# 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
34 Intro
B 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
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
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)
24 Sept Friday Laplace Eqn. Poisson's formula, Mean Values. Two lectures from 2020 recorded
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).
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)
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.
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.
47 Last Week with lectures. Repetiton, examples
25 Nov Thursday: Repetition
26 Nov Friday: Last Lecture Examples, Repetition
From week 46: Section 2.1 in Fritz John (SpringerLink; du må være innenfor NTNUs nett fysisk eller på VPN. )