## Operator Semigroups,

Sommersemester 2011

Place and time

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Vorlesung 2 std. | Di 13:10-14:40 | C209 | 1.3. |

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.

Intended audience

Majors in Mathematics (Vertiefungslehrveranstaltungen für den Studienschwerpunkt

*Analysis*– MANV), Physics, ...Assessment

Oral examination at the end of the course.

Literature

Some textbooks:

Looking forward to seeing you, Gerald Teschl
- K.-J. Engel und R. Nagl,
*A Short Course on Operator Semigroups*, Springer, New York, 2006. - L. C. Evans,
*Partial Differential Equations*, 2nd ed., Amer. Mat. Soc., Providence, 2010. - J. A. Goldstein,
*Semigroups of Linear Operators and Applications*, Oxford Univ. Press, Oxford, 1985. - A. Pazy,
*Semigroups of Linear Operators and Applications to Partial Differential Equations*, Springer, New York, 1983.