### Rand- und Eigenwertprobleme / Boundary and Eigenvalue Problems (Sommersemester 2005)

• Dozent*in:
• Veranstaltungen: Vorlesung (01578), Übung (01579)
• Semesterwochenstunden: 4+4
• Hörerkreis: Mathematics and other subjects with strong mathematical interests (ab 4. Semester)

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. 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).

# Übungsblätter

Termine
Vorlesung: Dienstag 11:30-13:00 Seminarraum 12
Freitag 9:45-11:15 Seminarraum 12
Übung: Mittwoch 14:00-15:30 Seminarraum 12

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, Gårdings inequality, Fredholms 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.

# Literaturhinweise

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.)