Welcome to the course on Differential operators and Function spaces,
Fall 2011, Block 2. The lectures will be given by Gerd Grubb, and the exercise
classes conducted by Kim Petersen.

We shall use the book by Gerd Grubb: "Distributions and
Operators", Graduate Texts in Mathematics, Springer Verlag 2009.

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.

Weakly scheduled hours:

Lecture 1 Monday from 15:00 to 17:00 in Auditorium 7

Lecture 2 Wednesday from 10:00 to 12:00 in Auditorium 8

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

Exercise class: Wednesday from
13:00 to 15:00 in 1-0-30 (DIKU)

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 21, 2011 at
15.15 in Aud. 7 at the HCØrsted Institute, Universitetsparken 5.

**Teaching period: **November 21 2011 – January 27 2012, in the weeks 47-51 of 2011 and
1-4 of 2012.

There is homework every week, and the
homework in weeks 50 of 2011 and 2 of 2012 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 14 and January
11, 2012. (These dates may be modified if some outer circumstances make
it necessary.)

In week 4 2012, on the last day of the course January
25, 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
answer (we do not accept copying). Write in Danish or English.

List of comments to the book: corrections .
Additional miscellaneous exercises: miscellaneous

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. Written homework is to be handed
in at the exercise session on Wednesdays.

PAST LECTURES:

Week 47, Nov. 21 and 23: 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 2.1. 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. (From Appendix B we shall in the course mainly use
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. As for Th. B.18 we use the concrete version Th. 2.5.)

Week 48, Nov. 28 and 30:
The rest of Sections 2.1-2.3 was covered and we began Chapter 3,
defining distributions, illustrated by the important examples: locally
integrable functions, the delta-measure, and their derivatives. Rules of
calculus were covered up until and including Lemma 3.6. (Remark 3.3 was only
briefly mentioned, Lemma 3.6 was proved for k=1.)

Week 49, Dec. 5 and 7: We continued Chapter 3, 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 was left as voluntary reading. Finally, we began Chapter 12 on
unbounded operators, up to and including Theorem 12.5.

Week 50, Dec. 12 and 14: In Chapter 12, the study of unbounded operators
in Hilbert space was covered essentially until the end of Sections 12.4. We
skipped Theorems 12.11 and 12.12, and the considerations on uniqueness in
Corollaries 12.22 and 12.25 were only lightly touched. Corollary 12.21 was
postponed, Remark 12.23 skipped. This was a week with obligatory exercises
(handed in on December 14).

Week 51, Dec. 19 and 21: Back to distribution theory, Chapter 4. On
December 19, the topic was realizations of differential operators and Sobolev
spaces, Sections 4.1-4.2 (some rules were given only in survey form). Because
of travels, the lectures 10-12 on December 21 were given by Heiko Gimperlein,
on Section 4.3. Lectures 15-16 were postponed into the new year, as an extra
hour 16-17 on January 4.

Week 1 (2012), Jan. 2 and 4: The lectures on January 2 were given by Kim
Petersen, on Section 4.4 until Theorem 4.29. On January 4 (lectures 10-12 and
15-17), we ended Section 4.4 and did Ch. 5, Sections 5.1-5.3, on basic
properties of the Fourier transformation for temperate distributions.

Week 2, Jan. 9 and 11: Chapter 6, Section 6.1-6.3, on x-independent
pseudodifferential operators, Sobolev spaces over R^n of all orders,
Sobolev’s Theorem and the Structure Theorem. This was a week with
obligatory exercises (handed in on January 11).

Week 3, Jan. 16 and 18: Sections 6.4 and 5.6. Lectures planned for
Monday 15-17 and Wednesday 10-12. Exercises on Wednesday used all three hours
13-16. The lectures Wednesday were cancelled because of illness.

Week 4, Jan. 23 and 25:
Question session (also with exercises) on January 23, written exam
January 25.

EXERCISES:

Week 47: 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.

Week 48: 2.2, 2.3, 2.4, __2.5__, B.3, B.10, __B.11__, B.12.

Week 49: 2.6, __2.7__, 3.1, __3.2(a)__, 3.3, 3.4, 3.5, __3.8__,
3.9.

Week 50: ** B.13**, 3.7, 3.10, 3.11, 3.12,
3.13, 3.17,

Week 51: 12.1, 12.3, 12.4, __12.8__,
12.10, __12.12__, 12.23, 12.35, 6.39 (this is posted in miscellaneous ).

Week 1 (2012): 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: ** 4.9**, 4.25,
5.1, 5.3, 5.4(a), 5.6, 5.7, 5.10,

Week 3: 4.13, 5.8, 6.1, 6.4, 6.5, 6.6(a,c), 6.22, 6.26, 6.28, 6.38.
Exercise class from 13h to 16h.

Week 4: Written exam.

Old examination sets: The exercises 6.33-35 were a 2-hour test in April
2006, 6.36-38 were a 3-hour test in April 2007, 6.41-43 were a 2-hour test in
April 2009, and 6.44-46 were a 3-hour test in January 2011. Most of the
exercises in Chapter 6 (from 6.7 on) are old exam problems, but the course was
larger in earlier versions, and exams were often take-home tests.

Eksamen finder sted på
CSS (Øster Farimagsgade 5) i lokale 7-01-30 i bygning 7 kl. 9-12. Se
evt. oversigtskort på

http://csc.ku.dk/hoejrebokse/om_dib/CSS_oversigtskort.pdf

Medbring og aflever de
obligatoriske sæt, så de kan forelægges for censor ved den
endelige bedømmelse.

Kære alle, her er
omsider resultatet af eksamen (med afleveringssættene vægtet 20%
hver og slutprøven vægtet 60%)

Nr. 1: 10. Nr. 2: 10. Nr. 3: 02. Nr. 4: 7. Nr. 5:
02. Nr. 9: 4.

Nr. 11: 7. Nr. 12: 7. Nr. 14:
12. Nr. 15: 10. Nr. 16: 7. Nr. 17: 4.

Det var godt at alle bestod.
Jeg takker for jeres indsats med kurset, og ønsker jer held og lykke
videre i studiet.

Venlig hilsen, Gerd.

**Pensum.**

Pensum
er baseret på Chapters 1-6, Chapter 12 og Appendices i bogen
”Distributions and Operators”.

Betegnelsen
"kursorisk" drejer sig om emner, der har været gennemgået
overfladisk eller nævnt uden bevis (som man altså ikke skal
stå til regnskab for), men hvor resultatet kan bruges i senere tekst og i
øvelser.

Alle
tekststykker betegnet ved Remark (indenfor nedennævnte afsnit) er
kursoriske.

Ch. 1:
Kursorisk.

Ch. 2:
Sections 2.1-2.3 indgår med fuld værdi, Section 2.4 er kursorisk.

Ch. 3:
Sections 3.1-3.4 indtil Th. 3.20 (og exklusive denne sætning). Her
følgende læst kursorisk: Lemma 3.6, Th. 3.14, Th. 3.16.

Ch. 4:
Sections 4.1-4.4, med kursoriske dele: Th. 4.9, Th. 4.10 vedrørende
R^n_+ og Omega, Cor. 4.11 vedrørende Omega, Th. 4.12 og Th. 4.25.

Ch. 5:
Sections 5.1-5.3, med kursoriske dele: Th. 5.4, Th. 5.5 og Th. 5.6 (da de er
forudsat bekendt).

Ch. 6:
Sections 6.1-6.4, med overspringelse af Lemma 6.7 og beviset for ikke-hele s i
Lemma 6.17 2^o.

Ch. 12:
Sections 12.1-12.5, med overspringelse af Th. 12.11 og Th. 12.12; her er Cor.
12.22 og 12.25 kursoriske.

App. A
er baggrundsviden. App. B er læst kursorisk, og vi bruger heraf
især Definition B.4, Th. B.5, Remark B.6, Lemma B.7, Remark B.8, Th. B.9,
Th. B.15 og Th. B.16. App. C medtages kursorisk.