Periodic media (e.g. structures which are constructed by

indefinite repetition of a basic unit cell) are becoming

increasingly important in technical applications, especially

in optical nanotechnology, which is thought to

be a key science in the 21st century.

The lecture gives an introduction to the mathematical analysis of

partial differential equations with periodic coefficients modeling

the acoustic or electromagnetic wave propagation in these media.

The lecture starts with an introduction to the physical situations

in which PDE's with periodic coefficients appear.

The first major part of the lecture will be devoted to the study of the Floquet Bloch (or Gelfand) transform, defined at first for compactly supported by

In particular, we are interested in the mapping properties of . Next we will explain how to use the

Floquet transform to transform elliptic problems with periodic coefficients

on the whole space to boundary value problems on a unit cell (Floquet-Bloch

decomposition). Throughout the lecture, we will make extensive use of

quadratic forms to streamline the presentation as much as possible, at the same

time imposing only weak regularity assumptions on the coefficients.

If enough time remains, we will discuss further spectral properties of periodic

differential operators, such as the absolute continuity of spectra or some

boundary value problems involving semi-infinite structures, where the usual

Floquet-Bloch theory cannot be applied.

I will try to make the lecture reasonably self-contained, but some familiarity

with functional analysis and Sobolev spaces will be helpful.

# Differential equations with periodic coefficients (Summer Semester 2011)

Lecturer: | Dr. Vu Hoang |
---|---|

Classes: | Lecture (0156200) |

Weekly hours: | 2 |

Lecture: | Monday 14:00-15:30 | 1C-03 |
---|

Lecturer | Dr. Vu Hoang |
---|---|

Office hours: | |

Room Allianz-Gebäude (05.20) | |

Email: duy.hoang@kit.edu |

**The lecture takes place twice a week:
Mo, 11:30-13:00 and Mo, 14:00-15:00,
room 1C-03**

# References

Kato, T. : Perturbation Theory for Linear Operators. Springer.

Kuchment, P : Floquet theory for Partial Differential Equations. Basel Birkhäuser Verlag.