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

# Boundary and Eigenvalue Problems (Summer Semester 2018)

Lecturer: | Dr. Rainer Mandel |
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Classes: | Lecture (0157500), Problem class (0157600) |

Weekly hours: | 4+2 |

Lecture: | Tuesday 8:00-9:30 | SR 3.69 |
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Friday 8:00-9:30 | SR 2.67 | |

Problem class: | Friday 14:00-15:30 | SR 3.69 |

Lecturer | Dr. Rainer Mandel |
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Office hours: by appointment | |

Room -1.019 Kollegiengebäude Mathematik (20.30) | |

Email: rainer.mandel@kit.edu | |

Problem classes | 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.

- Introduction (examples and motivation for eigenvalue and boundary value problems)
- Boundary value problems for ODEs
- Boundary value problems for 2nd order elliptic PDEs
- 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

Exercises

exercise sheet 1 solutions

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exercise sheet 11 solutions 1 solutions 2

exercise sheet 12 solutions

# 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