MA8105 Nonlinear Partial Differential Equations and Sobolev spaces · Spring 2025
About the course
Lecturers are Mats Ehrnström and Helge Holden.
The course covers mathematical methods and structures that are fundamental for the study of partial differential equations (PDEs), variational calculus, numerical methods etc. The main focus is on analytical tools: Functional analysis, 𝐿𝑝 and Sobolov spaces, compactness, modes of convergence, distributions, error estimates. We cover some applications to linear and nonlinear PDEs.
The lectures will be in English.
Textbook: H. Holden: Tools from the Toolbox. Functional Analysis for Differential Equations (manuscript – will be distributed at the beginning of the course)
Lectures
- Tuesdays at 8:15–10:00 in GL-RFB R57
- Thursdays at 10:15–12:00 in GL-RFB R57
The first lecture will be on Tuesday, January 7
| Week | Topic | Recommended exercises |
|---|---|---|
| 2 | Tue: We proved the inequalities (1.1)-(1.3) in Ch. 1. In Ch. 2 we covered the material up to the definition of the second dual (p.9). Thu: Ch. 2, p. 9-12. Showed that \(\ell^p\) are Banach spaces, showed that \((\ell^p)'=\ell^q\) when \(1<p<\infty\). Showed that \(\ell^p\) spaces are nested. Props. 2.3, 2.4 mentioned. | Ch. 2: 3, 6. |
| 3 | Pages 12–16. Notes: Theorems 2.6, 2.14 and and 2.17 are different versions of what is usually called Banach–Alaoglu's theorem; added Riesz representation theorem (classical version for Hilbert spaces); pay attention to so-called diagonal arguments, and how separability (of the space or a subspace of it) plays an important role in these constructions. | Ch. 2: 8, 10, 15. |
| 4 | Pages 16–21. Notes: Prop 2.20 about equivalence of compactness notions is a result used throughout analysis of PDEs. It has a correspondence in the weak topology, usually called Eberlein-Smulian (sic). There is no equivalent for the weak-* topology. The classical Arzelà–Ascoli result was proved for a family of equicontinuous and bounded (equibounded) real functions on a compact interval on the line. Pay attention to varying definitions concerning equiboundedness, as well as differences between pointwise and uniform equicontinuity, boundedness, and equiboundedness. | |
| 5 | Tue: Defined continuous and compact imbeddings (p. 21-22) and Lemma 2.26. Defined Frechet and Gateaux derivatives and showed by example that they need not both exist. Proved that if a function is Frechet differentiable, and it is also Gateaux differentiable (with the same derivative). Stated and started proving Krasnosel'skii's theorem. Thu: Completed the proof of Krasnosel'skii's theorem, and then proved Thm. 2.31. Did Example 2.33. Started Sec. 2.3 and stated Thm. 2.39 without proofs. | Ch. 2: 13, 16. |
| 6 | Pages 33–39. Grönwall's inequality, mentioned other forms of it. Prop 2.42: dependence on RHS by Grönwall. Just mentioned Theorem 2.43, but note the condition (2.93), which is famous. Section 2.4: Bi- and sesquilinear forms. Stampacchia is a generalisation of Lax-Milgram, did not lecture it. Important: Lax–Milgram and the example 2.48 on ODEs. Note that the brackets and conjugates should be shifted if working over \(\mathbb{C}\). | |
| 7 | Pages 43–54. History, definitions and operations on distributions. Mentioned Schwarz space \(S\) and tempered distributions \(S'\). Sequential convergence in D and E, by use of \(C^k\) semi-norms on compact sets. Covered essentially everything, with some additions and variations, except: no proof of Prop 3.7; sketch of proof of (3.26). Proved Thm 3.11 ('fundamental theorem of analysis for \(D'(\mathbb R)\)') up to the existence of \(U\). | Ch. 3: 1, 2. |
| 8 | Tue: Proved Thm 3.11. Defined convolutions of distributions. Ended by stating Thm 3.14 (Malgrange–Ehrenpreis) about the existence of fundamental solutions of liner PDEs with constant coefficients. Thu: We proved the structure theorem for distributions. We proved various inequalities in Ch. 4, up to and including Jensen's inequality. | Ch. 3: 4, 5. |
| 9 | Tue: We proved estimate (4.25). Then we proved the Kolmogorov–Riesz–Sudakov theorem 4.1 regarding compactness in Lebesgue spaces. Thu: We covered selected parts of Sec. 4.4. Specifically: Prop. 4.4, Remark 4.5, Thm. 4.6. Sec. 4.4.1: incl. Thm. 4.11 (no proof), Thm. 4.13. Sec. 4.4.2: incl. Dunford–Pettis (no proof), Thm. 4.20 (no proof). Sec. 4.4.3: Ex 4.23 (i) and (ii). | Ch. 4: 5, 7, 8, 13, 14. |
| 10 | Sec. 4.7: Thm 4.38, Thm 4.39, Examples of explicit test functions, Friedrichs mollifiers, Thm 4.41 with proof. Sections 5.1 and 5.2 (up to proof of Thm 5.4): Def 5.1, Thm 5.2, Thm 5.3 with proof. Extra material on test functions on arbitrary compact sets with open neighbourhoods, and partitions of unity (see Chapter I: Test functions, in Hörmander: The Analysis of Linear Partial Differential Operators I, and Appendix B2 in Tools). Thm 5.4 with proof. | Ch. 5: 1. There are exercises in Haroske & Triebel: Distributions, Sobolev Spaces, Elliptic Equations (Springer Ebooks); for example Exc 3.16 and 3.18. |
| 11 | Sec. 5.2 (cont'd)–5.3: Thm 5.5 with definition of \(C^k(\overline \Omega)\) (no proof). Thm 5.6. Thm 5.7 without proof. Prop 5.8. Appendix B3. Thm 5.9 stated for \(W^{k,p}\), sketch of proof for \(W^{1,p}\). Thm 5.10 and 5.11 stated as one, no proof. | No lecture Thursday March 13 |
| 12 | Tue: Sec. 5.4 Sobolev inequalities: Thm. 5.12-5.16. Thu: Thm. 5.16-5.18. | |
| 13 | Sect. 5.4 (cont'd)–5.5: Thms 5.19, 5.20 and 5.21 with proofs. Compactness of embeddings on compact domains: Thms 5.23 and 5.24 with proofs. Thm 5.25 with sketch of proof. Note that one needs \(C^k\)-boundary for an extension of the same regularity, and hence for the embedding theorems for \(k > 1\). | |
| 14 | Tue: Sec. 9.8 (The porous medium equation): pp. 189-191. Thu: Sec. 9.8: pp. 192-195. | Revised Sec. 9.8 |
| 15 | Tue: Sec. 9.8: pp. 196-198. Thu: Solitary solutions of nonlinear dispersive PDEs through constrained minimisation. Lecture 1. My (very) handwritten lecture notes. | Basically following Section 2 in John P. Albert, Concentration–compactness and the stability of solitary-wave solutions to nonlocal equations. Applied analysis (Baton Rouge, LA, 1996), 1–29. |
| 16 | Easter | |
| 17 | Tue: Dispersive PDEs, Lecture 2. Lecture notes for the first two lectures | |
| 18 | Tue: Dispersive PDEs, Lecture 3. In Simastuen Lecture notes for all three lectures | Lecture in Panopto |
Reference group
The group comprises Casper Evensen casper [dot] evensen [at] ntnu [dot] no, Johannes A. Kjær johannes [dot] a [dot] kjar [at] ntnu [dot] no og Robin Ø. Lien robin [dot] o [dot] lien [at] ntnu [dot] no.
Syllabus
We will cover (tentatively): Ch. 1, Sec. 2.1, 2.2 (part of), 2.3, 2.4, Ch. 3, Sec. 4.1-4.4, 4.7, Sec. 5.1-5.5, Sec. 9.8, Dispersive equations
Exam
The exam is oral (language English or Scandinavian). Grading will be Pass/Fail. The exam will be held between 9 am and 5 pm the following dates:
Thursday May 15 Tuesday June 3 Monday June 23
Please send an Email to both lecturers at the latest May 1 with your preferred times. Since there are not enough slots within one day to accommodate all students, we ask you to indicate all times possible.
The exam has two parts:
(20 min): One of the topics given below is randomly selected and the candidate then gives a presentation of it on the blackboard (no slides). Focus on the main parts of the topic, or if the topic is big, a selected part of it. There should be at least one proof. Notes are allowed – but watch the time – 20 min only.
(25 min): Examination in all of the curriculum. No notes.
Topics (from Tools from the Toolbox):
(A) Sections 2.1-2.2; (B) Chapter 3; (C) Sections 4.1-4.3; (D) Sections 4.4, 4.7; (E) Sections 5.1-5.3; (F) Sections 5.4, 5.5 (Thm. 5.23); (G) Sec. 9.8 or concentration–compactness/dispersive PDEs (see lecture notes above).