# 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.

## Lectures; past, present, and future

Week | Chapter/section | Remarks and notes |
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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 |
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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 |
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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.) |
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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.) |
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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. |
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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 |