# TMA4305 Partial Differential Equations 2017

## Messages

In reverse chronological order:

• (2017-11-29) You can find solutions to today's exam, along with the (corrected) problems at the top of the list under “Old exams” in the menu. If you find any mistakes or misprints in the solution, let me know.

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• (2017-11-23) Office hours: I will be available in my office Friday, 10–12 and 14–15. Also Monday 10–12.
• (2017-11-22) For the exam, you may bring one yellow A4 sheet of paper stamped by the Department of Mathematical Sciences. I kid you not! You can pick up a such sheet outside the department offices on the 7th floor of Sentralbygg 2. You may write whatever you want on the sheet, both front and back.
In other news, brief – and rough – answers to the first three exercises have appeared on the Exercises page. Look for green. More will appear soon.
• (2017-11-21) The syllabus / reading list / pensumliste (whatever you call it) is up on the Course information page.
I keeping getting questions regarding answers to the exercises; I am sorry that I haven't found time for them yet, but I should be able to start getting them posted soon. But first, I have to prepare the final lectures.
• (2017-11-05) Minutes from the second reference group meeting are posted. See the bottom of the course information page.
• (2017-09-08) I just posted a message via Blackboard. You should all have received a copy by email, and you can see it on Blackboard. Let me know if these assumptions are wrong. The intention is to use Blackboard for announcements, especially for announcements that I don't want the entire world to see or especially urgent ones. Otherwise, this web site is where the action is and will be.
• (2017-08-30) I fixed a wrong link to Fritz John's book (see Course information). I also updated the small note on quasilinear equations with what seems like a better approach (see the table below).
• (2017-08-26) I am reorganizing these pages. The first exercise is (finally!) available; see the menu!. The last two problems rely on material not in the book. I lectured on this last Thursday, and some very brief notes will appear here shortly.

## Lectures; past, present, and future

Week Chapter/section Remarks and notes
34 Ch 1–3 Read Ch 1 on your own. We may return to sections in Ch 2 as needed. For needed background on well-posedness theory for ODEs, consult for example my notes on dynamical systems, Ch 1, pp. 2–7. (No need to study the proofs.)
Notes: On first-order quasilinear equations (A5 for the screen, A4 for printing) Updated 2017-09-18
35 Ch 3, start Ch 4 Finished Ch 3 and the notes posted above.
Next, the classification of second-order equations; see the first two and a half pages of chapter 2 from Fritz John's book. You can find it on Springerlink; see the reference in the course information.
Started on the wave equation (Ch 4 in Borthwick): d'Alembert's solution, and Duhamel's principle (or method, as the book says).
36 Ch 4 More on the solution of the forced wave equation using Duhamel's principle: I made the solution formula more concrete, and proved again that it solves the problem. Resonance (example: $u_{tt}-u_{xx}=\cos(\omega t)\cos(x)$).
Darboux's formula and the wave equation in three dimensions (solved by Kirchoff's formulA); descent to two dimensions, and Huygens' principle.
Finally, the energy method in one space dimension.
37 Ch 4, 6 Review a small bit of higher dimensional calculus before we moved on to uniqueness by the energy method. Quick review of what we have learned about the wave equation.
Then we started working on Ch 6, the heat equation. Only got as far as deriving and defining the heat kernel (in one dimension), without actually mentioning the name; and wrote up the solution to the Cauchy (initial value) problem, so far only presented as an “educated guess”. Important concept along the way: Scale invariant solutions.
Notes: The energy method for wave equations. (A5 for the screen, A4 for printing) Updated 2017-09-13
38 Ch 6 The heat equation: Heat kernel in higher dimensions, initial value problem, Duhamel's principle.
Notes: Weak maximum principle for the heat equation (A5 for the screen, A4 for printing) This is a new note. It feels a bit unfinished, and may well contain mistakes. But I thought it important to get it out there. (Updated 2017-09-22 with two more exercises, and again 2017-09-25.)
39 Ch 9 The Laplace equation, strong and weak maximum principles for harmonic and subharmonic functions, Poisson equation, fundamental solution.
Why? I want to do more PDE theory “with bare hands” before we start building more abstract machinery.
Do read Edward Nelson's marvellous 1961 proof of Liouville's theorem! It is only nine lines of text, without a single formula. Available at DOI: 10.2307/2034412 (for JSTOR access, you must be within the campus network, either physically or using VPN).
Notes: Harmonicfunctionology (A5 for the screen, A4 for printing) (Extended 2017-10-04 (see below) and updated 2017-10-11.)
40 §12.5, §9.1 Green's function; general theory, explicit formula on the unit ball in $\mathbb{R}^n$, relation with Poisson's formula on the unit disk $\mathbb{D}\subset\mathbb{R}^2$
Notes: Harmonicfunctionology (see previous week, above) has been extended with notes on Green's functions and the Poisson kernel, and updated with a direct proof of the symmetry of Green's function.
41 Ch 7 After presenting a better better proof of the symmetry of Green's function, we are now working on Function spaces. (Very quickly, with enough details skipped to fill another course or two.) May even dip our toes into Ch 10 if there is time.
(The note on Harmonicfunctionology has seen yet another update; se above.)
42 Ch 10 Weak derivatives, weak solution of continuity laws (example: traffic flow with shocks)
43 Sobolev regularity, with a quick look at the Dirichlet principle (ch 11) at the end
44 Ch 11 Variational methods – and a quick peek into Ch 12
45
46 Ch 12 Distributions.
Note that sections 12.4–5 were already (mostly) covered in the “harmonicfunctionology” note.
47 Summary, overview etc … end of lectures