# MA8105 Nonlinear PDEs and Sobolev spaces · Spring 2021

## Messages

Newest on top:

- (2021-04-21)
**The lectures the last week of the semester, 22 and 23 April, will be in-person** - (2021-03-25)
**Effective today, we are unfortunately back to video lectures until further notice** - (2021-01-29) The Thursday lectures will be in F3 from now on (when possible).
- (2021-01-21)
**The lectures starting on Thu, 28 Jan will be physical and take place at Gløshaugen.** - (2021-01-15) The former home page has been move to “about”. This page will be for messages and notes on the lectures.

## Lectures

- 14 Jan (Harald): Some inequalities from
**Ch 1**.**2.1:**Basics on norms, completeness, inner products, bounded linear maps, dual spaces, bidual. Also outlined the proof of the Hahn–Banach theorem (app.**A**).

In the video recording, the right edge is obscured. Use the notes transcript to see the missing part.

Panopto video link, raw notes transcipt - 15 Jan (Harald):
**2.1:**A bestiary of sequence spaces, Hölder's inequality (and when it is an equality), Riesz representation theorem. Weak convergence, a quick look at Banach–Steinhaus (no proof), a brief look at the wild nature of the dual of \(\ell^\infty\).

In the video recording, the right edge is obscured. Use the notes transcript to see the missing part.

Panopto video link, raw notes transcript

**Recommended exercises:**2–4 on p. 37 (but I covered much of ex 4 in the lecture, actually). - 21 Jan (Helge): Started with uniform convexity on p. 12. Video and notes: Panopto video link, notes
- 22 Jan (Helge): Started with Prop. 2.9. Covered up to and including Alaoglu, and stopped in the middle of the proof of Prop. 2.20. Panopto video link, notes
- 28 Jan (Harald): Started by proving Thm 2.18 and prop 2.20 together. Ended with the Arzelà–Ascoli theorem (II, thm 2.23)). But did not prove that (1) and (2) are necessary.
- 29 Jan (Harald): Started with imbeddings (end of §2.1), moved on to §2.2 Calculus in Banach spaces.
*Skipped*Nemitski operators and Krasnosel'skii's theorem. Calculated the Fréchet derivative of inversion \(f(A)=A^{-1}\) on \(\mathcal{L}(X)\): The result is \(f'(A)B=-A^{-1}BA^{-1}\). (Compare the defivative of the real function \(f(x)=x^{-1}\).) - 4 Feb (Harald, in F3): start with Newton iteration, move on to implicit and inverse functions theorems.

Video recordings: one, two (Panopto). And a little note for what I did not have time for at the end. - 11 Feb (Helge): Finish §2.4, start §2.5 (Some results for Hilbert spaces). Panopto video
- 12 Feb (Helge): Example on the use of Lax-Milgram and Galerkin approximation. Start on distributions. Panopto video, notes
- 18 Feb (Helge): Derivative of distributions, the fundamental theorem of distributions. Panopto video
- 19 Feb (Helge): Convergence of distributions, integrals. Panopto video
- 25 Feb (Harald): More work with convolutions, also introduced the spaces \(\mathcal E=C^{\infty}\) and its dual \(\mathcal E'\) (distributions with compact support). Panopto video
- 26 Feb (Harald): Yet more distributions, dense imbeddings, just got started on fundamental solutions. Panopto video.

Unfortunately,**the first half did not get recorded**. I had forgotten one annoying aspect of how that works in zoom. Also, especially in the first (unrecorded) half, I forgot to reposition the camera a bit too much. My apologies.

As compensation, especially since I went outside the*Tools*a bit,**here are some notes**covering much of what I lectured this week – but not quite in the order of the lectures! - 4 Mar (Harald): Fundamental solutions for the wave equation and the Laplace operator. Lebesgue spaces, start on inequalities. Panopto video: one, two.

Notes for the technologically curious: My laptop would not see the camera in the auditorium. It connects by USB, but as far as I could tell, no device showed up on the USB bus. (Yes, I know that the B in USB stands for “bus”.) So I recorded the first half with the built-in laptop camera. It turned out acceptable, I think, especially after I moved the laptop – but not good. In the second half, I used an old iPhone 6s with a small tripod and the camo software, with a much better result. The photographer in me protests that the white balance is wrong, but I'll make sure to adjust that next time. - 5 Mar (Harald): (Summary forthcoming …) Quite mysteriously, the auditorium camera worked today: Panopto video.
- 11 Mar (Harald): (Summary forthcoming …) I had forgotten to plug the laptop into power, so it died ten minutes before the end. I did not say much in those ten minutes, and Helge will cover it again tomorrow. Panopto video
- 12 Mar (Helge): The rest of the proof of Kolmogorov–Riesz–Sudakov.Then convergence in measures, and some counterexamples. General convergence in Lp spaces for p strictly between 1 and infinity. Panopto video (I think that maybe only the first half was recorded.
- 18 Mar (Harald): Panopto video
- 19 Mar (Harald): Panopto video
- 25 Mar (Helge): There were serious technical issues with the equipment with the result that the lecture got delayed. Panopto video, notes. We covered partition of unity, and approximation of Sobolev functions by smooths functions
- 26 Mar (Helge): Panopto video, notes. We covered straightening out of boundaries, further approximations of Sobolev functions by smooth functions, and how to extend a Sobolev function defined on a bounded domain to the full space.
- 8 April (Helge): We covered the existence of Sobolev traces, and started on the study of functions with vanishing traces. notes, Panopto video.
- 9 April (Helge): We proved the Gagliardo-Nirenberg-Sobolev inequality. notes, Panopto video.
- 15 April (Harald): Some corollaries to the GNS inquality, Poincaré's inequality, Morrey's inequality (except for the conclusion and final statement of the result). I messed it up at the end, should not have dropped the averages (as Halvard S correctly pointed out in chat, but I did not see it in time). It turns out it was easy to fix, and I
*have*now fixed it, in the notes. So please look at those. Panopto video, fixed-up notes. - 16 April (Helge): Completed Morrey's theorem, and just briefly started on the Rellich-Kondrachov theorem. notes, Panopto video
- 22 April (Harald): The Porous Medium Equation (PME). My own notes (now at
**version 3**), inspired by similar notes from 2019 (see weeks 14–15 here) and also informed by the book*The Porous Medium Equation – Mathematical Theory*^{(PDF)}by Juan Luis Vázquez.

Panopto videos: Part 1, part 2. In the second part between minutes 35–40 or so, all the action was off camera. What you missed is the weak formulation of the problem. You can find it in the notes, and also Helge should cover it tomorrow. - 23 April (Helge): We finished the porous media equation, and completed the Rellich-Kondrachov compact imbedding theorem in the case with p<d. Panopto video (only the last 45 mins were recorded)