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Institute for Analysis

Workgroup Nonlinear Partial Differential Equations

Differential equations with periodic coefficients

Differential equations with periodic coefficients

Differential equations with periodic coefficients (Summer Semester 2011)

Lecturer: Dr. Vu Hoang
Classes: Lecture (0156200)
Weekly hours: 2

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 $f\in L^2(\R^d)$ by
$U f(x, k) = \frac{1}{(2\pi)^{d}} \sum_{m\in \Z^d} e^{i k\cdot m} f(x-m), \quad ( (x,k)\in (0,1)^d\times (-\pi,\pi)^d).$
In particular, we are interested in the mapping properties of $U$ . 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.

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

Lecturers
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.