# Partial Differential Equations

SS 2023    03.04.2023- 14.07.2023

## Klausur (written exam)

Do 20.07.2023  12:00-14:00  V2-205
Results of Klausur

Zweitklausur:
Do 28.09.2023 12:00-14:00  T2-141
Results of 2.Klausur

The exam scheet consists of 5 problems of 25 points each. For a full mark (Note 1,0) it is necessary to score 95-100 points (equivalent of 4 solved problems), for a pass mark (Note 4,0 - bestanden) it is necessary to score 45-50 points (equivalent of 2 solved problems).

The exam problems may contain all the topics from lectures and homework exercises, except for topics/exercises marked by *.

## Lectures

Mo 14:15-15:45   V2-205,  Do  12:15-13:45  V2-205

Lecture notes (final version)

## Exercises

The problems that are marked by * are additional. They do bring points to those who solve them but they do not contribute to the maximal possible number of points.
Deadline for submission of homeworks is always Friday. Joint submission of homework by groups of students is not allowed.
At least 50% of points for homework is required for admission to the exam.

## Contents of the course

0. Introduction
Examples and origin of PDEs: Laplace equation, heat equation, wave equation, Schrödinger equation.
Quasi-linear PDEs of second order and change of coordinates.
Classification of linear 2nd order PDEs: elliptic, parabolic, hyperbolic.

1. Laplace equation and harmonic functions
Maximum principle and uniqueness in the Dirichlet problem.
The Green function in a ball.
Poisson formula and solvability of the Dirichlet problem in a ball.
Harnack inequality and other properties of harmonic functions.
Sequences of harmonic functions (Harnack theorems).
Discrete Laplace operator on graphs.
Separation of variables in the Dirichlet problem.
Variational problem and the Dirichlet principle.

2. Heat equation
The heat kernel.
Existence of bounded solutions of the Cauchy problem.
Maximum principle and uniqueness in the Cauchy problem.
Mixed problem and separation of variables.

3. Wave equation
Cauchy problem in dimension 1.
Energy and uniqueness.
Mixed problem for the wave equation.
Sperical means.
Cauchy problem in dimensions 2 and 3.

4. The eigenvalue problem
Distributions and Sobolev spaces.
Weak Dirichlet problem and Green operator.
Compact embedding theorem,
Eigenvalues and eigenfunctions of the weak Dirichlet problem.
* Higher order weak derivatives of weak solutions and eigenfunctions.
* Sobolev embedding theorem and smoothness of weak solutions and eigenfunctions.

## Literature

1. Courant R., Hilbert D., Methods of mathematical physics, Vol. 2. (Methoden der mathematischen Physik, Band 2)
2. DiBenedetto E., Partial differential equations, Birkhäuser, 2010
3. Burg K., Haf H., Wille F., Meister A., Partielle Differentialgleichungen und funktionalanalytische Grundlagen, Vieweg+Teubner, 2010
4. Drabek P., Holubova G., Elements of partial differential equations, De Gruyter, 2014
5. Evans L.C., Partial differential equations, Graduate Studies in Mathematics 19, AMS, 1997, 2008, 2010
6. Farlow S.J., Partial differential equations for scientists and engineers, John Wiley & Sons, 1982.
7. Hattori H., Partial differential equations: methods, applications and theories, World Scientific Publ., 2013
8. Jost J., Partial differential equations, Graduate Texts in Mathematics 214, Spinger, 2013.
9. Komech A., Komech A., Principles of partial differential equations, Springer, 2009
10. Leis R., Vorlesungen über partielle Differentialgleichungen zweiter Ordnung, Mannheim : Bibliograph. Inst., 1967
11. Olver P.J., Introduction to partial differential equations, Undergraduate Texts in Mathematics, Springer, 2014
12. Petrowskij I.G., Vorlesungen über partielle Differentialgleichungen , Leipzig : Teubner, 1955
13. Pinchover Y., Rubinstein, J., An introduction to partial differential equations, Cambridge Univ. Press, 2007
14. Schweizer B., Partielle Differentialgleichungen, Springer 2013
15. Shearer M., Levy R., Partial differential equations : an introduction to theory and applications Princeton Univ. Press, 2015

and many more.