Vorlesung: Partielle Differentialgleichungen (PDG1) (WiSe 2020/21)

[UPDATE 07.10.2020:] Due to the present situation, this course will be online (uploaded videos, lecture notes, homework exercises; all details now on uni2work). It will start 03 November 2020.
For updated information, check back here, on LSF, and on uni2work (where all material will be uploaded).
A brief video introduction can be found here.


To access the course material (videos, lecture notes, Zoom data, exercise sheets etc), and to be able to hand in homework, you need to sign up in uni2work.

Lecture (Vorlesung):
Tue 14-16 & Wed 12-14: Online (videos) and via Zoom; see uni2work.   LSF

Exercises (Ãœbungen):
Via Zoom; see uni2work.   LSF

Tutorials (Tutorien):
Via Zoom; see uni2work.

Synopsis (Kurzbeschreibung):
This course gives an introduction to Partial Differential Equations (PDEs), a vast area within Analysis. PDE's play an important role in applications of Mathematics to other sciences (most prominently in Physics and Engineering, but also in Biology and Financial Sciences), as well as in Pure Mathematics (Analysis, Geometry, Stochastics; Algebra less).
Among other things, we will study: the method of characteristics for (non-linear) first-order PDEs, the classification of linear 2nd order PDEs in elliptic, parabolic, and hyperbolic equations, explicit classical solutions for the most prominent such equations (Laplace and Poisson equations, heat equation, wave equation), including boundary value problems and Cauchy problems.

(Die Vorlesung führt in die Theorie der partiellen Differentialgleichungen ein. PDG'en spielen eine zentrale Rolle sowohl in vielen Anwendungsgebieten der Mathematik, als auch in der reinen Mathematik. Behandelt werden, unter anderem, die Charakteristikenmethode, die Typeneinteilung in elliptische, hyperbolische und parabolische Differentialgleichungen, explizite Lösungsmethoden für die wichtigsten Typen linearer PDG'en zweiter Ordnung (Laplacegleichung, Poissongleichung, Wellengleichung und Wärmeleitungsgleichung), Randwert-Probleme, Cauchy-Probleme.)

Audience (Hörerkreis):
Bachelor students of Mathematics (WP16), Master students of Mathematics (WP2), Master students of 'Finanz- und Versicherungsmathematik' (WP49), TMP Master.

Credits:
9 (6+3) ECTS.

Prerequisites (Vorkenntnisse):
Analysis I-III, Lineare Algebra I-II.
You find a handout with the needed facts (without proofs, and to be updated!) in uni2work.

Language (Sprache):
English.

Exam (Prüfung):
There will be a written exam (Es wird eine schriftliche Klausur geben).
There will be a written re-exam (Es wird eine schriftliche Nachklausur geben).
All information in uni2work.

Content (Inhalt):

0. Introduction and Motivation

1. Transport Equations

2. The Laplace and Poisson Equations

   2.1 Boundary Value Problems
   2.2 Gauß, Green & Spherical Means
   2.3 Mean Value Properties, Maximum Principles & Harnack
   2.4 Regularity & A Priori Estimates
   2.5 Green's Function & Poisson's Kernel
   2.6 Existence: Perron's Method
   2.7 The Poisson Equation
   2.8 Energy Methods

3. The Heat Equation

   3.1 Initial Value Problems
   3.2 Mean Value Property, Maximum Principle & Uniqueness
   3.3 Regularity & A Priori Estimates
   3.4 Energy Methods

4. The Wave Equation

   4.1 R^1 (d'Alembert) & half-line
   4.2 Spherical Means & R^3 (Kirchhoff)
   4.3 Method of Descent & R^2 (Poisson)
   4.4 General R^d: Odd & Even d
   4.5 Huygens' Principle & Finite Propagation Speed
   4.6 Nonhomogeneous Problem (Duhamel)
   4.7 Energy Methods

5. Fourier Transform and PDE

6. Method of Characteristics

Literature: In uni2work you will find a copy of the notes from the lecture (to be updated as we go along).
Above you will find a short list of content of the lecture.
The lecture will mainly follow the books by Evans, and Arendt & Urban mentioned below.

(Auf uni2work wird es eine Mitschrift der Vorlesung geben. Oben eine Kurzübersicht der Vorlesung. Die Vorlesung wird größtenteils auf folgenden zwei Büchern (von denen mehrere Exemplare in der Bibliothek vorhanden sind) basieren:)

• [E] L. C. Evans, Partial Differential Equations: Second Edition, AMS, Providence, RI, 2010. (Extracts available online in uni2work!)
• [A-U] W. Arendt, K. Urban, Partielle Differenzialgleichungen, Springer Spektrum, 2018. (Login with your Campus-account.)

Supplementary literatur (Ergänzende Literatur):

• E. DiBenedetto, Partial Differential Equations (2nd edition), Birkhäuser Cornerstones, 2010.
• M. Renardy, R.C. Rogers, An Introduction to Partial Differential Equations, Springer, 2004. (NOT available online!)
• J. Jost, Partial Differential Equations, Springer, 2013.
• B. Schweizer, Partielle Differentialgleichungen, 2. Auflage, Springer Spektrum, 2018.
• F. John, Partial Differential Equations,Springer, 1982. (NOT available online!)
• J. Rauch, Partial Differential Equations, Springer, 1991.
• G.B. Folland, Introduction to Partial Differential Equations, Second Edition, Princeton University Press, 1995.
• P.J. Olver, Introduction to Partial Differential Equations, Springer, Cham, 2014.
• Q. Han, A Basic Course in Partial Differential Equations, AMS, 2011.
• W. Craig, A Course on Partial Differential Equations, AMS, 2018.

Here a longer liste.

Office hours (Sprechstunde):
Via Zoom; see uni2work.

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Letzte Änderung: 02 March 2021 (No more updates).

Thomas Østergaard Sørensen