# Lecture course: Partial differential equations

## Tue, Thu 10 - 12 in B 004

**Tutorials organised by: S. Sonner**(you will find the problem sheets here)

### NEWS

**10.03.10**You may pick up your certificates and inspect your graded exams in Mrs Warcholik's office. Only diploma students are allowed to take their exams home.

__Bachelor students please note:__Contrary to what I was hoping, I am obliged to inform the examination office about the scores and grades of all those who took part in any of the exams.**10.02.10**__Reminder:__The second part of the exam will take place on 18 February 2010 at 10:15 a.m. in room A 213 of the LMU main building (Geschwister-Scholl-Platz 1).**You are allowed to bring one DIN A4 sheet of paper with notes of your own choice (but no other resources).****14.01.10**Summer term 2010: This course will be continued next term (*Partial Differential equations II*). I will also offer a student seminar on*Random Schrödinger operators*.**18.12.09**Next lectures on Tue, 22 Dec 09 and Thu, 7 Jan 10.**18.12.09**Numbering corrected in script (Theorem 5.11)**10.12.09**Error correction on p. 59 in the script: Cavalieri's principle has been used incorrectly in the proof of Thm. 4.8. This part has been replaced by a transformation to spherical coordinates.**23.11.09**__Reminder:__the first part of the exam will take place on Thursday, 3 December 2009 at 10:15 in B 004. Duration: 90 minutes. No Script or other tools allowed. This first part will count 1/3 of the final grade.**05.11.09**The script has been updated. Please note the assumption on*f*in Thm. 2.25 (removing some confusion in the lecture today) and a more detailed (and corrected) version of the very end of the proof of Lemma 2.26 (top of page 30).

**Synopsis**

The course aims to give an introduction to

**P**artial

**D**ifferential

**E**quations, a vast area within Analysis. PDE's play an important rôle in applications of Mathematics to other sciences, most prominently in Physics and Engineering, but also in Biology and Financial Sciences. We will discuss explicit classical solutions for the most prominent linear, second-order PDE's, the method of characteristics for (non-linear) first-order PDE's and existence and regularity of weak solutions of second-order elliptic boundary value problems.

**Prerequisites**

Analysis I-III, Linear Algebra I-II, Functional Analysis

**Audience**

Students of Mathematics (Bachelor, Diploma, Lehramt), Financial Mathematics (Diploma), Physics (Bachelor, Diploma), Elite-Master Course Theoretical and Mathematical Physics (TMP)

**Literature**

- L. C. Evans,
*Partial differential equations*, AMS, Providence, RI, 1998 - M. Dobrowolski,
*Angewandte Funktionalanalysis*, Springer, Berlin, 2006 - G. B. Folland,
*Introduction to partial differential equations*, Princeton University Press, Princeton, NJ, 1995 - D. Gilbarg, N. S. Trudinger,
*Elliptic partial differential equations of second order*, Springer, Berlin, 1983 - F. John,
*Partial differential equations*, Springer, Berlin, 1982 - R. A. Adams,
*Sobolev spaces*, Academic, New York, 1975

- W. Rudin,
*Real and complex analysis*, MacGraw-Hill, New York, 1987

[German edition:*Reelle und komplexe Analysis*, 2. verbess. Aufl., Oldenbourg, München, 2009]

**Contents (as pdf with page numbers)**

- Introduction and overview (pdf)
- Continuity equation (pdf)
- Laplace and Poisson equation
2.1. Motivation and mathematical preparations (pdf)

2.2. Harmonic functions (pdf)

2.3. Poisson problem in ℝ^{d}(pdf)

2.4. Poisson problems in domains with boundaries (pdf)

2.5. Energy methods (pdf)

- Heat equation
3.1. Heat equation in ℝ

^{d}(pdf)

3.2. Uniqueness and regularity of solutions (pdf)

- Wave equation
4.1. Solution by spherical means (pdf)

4.2. Energy methods (pdf)

- Local existence theory
5.1. Preparations and strategy (pdf)

5.2. The quasilinear case (pdf)

5.3. The general first-order case (pdf)

- Sobolev spaces
6.1. Basic properties (pdf)

6.2. Approximation by smooth functions ("H=W") (pdf)

6.3. More on weak derivatives (pdf)

6.4. Extension and traces (pdf)

6.5. Sobolev inequalities (pdf)

6.6. Compact embeddings (pdf)

- Elliptic equations of second order
7.1. Existence of weak solutions (pdf)

7.2. Regularity of weak solutions (pdf)

*(I am happy to receive feedback on misprints, etc., in order to improve future updates)*