# MA8105 Nonlinear PDEs and Sobolev spaces · Spring 2023

## About the course

- 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, \(L^p\) 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 10:15–12:00 in GL-KH KJL23
- Thursdays at 08:15–10:00 in GL-KH KJL21
**NOTE: New day and room! Effective week 5.**

- Tuesday week 3, 17 January (Harald): Here are some notes on chapter 1 (updated after the lecture) in the book – since the book has no proofs in ch 1, and I would like to provide at least a hint of some. Also, a shameless plug for one of my own notes on functional analysis, where you can find some proofs not found in the book for this course. (
*Not*considered mandatory reading, of course.)

From Ch 1: Inequalities. Started on Ch 2. - Wednesday week 3, 18 January (Harald): Some notes related to Ch 2 (updated after the lecture).

Covered some material in the beginning of Ch 2 – more detailed description coming soon. - Tuesday week 4, 24 January (Helge): We introduced uniform convexity, and showed Eberlein-Smuljan (Thm. 2.6). Prop. 2.8 wasn't proved, but we proved Prop. 2.9.
- Wednesday week 4, 25 January (Helge): We define weak-star convergence, and go through Ex. 2.12. We do the proof of The. 2.14 and Prop. 2.16. We did not mention Alaoglu's thm., but we gave the complete proof of Arzela-Ascoli thm. (Thm. 2.21).
- Tuesday week 5, 31 January (Helge):
- Thursday week 5, 3 February (Helge):
- Tuesdagy week 6, 7 February (Harald): More fundamentals about distributions, continuity and convergence of test functions, support of distributions, distributions with compact support, convergence of distributions up to the trouble with rapid oscillations.
- Thursdag week 6, 9 February (Harald): Differentiation of distributions, weak derivatives of locally integrable functions, integrating distributions on \(\mathbb{R}\), convolution of distributions with test functions, associativity (\((T*\varphi)*\psi=T*(\varphi*\psi)\) and therefore \((T*\varphi)*\psi=(T*\psi)*\varphi\)).

Unfortunately, I got a bit lost in the proof, and ended up with a sign mistake and some minutes overtime. ➔Here is a note that fixes the problem (except for the overtime). The note also includes some preliminaries to set the stage, and as a bonus, it contains something I did not have the time for: The proof of Proposition 3.7. I think the proof in my note is more direct and simple, but I also include some thoughts on why the book's proof works at the end. - Tuesday week 7, 14 February (Helge): We did the example on p. 49f. After that we continued on p. 54, and ended in the middle of Example (v) on p. 57.
- Thursday week 7, 16 February (Helge): Finished Example (v), and continued with convolution of functions with distribution, and the convolution of two distributions. Introduced fundamental solutions, and mentioned Malgrange-Ehrenpreis theorem. Computed the fundamental solution for the Poisson equation.
- Tuesday week 8, 21 February (Helge): Computed the fundamental solution for the wave equation. We did the structure theorem for distributions. We started on Ch. 4, and did the Holder inequality.
- Thursday week 8, 23 February (Helge): We proved the generalized Holder inequality. Furthermore, we showed a few other inequalities about Lp spaces. We showed that Lp spaces are nested if the measure is finite. Finally, we showed the Minkowski, the Chebychev, and the Jensen inequalities, and a fundamental inequality for convolutions, essentially completing Sec. 4.2.
- Tuesday week 9, 28 February (Helge): We proved Kolmogorov–Rise–Sudakov (Thm. 4.1), and convergence in measure, proving Prop. 4.2, part (i), but not (ii) and (iii). Remark 4.3 was not discussed.
- Thursday week 9, 2 March (Helge): We proved Brezis–Lieb (Thm. 4.11), and studied convergence in L1. We stated, but did not prove Dunford–Pettis (Thm. 4.16).
- Tuesday week 10, 7 March (Helge): We did not prove Prop. 4.19 or Thm. 4.20. Studied Ex. 4.23 & 4.24 in detail. You are strongly encouraged to do Exercise 11&12 in Ch. 4.
- Thursday week 10, 9 March (Helge): We started with Thm. 4.25. Sections 4.5-4.8 were not covered.
- Tuesday week 11, 14 March (Harald): Ch 5, up to and including Thm 5.4, App B.2 (partions of unity). Also, an example: \(f(x)=\ln \lvert x \rvert\) belongs to \(W^{1,p}_{\text{loc}}(\mathbb{R}^d)\) if \(1 \le p < d\). By setting it to zero for \(\lvert x \rvert >1\), we get a function in \(W^{1,p}(\mathbb{R}^d)\). No need to smooth it out, as I indicated in the lecture. Combining translates of this function, we can create a function in \(W^{1,p}(\mathbb{R}^d)\) which blows up at every point with rational coordinates. Lesson: These functions can look quite bad, despite having weak derivatives.
- Thursday week 11, 16 March (Helge): We covered Thm. 5.5, 5.6, 5.7.
- Tuesday week 12, 21 March (Harald): Prop 5.8 and Theorem 5.9.
- Thursday week 12, 23 March (Helge): Theorems 5.10 and 5.11. We did not complete the proof of Thm. 5.11, but will wrap it up in the next lecture.
- Tuesday week 13, 28 March (Helge): Theorem 5.11 completed. We did the proof of Theorem 5.11 in the case \(p=1\).
- Thursday week 13, 30 March (Helge): We completed the proof of Theorem 5.12. In addition, we proved Theorems 5.13, 5.14, 5.15. We defined Hölder spaces, and stated Theorem 5.16 and Theorem 5.17 (Morrey's inequality).
- Week 14 is Easter holiday.
- Tuesday week 15, 11 April: no lecture
- Thursday week 15, 13 April (Helge): We proved Theorems 5.17 and 5.18.
- Tuesday week 16, 18 April (Helge): We will do the cases \(p=d\) (Thm. 5.21) and \(d=1\) (Thms. 5.19 and 5.22). We also presented the first part of Thm 5.23, and part of the proof of part (i).
- Thursday week 16, 20 April (Helge): We completed the proof of Thm. 5.23. We started discussing fundamental properties of the generalized porous medium equation \(u_t=\Delta \phi(u)\) from notes by Harald (updated to
**v8**2023-04-26). We covered p. 1-5. - Tuesday week 17, 25 April (Helge): We covered pp. 6-10.
- Thursday week 17, 27 April (Helge): We proved existence of weak solution (up to p. 17).
- Tuesday week 18, 2 May (Helge):
**[last lecture]**We showed uniqueness of solutions to the porous medium equation.

## Exam

The exam is oral (language English or Scandinavian). Grading will be Pass/Fail. Proposed dates for the exam are 1–3 June.

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. Proofs are encouraged.

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 and notes on the porous medium equation):

**(A)** Section 2.1; **(B)** Chapter 3; **(C)** Sections 4.1-4.3; **(D)** Sections 4.4; **(E)** Sections 5.1-5.3; **(F)** Sections 5.4, 5.5 (Thm. 5.23); **(G)** Porous medium equation.