Internet Seminar 2000 

Semigroups generated by elliptic operators.

The most elegant way to treat parabolic equations is given by the so-called variational method. It consists of associating to each coercive sesquilinear form the generator of an analytic semigroup. Elliptic operators can be treated most efficiently by this method.

But the electronic reading course will start slowly, by a systematic study of the most important example, the Laplacian with diverse boundary conditions. Only elementary background in Calculus and Functional Analysis is needed. Basic properties of semigroups and of Sobolev spaces will be established in the course. The main goal is a systematic study of the heat equation on a domain in ⁿ. We start with L²-theory, but will extend the results to L^p and also to C-spaces. Some elementary results of potential theory will be developed in order to understand the Laplacian on C₀().

In a second part, variational methods will be studied systematically. We will see how criteria by Beurling-Deny allow one to pass from L²() to L^p() and how heat kernels can be obtained. These methods allow one to treat general elliptic operators. Many of these topics will be treated as projects.

A basic text will be distributed to the participants. But the following books give an idea of the subject treated in the course.

References:

[Da1]	E.B. Davies: Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics 92, Cambridge University Press (1989).
[Da2]	E.B. Davies: Spectral Theory and Differential Operators, Cambridge studies in advanced mathematics, 42, Cambridge University Press (1995).

Lecturer:	Wolfgang Arendt
Internet trainers and coaches:	Ralph Chill, Markus Haase,
	Sonja Thomaschewski.