Junior Research Group "Nonlinear Helmholtz Equations"

Secretariat
Kollegiengebäude Mathematik (20.30)
Room 3.029

Karlsruher Institut für Technologie
Institut für Analysis
Englerstraße 2
76131 Karlsruhe
Germany

Office hours:
Mon-Fri 10-12, Tue+Wed 14-16

Tel.: +49 721 608 42064

Fax.: +49 721 608 46530

# Boundary and Eigenvalue Problems (Summer Semester 2018)

 Lecturer: PD Dr. Rainer Mandel Lecture (0157500), Problem class (0157600) 4+2

This Lecture provides an introduction to boundary value problems for second order elliptic boundary problems arising in Mathematical Physics. The students should have the basics in analysis (Calculus, Lebesgue integration, ODE theory) as well as functional analysis. Moreover, some interest in partial differential equations is desirable.

Seminar Announcement
Seminar Announcement with details

 Lecture: Problem class: Tuesday 8:00-9:30 SR 3.69 Friday 8:00-9:30 SR 2.67 Friday 14:00-15:30 SR 3.69
 Lecturer PD Dr. Rainer Mandel Office hours: by appointment Room -1.019 Kollegiengebäude Mathematik (20.30) Email: rainer.mandel@kit.edu M.Sc. Peter Rupp Office hours: Wednesday, 14.00-15:30 Room 3.026 Kollegiengebäude Mathematik (20.30) Email: peter.rupp@kit.edu

The whole course deals with the linear theory of 2nd order elliptic PDEs in the framework of Sobolev spaces.

1. Introduction (examples and motivation for eigenvalue and boundary value problems)
2. Boundary value problems for ODEs
3. Boundary value problems for 2nd order elliptic PDEs
4. Eigenvalue problems for 2nd order elliptic PDEs

The 3rd part basically treats the following topics:

• Weak derivatives and weak formulation of boundary value problems
• Sobolev spaces: Poincaré's inequality, Sobolev's imbedding theorem, Extension Theorem, Trace Theorem
• Solvability of elliptic boundary value problems via Fredholm operator theory
• Qualitative properties of solutions: regularity (=smoothness) and positivity

The 4th part deals with:

• Spectral theory for compact selfadjoint operators in Hilbert spaces
• Existence of an orthonormal basis of eigenfunctions

# References

• L. C. Evans: Partial Differential Equations
• D. Gilbarg, N. Trudinger: Elliptic Partial Differential Equations of Second Order
• R. A. Adams, J. F. Fournier: Sobolev Spaces