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Workgroup Nonlinear Partial Differential Equations

Secretariat
Kollegiengebäude Mathematik (20.30)
Room 3.029

Address
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 2010)

Lecturer: Prof. Dr. Wolfgang Reichel
Classes: Lecture (1575), Problem class (1576)
Weekly hours: 4+2


The lecture this week Wednesday (June 30th) is canceled and shifted to Tuesday, July 6th, 9:45--11:15 in 1C-01.

Current evaluation of the course

as .pdf file


Topics

The course consists of topics on linear elliptic partial differential equations, where the differential equations are coupled with boundary conditions. Boundary value problems of such kind appear e.g. in modeling of reaction-, convection- and diffusion processes. Special cases of boundary value problems are so called eigenvalue problems, which appear e.g. in quantum mechanics or in vibrations of elastic materials. If time permits we will also investigate basic properties of time-dependent problems (parabolic initial-boundary value problems) like the heat equation, which models heat conduction.

In the course I will cover results on existence of weak solutions in Sobolev spaces, estimates of such solutions together with qualitative and regularity properties.

1. Motivation & examples
2. Explicit solution of the Poisson boundary value problem on balls
3. Weak derivatives and Sobolev spaces

  • Poincaré and Sobolev inequalities, imbedding theorems

4. Elliptic boundary value problems

  • Maximum and comparison principles, existence results, Fredholm alternative
  • Regularity properties of solutions

5. Elliptic eigenvalue problem
Eigenvalues, eigenfunctions, completeness, variational characterization
6. Parabolic initial-boundary value problems

Audience: Mathematicians, physicists, engineers;

Prerequisites: Analysis I--III (or similar lectures), basics in functional analysis


Schedule
Lecture: Monday 9:45-11:15 1C-04
Wednesday 14:00-15:30 1C-03
Problem class: Thursday 15:45-17:15 1C-04
Lecturers
Lecturer Prof. Dr. Wolfgang Reichel
Office hours: Monday, 11:30-13:00 Before you e-mail: call or come!
Room 3.035 Kollegiengebäude Mathematik (20.30)
Email: Wolfgang.Reichel@kit.edu
Problem classes Dr. Rainer Mandel
Office hours: by appointment
Room -1.019 Kollegiengebäude Mathematik (20.30)
Email: rainer.mandel@kit.edu

Problem sheets

Handouts

References

  • L.C. Evans: Partial Differential Equations, American Math. Soc. 1998
  • Gilbarg & Trudinger: Elliptic Partial Differential Equations of Second Order, Springer 1998
  • Renardy & Rogers: An Introduction to Partial Differential Equations, Springer 1992
  • Walter Strauss: Partial Differential Equations - An Introduction, John Wiley 1992
  • Gerald B. Folland: Introduction to Partial Differential Equations, Princeton Univ.Press, 1995