done fully.

Week
4: Final written test Wednesday from 9 to 12 in Mariendals-Hal
B, Mariendalsvej 21 C,

**List of contents of the course
(curriculum, pensum).**

The curriculum is based on Chapters 1-6, Chapter 12 and the Appendices.

Some parts are labelled "superficial" (in Danish
"kursorisk"). This means that the results have been mentioned
(usually) without proof; they have been and can be used in exercises.

The text appearing in Remarks has the superficial status.

App. A is background knowledge. App. B has been read superficially, and
we are mainly using Definition B.4, Th. B.5, Remark B.6, Lemma B.7, Remark B.8,
Th. B.9, Th. B.15 and Th. B.16. App. C is included superficially.