## Operator Semigroups, Winter Semester 2016/17

Place and time
 Type: Time: Place: Start: Vorlesung 3 std. Mo 09:45-11:15Di 09:15-10:00 SR12 3.10.
Course aims
In the course on ordinary differential equations it is shown how a first order system d/dt u(t) = A u(t) can be soved by virtue of the matrix exponential: u(t) = exp(t A) u(0). In particular, as a function of t this give rise to a one-parameter group.

Similarly, partial differential equations like the heat equation fit within this framework if the matrix A is replaced by the Laplace operator. However, since the Laplace operator is not bounded, the series for the exponential will not converge and this raises the question how the exponential should be defined in this case. Moreover, solutions of the heat equation exist in general only for positive times and the solution operator can form at best a semigroup.

Giving a solid mathematical foundation to this approach will be the aim of the present course.

Content
Theory of operator semigroups and applications to partial differential equations: generators, Hille-Yoside-Theorem, perturbation theory, dissipative operators, etc. We covered Lectures 1-9 from [HMMS].
Intended audience
Majors in Mathematics (Vertiefungslehrveranstaltungen für den Studienschwerpunkt Analysis – MANV), Physics, ...
Assessment
Oral examination at the end of the course.
Literature
Some textbooks:
1. K.-J. Engel und R. Nagl, A Short Course on Operator Semigroups, Springer, New York, 2006.
2. L. C. Evans, Partial Differential Equations, 2nd ed., Amer. Mat. Soc., Providence, 2010.
3. J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Univ. Press, Oxford, 1985.
4. D. Hundertmark, M. Meyries, L. Machinek, R. Schnaubelt, Operator Semigroups and Dispersive Equations, Lecture Notes, 2013.
5. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
Looking forward to seeing you, Gerald Teschl