Vorlesung: Partielle Differentialgleichungen II (PDG2) (SoSe 2023)

Lecture (Vorlesung):
Tue 14-16 (in B 006) & Wed 12-14 (in B 004).   LSF

Exercises (Ãœbungen):
See separate webpage (Moodle).   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 (WiSe2022/23).
It can also be taken as a follow-up to any other introductory lecture on Partial Differential Equations (PDEs).

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, and motivated bachelor students.

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).
See separate webpage (Moodle).

Content (Inhalt):

0. Introduction and motivation

1. Weak derivatives & Sobolev spaces

   1.1 Weak derivatives
   1.2 Approximation by smooth functions
   1.3 Extensions
   1.4 Restriction & traces
   1.5 Sobolev inequalities & embeddings
   

2. Linear 2nd order elliptic PDE: Existence

   2.1 Weak formulation & weak solutions
   2.2 Lax-Milgram & general elliptic PDE
   2.3 Fredholm alternative for elliptic PDE
   2.4 Inhomogeneous BVPs
   

3. Linear 2nd order elliptic PDE: Regularity

   3.1 Motivation & heuristic
   3.2 Difference quotients & H^2-regularity
   3.3 Boundary regularity
   

4. C_0 - semigroups & evolution equations

   4.1 C_0 - semigroups
   4.2 Operators, integration, differentiation
   4.3 Semigroups, generators & Hille-Yosida
   4.4 2nd order parabolic PDE
   4.5 Inhomogeneous & semilinear parabolic PDE
   4.6 2nd order hyperbolic PDE
   

Literature:
In Moodle 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 book by Evans:

• [E] L. C. Evans, Partial Differential Equations: Second Edition, AMS, Providence, RI, 2010.
  (Extracts available online in Moodle!)

Supplementary literature (Ergänzende Literatur):

• W. Arendt, K. Urban, Partielle Differenzialgleichungen, Springer Spektrum, 2018.
• 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.

Here is a longer list.

Office hours (Sprechstunde):
See Moodle.

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Letzte Änderung: 03 August 2023 (No more updates).

Thomas Østergaard Sørensen