# TMA4305 Partielle differensialligninger 2021

## Meldinger

OFFICE HOURS BEFORE THE EXAM: Tuesday 30.XI in room 1152 SB II, from 2 o'clock to 4 o'clock.

Lectures will be in English.

Reference Group Meeting 17.IX.2021 and 26.XI.2021.

Exam 2.XII.2021 Text and Solutions. Grading according to the normal scale: 0 - 43 F; 44 -57 E; 58 - 70 D; 71- 83 C; 84-96 B; 97 -110 A. The eleven problems have the same weight.

## Forelesninger

**B** below means the book of Borthwicks.

Uke | Dato | Ref | Hva |
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34 | IntroB 1, 2.6, 3.4 | Introduction to the course, | |

We will cover material in Ch. 2 when we need it | |||

25 aug | Wed: Introduction, existence and uniqueness of solutions for ODEs. 1st order quasilinear equations in two variables. | ||

26 aug | Thu: Examples on quasilinear equations, in particular, Burgers' equation. Derivation of the linear wave equation. Started derivation of d'Alembert's solution. | ||

35 | B 2.5, 3.3, 4.1–2, notat | Mer generelle kvasilineære systemer, bølgeligningen | |

⮕ Notat: First order quasilinear equations ( A5 for skjerm, A4 for utskrift)Oppdatert 2020-08-30 Notes on weak solutions Rankine-Hugoniot |
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2 sept | Torsdag: d'Alembert's solution finished. Duhamel's Principle. Energy considerations, uniqueness | ||

3 Sept | Friday: Darboux's equation. Kirchhoff's formula, n = 3. | ||

36 | B 4.3, 4.4, 4.6, 4.7 | Bølgeligningen | |

9 Sep | Thursday: Poisson's formula via descent from Kirchhoff's, randverdiproblemer | ||

10 Sep | Friday: Local energy, uniqueness, past light cone. Derivation of the Heat Eqn. | ||

37 | B 6.1–4 | The Heat Equation | |

16 Sep | Thursday: Heat Kernel, solution of the initoal value problem (the Cauchy Problem)From Fouriertransform | ||

17 sep | Friday: Uniqueness via energy considerations. Duhamel's principle again. (Weak) Maximum Principle. B.9.5 |
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38 | B 6.4, 9.5, 9.1, 9.2, notes 2020 | Heat Eqn, the Laplace Eqn. | |

23 Sept | Thursday: Short Summary of Wave Eqn, Maximum Principle in unbounded domain | ||

⮕ Notat: Weak maximum principle for the heat equation (A5 for skjerm · A4 for utskrift) |
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24 Sept | Friday Laplace Eqn. Poisson's formula, Mean Values. Two lectures from 2020 recorded |
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Maksimumsprinsippet for et ubegrenset område (Panopto) | |||

Harmoniske funksjoner og middelverdiegenskapen (Panopto) | |||

⮕ Notat: Harmonicfunctionology (A5 for skjerm · A4 for utskrift) · Ny versjon 2020-10-08 β2: Sterkt forenklet utledning av Theorem 22 («β2» fordi jeg ikke har korrekturlest det veldig nøye). |
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39 | B Ch. 7 | ||

30 sep | Thu: Gave formal definitions of vector spaces, inner products, and norms. Gave a brief introduction to Lebesgue spaces, incl Holder inequality. | ||

1 okt | Fri: Proved Minkowski inequality. Discussed completeness and orthonormal bases. Started on self-adjointess. | ||

40 | B 7.6, 10.1, 3.4, 10.2 | ||

7 okt | Thu: Completed self-adjointness for the Laplace operator. Introduced weak derivatives, and studied characteristics for quasilinear equations. | ||

8 okt | Fri: Studied characteristics in more details, example from traffic flow. Proved the Rankine-Hugoniot relation. | ||

41 | B 10.3, 10.4, 10.5; 9.4 | Sobolev spaces, Weak solutions | |

14 Oct. | Thursday: Sobolev spaces. The proof in 10.4 was skipped. | ||

15 Oct. | Friday: Sobolev's inequality. Maximum Principle 9.4 (-L now +L) |
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42 | B 10.6, 11.1, 11.2, 11.3 (first part) | Weak solutions, variational methods | |

21 Oct | Thu: Weak formulations of evolution equations (Sec. 10.6) | ||

22 okt | Fri: Variational methods, Laplace equation, Poissson equation, Dirichlet principle, Poincare inequality | ||

43 | B 11.3 (last part), 11.4, 11.5 (partial) | Poisson equation, elliptic regularity, spectral theorem | |

28 Oct | Thu: Coercivity and boundedness, existence of unique weak solution of the Poisson equation | ||

29 Oct | Fri: Elliptic regularity, existence of lowest eigenvalue for the Dirichlet Laplacian. Rellich's theorem without proof | ||

44 | B 11.4–11.7 | Elliptisk regularitet, egenverdier for Laplaceoperatoren, … | |

??.? | Onsdag: Elliptisk regularitet, kompakthet | ||

??.? | Fredag: Spektralteoremet for Laplace-operatoren med Dirichlet randbetingelser | ||

⮕ Håndskrevet notat: kompakthet (fordi det ikke er dekket så godt i Lineære metoder?)– se også avsnitt 2.8 i C. Heil: Metrics, Norms, Inner Products, and Operator Theory. |
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45 | B 12.1–12.4 | Theory of distributions | |

11 Nov. | Thursday: Distributions, Dirac's Delta | ||

12 Nov. | Friday: Distributions, Fundamental Solutions | ||

46 | B 12.5 | Distributions, Green's Functions | |

18 nov | Thursday: Green's function, fundamental solutions | ||

19 nov | Friday: Green's functions. The "confusing" B 12.6 is replaced by the Fundamental Solution of the Heat Eqn (… = delta(x,t)). Characteristics and classification of 2nd order quasilinear eqns, F. John: section 2.1. |
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47 | Last Week with lectures. Repetiton, examples | ||

25 Nov | Thursday: Repetition | ||

26 Nov | Friday: Last Lecture Examples, Repetition |
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From week 46: Section 2.1 in Fritz John (SpringerLink; du må være innenfor NTNUs nett fysisk eller på VPN. ) |