On this page we list the detailed plans for past and immediately upcoming lectures and exercise classes, modifying the content as the course proceeds. Homework for the following week will be normally announced on or before Wednesday.
PAST LECTURES:
Uge 6, 3.2 og 6.2: After some introductory remarks as in 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 sets. 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 space of test functions was introduced. We covered Section 2.1 and part of 2.2.
Uge 7, 10.2 og 13.2: Having defined the testfunction space and its topology, we can now begin to define distributions and consider some important examples. This is based on Sections 3.1 and 3.2. On Friday, Jacob Stordal Christiansen took over lecturing and went through the approximations theorems for R^n in Section 2.3.
Uge 8, 17.2 and 20.2: Jacob first finished Section 2.3 and also proved the Du Bois-Reymond lemma (from Section 3.1). Then we started on the material in Chapter 12 on unbounded operators in Hilbert space. This material will be needed when we begin to apply Distribution Theory. Sections 12.1, 12.2, and 12.3 (except for Thm 12.11 and Thm 12.12) were covered in this week.
Uge 9, 24.2 and 27.2: We continued the study of unbounded operators and went through Sections 12.4 and 12.5. In particular, the Lax-Milgram construction (Thm 12.18) was explained and we saw that a lower bounded (densely defined, symmetric) operator always has a selfadjoint extension. In fact, the Friedrichs extension (Thm 12.24) has the same lower bound as the operator itself. Afterwards, we returned to Section 2.4 and established a 'partition of unity' result (Thm 2.17). This result will be useful in Chapter 3.
Uge 10, 3.3 and 6.3: Gerd resumed the lectures, and continued in Chapter 3 with Sections 3.3 and 3.4, ending before Theorem 3.20; the proof of Theorem 3.16 was skipped and that of Theorem 3.18 was only summarily explained. Section 3.5 contains a quicker formulation of some of the same results, and is voluntary reading for those who have a good background in functional analysis. We started on Chapter 4, up to and including Lemma 4.6 on Sobolev spaces.
Uge 11, 10.3 og 13.3: We went through Sections 4.2, 4.3 and part of 4.4, continuing the discussion of Sobolev spaces while omitting heavier proofs: that of Th. 4,9, the part of Th. 4.10 for subsets, Th. 4.12, some details in Th. 4.17 4.18 and 4.23, and Th. 4.25. The aim was to discuss the treatment of differential operator problems by the help of the variational theory from Chapter 12. A one-dimensional example (Example 12.20 and exercise 12.25) was included in the lectures Friday, and we did part of Th. 4.27 on the Friedrichs extension of the Laplacian on an open set in n-space.
Uge 12, 17.3 og 20.3: Chapter 4 was completed. Next, we did Sections 5.1-5.3 of Chapter 5 on Fourier transformation and temperate distributions. Finally, Chapter 6 was read up to and including Theorem 6.3.
Uge 13, 24.3 og 27.3: We continued in Chapter 6, seeing how the Fourier transform is used in the study of Sobolev spaces. We showed Sobolev's theorem (on the injection of Sobolev spaces with large exponent into classically differentiable functions), the duality between H^s and H^{-s}, and the Structure Theorem on how distributions are sums of derivatives of continuous functions. Lemma 6.7, and the proof of Lemma 6.17 for noninteger s, were skipped. Finally we looked again at operators, in particular elliptic differential operators, until and including the statement of Theorem 6.24.
UPCOMING LECTURES:
Uge 14, 31.3 og 3.4: On Tuesday March 31, the first hour will be used for a final lecture with summaries and perspectives, and the second hour will be used for questions (extended into the afternoon hour if needed). On Friday, there is the classroom test in A106 (from 10.00 to 12.00) instead of lectures.
EXERCISES:
Underlined exercises are homework, to be handed in that day, at the exercise class.
Uge 6, 3.2: 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.
Uge 7, 10.2: 2.2, 2.3, 2.4, 2.5, B.3, B.10, B.11, B.12. Details from Appendix A on L_p spaces.
Uge 8, 17.2: 2.6, 2.7, 3.1, 3.2(a), 3.3, 3.5, 3.8, B.13. The homework this week counts as obligatory homework.
Uge 9, 24.2: 3.12, 3.17, 12.1, 12.3, 12.4, 12.8, 12.10, 12.13.
Uge 10, 3.3: 3.9, 12.11, 12.12, 12.18, 12.23, 12.30. 12.31, 12.32. The homework this week counts as obligatory homework.
Uge 11, 10.3: 12.35, 12.36, 3.4, 3.7, 3.10, 6.9, 6.23, 6.31.
Uge 12, 17.3: 4.2, 4.3, 4.4, 4.7, 4.13(a)-(c), 4.14, 12.37, 6.10. [Supplementing information: In 4.14 and 12.37 one can use the information in Remark 4.21 (proved in Exercise 5.10). For 6.10: This is a case where S_0 has only one selfadjoint extension.] The homework this week counts as obligatory homework.
Uge 13, 24.3: 4.13(d), 4.23, 5.1, 5.3, 5.6, 5.7, 6.33, 6.36, 6.37.
Uge 14, 31.3: 4.9, 5.10, 6.1, 6.4, 6.5, 6.11, 6.22, 6.38.
The final classroom test will be from 10.00 to 12.00 on Friday April 3, in room A106.
The exercises in Chapter 6, from number 6.7 on, have been used for finals in courses like this one through the years (with varying content). The longer exercises were used in take-home exams. In 2006, 6.33, 6.34 and 6.35 were used in a 2 hour classroom test; in 2007, 6.36, 6.37 and 6.38 were used in a 3 hour classroom test.
