Vorlesung: Partielle Differentialgleichungen II (PDG2) (SoSe 2021)
[UPDATE 15.02.2021:] Due to the present situation, this course will be online (uploaded videos, lecture notes, homework exercises, zoom sessions; all details on uni2work).
It will start 13 April 2021 (as planned).
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.
You will need to sign up (starting April 01) at uni2work to get access to the course material.
Lecture (Vorlesung):
Tue 14-16 & Wed 10-12: Online (videos) and via Zoom; see uni2work. LSF
Exercises (Ãœbungen):
Via Zoom; see uni2work. LSF
Tutorials (Tutorien):
There are NO tutorials!
Synopsis (Kurzbeschreibung):
This lecture is a continuation of the introductory lecture 'Partielle Differentialgleichungen' (PDG1) in the past semester (WiSe2020/21).
It can also be taken as a continuation of any other introductory lecture 'Partielle Differentialgleichungen' (PDG1).
We will study existence and regularity of weak solutions to elliptic equations. This will also involve the study of weak derivatives and Sobolev spaces (on domains). We will then apply this to the study of (parabolic and hyperbolic) evolution equations.
Audience (Hörerkreis):
Master students of Mathematics (WP 40), Master students of `Finanz- und Versicherungsmathematik' (WP 27), TMP-Master.
Credits:
9 (6+3) ECTS.
Prerequisites (Vorkenntnisse):
Analysis I–III, Linear Algebra I–II, Functional Analysis, PDG1 (in some form; approximately p. 1–90 in [E] Evans (see below)).
Language (Sprache):
English. (Die mündliche Prüfung kann auch auf Deutsch gemacht werden).
Exam (Prüfung):
There will be an oral exam (dates to be announced) (Es wird eine mündliche Prüfung geben).
The exam will be online. See separate webpage (uni2work).
Content (Inhalt):
- Introduction and motivation
- Weak derivatives & Sobolev spaces
- Linear 2nd order elliptic PDE
- Nonlinear elliptic equations
- Evolution equations & C_0 - semigroups
In uni2work you will find a copy of the notes from the lecture.
Above you will find a short description of the content of the lecture.
The lecture will mainly follow the books by Evans, and Arendt & Urban mentioned below.
- [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.)
- H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.
- V. Maz'ya, Sobolev spaces, with Applications to Elliptic Partial Differential Equations, Springer, 2011.
- D. D. Haroske, H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, EMS, 2007.
- G. Grubb, Distributions and Operators, Springer, 2009.
- R. Precup, Linear and Semilinear Partial Differential Equations, De Gruyter, 2013.
- G. Leoni, A First Course in Sobolev Spaces: Second Edition, AMS, 2017.
- K. B. Sinha, S. Srivastava, Theory of Semigroups and Applications, Springer, 2017.
Office hours (Sprechstunde):
Via Zoom; see uni2work.
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Letzte Änderung: 28 July 2021 (No more updates).
Thomas Østergaard Sørensen