# Why do mathematicians study partial differential equations

### Partial differential equations I.

(Lecture in winter semester 2018/19)

**Lecturers:**Thomas Schmidt (lecture, exercises), Lars Poppe (exercises).

**Lecture dates** (first lecture on October 16):

- Tue, 8-10, H6 and Thu, 12-14, H4

**Exercise groups** (first exercise on October 24/25):

- Wed, 16-18, room 432, Lars Poppe
- Thu, 14-16, room 432, Thomas Schmidt

**Relevance and audience:** It is a lecture in the elective (compulsory) area in all Master’s degrees in mathematics (M.Sc. Mathematics, M.Sc. Mathematical Physics, M.Sc. Technomathemtics, M.Sc. Business Mathematics), which all students attend with an interest in analysis is highly recommended. The course can also be taken in the bachelor's degree in mathematics (from the 5th semester); it can then either be included in the elective area of the Bachelor or "saved up" for a later Master's course. Interested listeners from other courses are of course welcome.

**Previous knowledge:** The lecture builds on the basic lectures on analysis and linear algebra. Basic knowledge of Lebesgue integration (e.g. from higher analysis or mathematical stochastics) is also required.

**Credit points:** The lecture and the associated exercises form a module worth 12 ECTS.

**Lecture contents:** Partial differential equations are equations in which an unknown function of several variables and a finite number of (partial) derivatives of this function occur. The theory of these equations is very rich, can be approached in a variety of ways, and interacts with many different areas of analysis, mathematics, and physics. In this introductory lecture, the richness of the theory is to be demonstrated using the following three model equations:

- Laplace equation (with harmonic functions as solutions), Poisson equation,
- Thermal equation,
- Wave equation.

**Lecture notes:** Preliminary final version as PDF.

**Literature:** Well-known books (of different levels and scope) are:

- S. Axler, P. Bourdon, W. Ramey, Harmonic Function Theory, Springer, 2001,
- L.C. Evans, Partial Differential Equations, American Mathematical Society, 1998,
- D. Gilbarg, N.E. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2001,
- J. Jost, Partial Differential Equations, Springer, 2013,
- J. Rauch, Partial Differential Equations, Springer, 1991,
- F. Sauvigny, Partial Differential Equations (2 volumes), Springer, 2012,
- M.E. Taylor, Partial Differential Equations (3 volumes), Springer, 1996

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