# TMA4305 Partial Differential Equations 2018

## Messages

- (2018-12-11) Solutions to today's exam can be found under “old exams” in the menu. Please let me know if you find misprints or, even worse, mistakes!

## 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 2018-09-15; but only the document font is changed) |
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35 | Ch 3 end, notes Ch 4: 4.1, 4.2, 4.4 | Finish Ch 3 and the notes posted above. Start on the wave equation (Ch 4 in Borthwick): Characteristics and d'Alembert's solution (§4.2), and Duhamel's principle (or method, as the book says – §4.4).Note: A mistake near the end of the Tuesday lecture is corrected here. |

36 | Ch 4: 4.4, 4.6 | Started with a direct proof that the formula resulting from Duhamel's principle (§4.4) really solves the problem, then moved on to three and two space dimensions (Darboux's formula, Kirchoff's and Poisson's solutions, Huygens' principle (§4.6).Note: A mistake near the end of the Thursday lecture is corrected here. |

37 | Ch 4: 4.3, 4.7 Ch 6: 6.2–3 | Boundary value problems for the wave equation (§4.3), and the energy method for proving uniqueness (§4.7).Peter Lindqvist lectured on Thursday: He computed the domain of dependency for the wave equation, using the energy method. Then he started on Ch 6 (the heat equation), finding the solution for the one-dimensional case by convolution with the heat kernel (§§6.2–3) |

Notes: On the energy method for wave equations (A5 for the screen, A4 for printing) (updated 2018-11-20) |
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38 | Ch 6 | Continued into ch 6. On Tuesday, I introduced the n-dimensional heat kernel, applied it first to the initial-value problem for the homogeneous heat equation, then Duhamel's principle for the non-homogeneous equation. I ended in the middle of the proof that this actually solves the equation. For easy reference, here is a snapshot of where we ended up (the thumbnail links to the full size image). Note that there is a term missing in the top line of the blackboard image above! It comes from differentiating wrt the upper limit in the outer integral. I corrected that on Thursday, then I finished the proof.Next, we moved on to the parabolic maximum principle and questions of uniqueness of solutions on bounded domains. |

Notes: On the parabolic maximum principle (A5 for the screen, A4 for printing) |
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39 | Notes Ch. 9: 9.2–9.3 | We turned to the study of the Laplace equation and its non-homogeneous version, the Poisson equation. Followed the notes (Harmonicfunctionology), and ended just before the section on Green's function. I have a few extra remarks about the lecture that are too long for this table, so I put them on a separate page.Why so soon? Because I want to get more out of “elementary” (not necessarily easy) methods before we start developing the heavy machinery needed later. |

Notes: Harmonicfunctionology (A5 for the screen, A4 for printing) (updated 2018-10-02)The notes can be seen as a replacement or alternative to §§9.2–9.3. |
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40 | Notes Ch. 9 Ch. 12: 12.5 | Finished the notes, and also the rest of Chapter 9. We start with Green's functions in Harmonicfunctionology. |

41 | Ch 7: 7.1–7.5 Ch 10: 10.2 | (Peter Lindqvist lectured.) Function spaces. I consider this background material – much of it will be known to many of you already. You will need to understand the concepts for what comes, but beyond that, there is little or no need to study proofs. Weak solutions of continuity equations; the Rankine–Hugoniot shock condition. |

42 | Ch 10 | Weak derivatives and weak solutions to PDEs. Sobolev spaces and boundary traces (see note). Sobolev regularity. I have shown that the Sobolev regularity theorem can be reduced to the periodic case of \(H^k(\mathbb{T}^n)\), and outlined the basic idea of how the theory of Fourier series will finish the proof. |

Notes: Boundary traces (A5 for the screen, A4 for printing) |
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43 | Ch 10: 10.4–10.5 Ch 11:1 11.2–11.4 | Finish the Sobolev regularity theorem. Move on to the Dirichlet principle and weak solutions to the Poisson equation; elliptic regularity. |

44 | Ch 11: 11.4–6 | Elliptic regularity (continued), spectral theorem for the Dirichlet Laplacian. |

45 | Ch 11: 11.6, 11.8 Ch 12: 12.2–12.4 | The min-max theorem. Distribution theory. Note that we already covered 12.5 (Green's function) earlier in the Harmonicfunctionology note. |

46 | Ch 12: 12.6 | More distribution theory with applications to PDEs. (Only the Tuesday lecture this week. The lecture on Thursday, 15 November is cancelled.) |

47 | Catch loose ends (if any), summary, overview … end of lectures |