Differentialoperatorer og Funktionsrum (DifFun10)


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 Nam in the second week.


Course description

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.



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 Laplace operator. - Have a good Christmas vacation!

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 Laplace operator without a constant, and the treatment of functions like 1/t.) In Chapter 6, we started the discussion of x-independent pseudodifferential operators and did the part of Theorem 6.3 that relates directly to Theorem 12.13 on multiplication operators.

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.


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, Frederiksberg.

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.



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 Nam will conduct the exercises this week.

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, Frederiksberg.


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.

Ch. 1: Superficial.

Ch. 2: Sections 2.1-2.3 with full value, Section 2.4 superficial.

Ch. 3: Sections 3.1-3.4 until Th. 3.20 (this theorem is not included), with superficial parts: Lemma 3.6, Th. 3.14, Th. 3.16.

Ch. 4: Sections 4.1-4.4, with superficial parts:Th. 4.9, Th. 4.10 concerning R^n_+ and Omega, Cor. 4.11 concerning Omega, Th. 4.12 and Th. 4.25.

Ch. 5: Sections 5.1-5.3, with superficial parts: Th. 5.4 3^o and Th. 5.5. Section 5.6 superficial.

Ch. 6: Sections 6.1-6.4, omitting the proof for noninteger s in Lemma 6.17 2^o.

Ch. 12: Sections 12.1-12.5, skipping Th. 12.11 and Th. 12.12, and reading Cor. 12.22 and 12.25 superficially.

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.