# TMA4305 Partial Differential Equations 2019

## Messages

• (2020-08-06) The solutions to today's exam problems are available. See “Old exams” in the menu.
• (2019-12-04) The solutions to today's exam problems are available. See “Old exams” in the menu.

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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

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
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 (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
These notes are an alternative to §9.5
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.
Notes: Harmonicfunctionology
These notes are related to §§9.1–9.4
39 Notes
Ch 9: 9.1–9.4
Notes on the Laplace equation (continued)
Wednesday: Solution of the heat equation in the (2-dimensional) unit disk, Poisson kernel (B §9.1). Then back to my notes; started on the Poisson equation.
Thursday: The fundamental solution $\Phi$ for the Poisson equation and its basic properties; solution $u=\Phi*f$ for the Poisson equation $-\Delta u=f$ where $f \in C^2_c(\mathbb{R}^n)$; and started on an introduction to Green's functions by deriving the formula $u(y) = -\int_{\Omega} \Phi(x-y)\,\Delta u(x)\,d^n x + \int_{\partial\Omega}\bigl(\Phi(x-y)\partial_nu(x)-u(x)\partial_n\Phi(x-y)\bigr) .$ Briefly stated how to get rid of the troublesome $\partial_nu$ term in the boundary integral by adding a harmonic function to $\Phi(x-y)$.
40 Notes Poisson integral formula in 2 or more dimensions, Perron's method (from updated “harmonicfunctionology” notes, see above).
Wednesday: The Dirichlet problem in balls, solved by Poisson's integration formula.
Thursday: The Dirichlet problem in bounded domains. Showed that the pointwise supremum of all subsolutions is harmonic. (Note that the note was updated after the lecture; see above.)
41 Notes
7.1–7.4
Finish “Harmonicfunctionology” notes with a look at barriers and boundary values; then back to the book, where we start work on the more functional analytic part of the course. We start with a quick introduction (review for some) of function spaces.
Wednesday: Finished the “Harmonicfunctionology” note, including some material that I ought to include: A barrier for two dimensions, and an example in three dimensions. (Now in the notes.)
Thursday: Chapter 7: Lebesgue measure and integral, $L^p$ spaces. Also $L^p_{\operatorname{loc}}(\Omega)$, not mentioned in the book.
42 7.5–7.6
Start Ch 10
Finish Ch 7. Move on to weak derivatives and weak solutions to PDEs, Sobolev spaces.
Wednesday: §7.5 (orthonormal bases) real quick, §7.6 (selfadjointness for the Laplace operator) even quicker; but I left the discussion on eigenvalues and -functions for later.
§10.1 (test functions and weak derivatives), skipped §10.2 (for now), beginning of §10.3 (Sobolev spaces).
Thursday: §10.3, §10.4 (Sobelev spaces and Sobolev regularity).
43 10.2, 10.5, 11.1–3
Notes
Sobolev spaces and boundary traces. Weak solutions (perhaps including a brief intro to distribution theory).
Wednesday: Weak solutions of continuity equations – The Rankine–Hugoniot shock condition, weak solutions of the Posisson equation, Dirichlet's principle.
Thursday: Dirichlet's principle contiunued, the Poincaré inequality, existence of weak solutions by minimising the Dirichlet functional. Boundary traces.
Notes: Boundary traces.
44 Ch 11
11.4–5
Elliptic regularity, spectral theory for the Dirichlet Laplacian.
Wednesday: §11.4 – Elliptic regularity.
Thursday: §11.5 – The spectral theorem for the Dirichlet Laplacian
45 Ch 11, 12
11.6, 11.8, 12.2–3
The min-max theorem, Rellich's theorem, Euler–Lagrange equations, distribution theory, applications to PDE
Wednesday: Finishing remarks on the Spectral theorem for the Dirichlet Laplacian
Notes, §11.6 – compactness and Rellich's theorem
Thursday: §11.8 Euler–Lagrange equations (generalising the Dirichlet principle)
Started Ch 12 on distributions
Notes: Compactness in Banach spaces
Handwritten note (I may type it up later) meant to cover a possible gap in students' background knowledge .
46 Ch 12 Distribution theory, applications to PDE
Wednesday: §12.4 Fundamental solutions, convolutions. I defined the support of a distribution as the smallest closed set so that $(u,\phi)=0$ whenever the support of $\psi$ is disjoint from the set. I also stated (without proof) that distributions with compact support have finite order, in that $(u,\psi)$ can be bounded by a finite number of derivatives of $\psi$. Moreover, such a distribution is a finite linear combination of derivatives of functions. (This is mostly useful background material. I really should write it up, briefly.)
Thursday: A bit more on convolutions, then §12.6 time-dependent distributions, applied to conservation laws (rederived the Rankine–Hugoniot shock condition) and the 1-D wave equation / wave kernel
47 Summaries Final week of lectures. Summaries / repetition, maybe go through some exam problems
Wednesday: Summarised some basics regarding the divergence theorem and partial integration; conservation laws; the wave equation; just barely got started on the heat equation.
Thursday: Continued summarising: More on the heat equation; then harmonicfunctionology (mean value property, sub- and superharmonic functions, weak and strong maximum principles, the fundamental solution, Green's function, Poisson kernels, the Dirichlet principle, Poincaré's inequality, … and then I ran out of time.
End of lectures.