### 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.**

# 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 | PD Dr. Rainer Mandel | |

Office hours: by appointment | ||

Room -1.019 Kollegiengebäude Mathematik (20.30) | ||

Email: rainer.mandel@kit.edu |

# Problem sheets

- Problem sheet 1
- Problem sheet 2
- Problem sheet 3
- Problem sheet 4
- Problem sheet 5
- Problem sheet 6
- Problem sheet 7
- Problem sheet 8
- Problem sheet 9, The corrected version: Problem sheet 9 corrected
- Problem sheet 10 In exercise 2 you may use the "general fact" .
- Problem sheet 11, The corrected version: Problem sheet 11 corrected
- Problem sheet 12
- Problem sheet 13
- Problem sheet 14

# 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