Welcome to the course "boundary and eigenvalue problems"!

I have a script for the course, which was written some years ago when I gave this course previously. However, this script is written in German. Nevertheless, I would like to use it for the online lectures. I will translate important key expressions into English and write these into the script text. Furthermore I will SPEAK in English, and thus I hope that everybody can easily follow the lectures without knowledge of the German language. (For the German students, this way of proceeding is perhaps even more convenient than an English script.)

Any foreign student who has difficulties with this is encouraged to tell me.

In the folder "Vorlesungsmaterial" on the ILIAS platform for online lectures I will upload the script, together with the aforementioned hand-written remarks and translations, as a PDF file. This will be done in pieces, which I create in parallel to the online-lectures.

The online lectures themselves, i.e. the MP4-videos with my spoken lectures, can be found in the sub-folder "Aufzeichnungen" of the folder "Vorlesungsmaterial" on the ILIAS platform. Since there is no time needed for writing on the blackboard, these MP4-videos cover the same amount of material during a much shorter time, compared with the "real" lectures in a lecture room with blackboard. For example, the video "Session 1", which is already available and takes about half an hour, contains approximately the same amount of material which took a whole 1.5-hours lecture when I gave the course previously. This higher "speed" of the lectures should however be no disadvantage for you (the participants), since you can replay the video as often as you like, and you can pause it at any point.

Good luck to all of us for a successful lectuer course!

Michael Plum

**Important Information:**

- Due to the current situation, the lecture and exercise will start online via the ILIAS platform.

- Lecture and exercise courses will be made available in the form of MP4 recordings.

- Please join the course
*Tutorial for 0157500 (Boundary and Eigenvalue Problems)* on the ILIAS platform for the exercise class.

A boundary value problem consists of an elliptic (or ordinary) differential equation posed on some domain, together with additional conditions required on the boundary of the domain, e.g. prescribed values for the unknown function. In the case of an ordinary differential equation, these boundary conditions are posed at both ends of the underlying interval (in contrast to initial value problems). Typical origins of boundary value problems are steady-state (i.e. time-independent) situations in physics and engineering.

An eigenvalue problem for a differential equation is a linear and homogeneous boundary value problem depending (typically linearly) on an additional parameter, and one is interested in values of this parameter such that the boundary value problem has nontrivial solutions. Eigenvalue problems arise e.g. after separation of variables in time-dependent problems (thus describing many vibrational situations, including quantum mechanics) or in stability and critical value theory for mathematical and physical systems.

The lecture course will start with a series of examples for occurrence of boundary value problems in mathematical physics, followed by the (comparatively simple) existence theory for ordinary linear regular boundary value problems. A large part of the lecture course will then be covered by an existence theory for linear elliptic boundary value problems; for this purpose, weak formulations of boundary value problems, Sobolev spaces, trace theory, the Lax-Milgram Lemma, Gårdings inequality, Fredholms Alternative, and other tools will be introduced. In a natural way, this theory connects to eigenvalue problems. Based upon the Spectral Theorem for compact symmetric operators in Hilbert spaces, and on the existence theory for linear boundary value problems, an eigenvalue theory for symmetric elliptic differential operators will be presented. If time permits, the lecture course closes with some numerical methods for boundary and eigenvalue problems (Galerkin, Finite Elements).

The lecture course addresses students in their fourth semester (second year) or higher, with substantial knowledge in analysis and linear algebra. The course is suitable for students of mathematics, and for students of other subjects who have strong mathematical interests.

The lectures will be accompanied by exercise lessons. Attendance of these exercises is strongly recommended to all participants.

**Exercise Sheets:**

Exercise sheet 1, to be explained on 29.04.2020 (ILIAS)

Exercise sheet 2, to be explained on 06.05.2020 (ILIAS)

Exercise sheet 3, to be explained on 13.05.2020 (ILIAS)

Exercise sheet 4, to be explained on 20.05.2020 (ILIAS)

Exercise sheet 5, to be explained on 27.05.2020 (ILIAS)

Exercise sheet 6, to be explained on 03.06.2020 (ILIAS)

Exercise sheet 7, to be explained on 10.06.2020 (ILIAS)

Exercise sheet 8, to be explained on 17.06.2020 (ILIAS)

Exercise sheet 9, to be explained on 24.06.2020 (ILIAS)

Exercise sheet 10, to be explained on 01.07.2020 (ILIAS)

Exercise sheet 11, to be explained on 08.07.2020 (ILIAS)

Exercise sheet 12, to be explained on 15.07.2020 (ILIAS)

Exercise sheet 13, to be explained on 22.07.2020 (ILIAS)

Holiday exercise sheet

# Examination

The examination will be in the form of a written exam, which takes place on 23.09.2020, 12:00-14:00, in 30.22, Gaede Hörsaal. The exam can be registered in the CAMPUS system.

**Please note that the end of registration is on August 15, 2020.**

# References

A. Friedman: Partial Differential Equations

(general elliptic PDE of order 2m, but smooth data only)

D. Gilbarg, N. Trudinger: Elliptic Partial Differential Equations of Second Order

(elliptic PDE of second order, mainly Dirichlet b.c.)

R. A. Adams: Sobolev Spaces

(no PDEs, but excellent and general introduction into Sobolev spaces, an essential tool in PDE theory)

L. C. Evans: Differential Partial Equations