# TMA4305 Partial Differential Equations 2019

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• (2019-09-06) There is no exercise class next week (September 13). I have posted exercises, however, and I will post a solution later.
• (2019-09-05) Another note added below. I had forgotten I made that note, but here it is! Look for the green background as always for notes.
• (2019-08-22) As I said in today's lecture, I made a short note to supplement the book. You find it below (look for the green background).
• (2019-08-21) It is now clear that the lectures will be in English.
As a consequence, the exam will be given in English as well.
• (2019-06-03) I will be teaching the course in the fall of 2019. The lecture will be in English if required by at least one student. If you plan on taking the course, and need the lectures to be in English, drop me a line.
Harald Hanche-Olsen

### Lectures; past, present, and future

Week Chapter/section Remarks and notes
34 Ch 1–3
Notes
Read Ch 1 on your own. Also have a look at Ch 2 on your own – but we will return to it when needed (especially the higher dimensional calculus). 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.)
Covered introductory and preliminary material, including some notation.
The concept of well-posed problem [Jaques Hadamard 1902]: Solutions exist, are unique, and depend continuously on data.
Basic theory of transport equations and conservations laws in one space dimension, balance law in multiple dimensions.
Notes: Collisions of characteristics (A5 for the screen, A4 for printing)
35 Ch 3 end
Ch 4: 4.1–4.4
Start on the wave equation (Ch 4 in Borthwick): Characteristics and d'Alembert's solution (§4.2), and Duhamel's principle (§4.4) (if we get that far)
Wednesday: Finished Ch 3 with a look at a traffic model.
Also proved that the characteristic method applied to the IVP $u_t+a(t,x,u)u_x=f(t,x,u)$ with $u(0,x)=g(x)$ actually solves the problem. I used differentials to handle the algebra; much easier than transforming partial derivatives between coordinate systems.
Thursday: Wave equation, D'Alembert's solution, Duhamel's principle.
36 Ch 4: 4.6, 4.7
Notes
The wave equation in higher dimensions, energy method for uniqueness
Wednesday: Solution formulas for the wave equation in 3 and 2 space dimensions. Huygens' principle in 3D, not in 2D.
(A similar technique works for $2k+1$ dimensions, with subsequent descent to $2k$ dimensions.)
Thursday: Proof that the solution formula in three dimensions actually solves the problem.
Energy methods for uniqueness and domains of dependence in the wave equation.
Notes: On the energy method for wave equations (A5 for the screen, A4 for printing
These notes may be thought of as an alternative to, and expansion of, §4.7)
37 Notes
Ch 6
A bit more from the above notes on the energy method.
Start chapter 6 (heat equation)
Wednesday: Proved Theorem 2 from the end of the note (above) on the energy method for wave equations. The current version in that note is too complicated. I will update it to match the way I lectured it.
Started on the heat equation, and derived the formula for the standard heat kernel in $n$ space dimensions.
Thursday: The solution of the initial value problem for the heat equation, and Duhamel's principle for the non-homogeneous heat equation.
As part of the proof I did a partial integration, moving the Laplacian from one term to another. This is really an application of Green's second identity (B Theorem 2.11): The boundary integrals vanish when we integrate over a large enough region containing the support of $f$.
38 Notes
Ch 9: 9.5
and more
Notes on the parabolic maximum principle (see below)
Perhaps start working on Ch 9 and my associated notes.
Wednesday: The weak maximum principle for the heat equation, on bounded and unbounded domains. Corollaries are continuous dependence of data, uniqueness.
Notes: Weak maximum principle for the heat equation (A5 for the screen, A4 for printing)
These notes are an alternative to §9.5
Updated 2019-09-18 with an even better version of the maximum principle on $\mathbb{R}^n$ than the one in the lecture!
Updated 2019-09-19 (removed an erroneous example)
Thursday: Started notes on the Laplace equation (below). Covered the mean value property, and proved its equivalence to harmonicity. Regularity (harmonic functions are $C^\infty$, and the strong and weak maximum principle.
New notes: Harmonicfunctionology (A5 for the screen, A4 for printing)
These notes are related to §§9.1–9.4
39 Notes
Ch 9: 9.1–9.4
Notes on the Laplace equation (continued)