PD Dr. Martin Väth
Preliminary Schedule, Summer 2017
- Monday, 16.15-17.45, SR E.31, Arnimallee 7
- Thursday 12.15-13.45, 017, Arnimallee 6 (Pi building)
On Thursday, April 27, 2017, 12.15-13.45, there will be a lecture
(not an exercise) in Room 017 of Arnimallee 6 (Pi building).
Please register (in addition to the standard registration) into KVV.
This is not a substitute for the standard registration and has no obligations;
for instance, you need not be afraid to register into KVV if you are not
sure yet whether you will join the course, but it will help us in
Instructions to do this:
- Go to the KVV homepage.
- Login with Direkter KVV-Login with your ZEDAT account.
(If you do not have a ZEDAT account and cannot easily get one,
please ask after lecture or exercises.)
- Go to Membership, choose Joinable Sites, choose the lecture
(Differentialgleichungen III) and press the join button.
The lecture is a continuation of part 1 and 2.
As such, it is assumed that participants are familiar with
- Basic concepts of ODEs and dynamical systems (e.g. notions and criteria
- Fundamentals of the theory of PDEs (e.g. basic examples and features
of elliptic, parabolic, and hyperbolic equations, the notion of semilinear
and quasilinear equations)
- Advanced basics from analysis and functional analysis, e.g.
the notion of the Lebesgue integral and Bochner integral (i.e. an analogue of
the Lebesgue integral for vector functions)
- Fundamentals of C0 semigroup theory (linear)
The main aim of the lecture is to extend the semigroup theory, mainly to
analytic semigroups. While C0 semigroups apply to a huge class
of problems (parabolic and hyperbolic equations as well as delay equations),
the main application of the more specialized theory of analytic semigroups
is for parabolic equations. It employs methods of complex analysis
(Funktionentheorie) to obtain much stronger results (compared to the
more general C0 semigroup theory). In particular, one can
conveniently work in so-called fractional power spaces of the operator
generating the semigroup, and due to this the main obstacles
arising in the theory of C0-semigroups can be avoided.
For instance, semilinear parabolic equations are usually linearizable in
these fractional power spaces while this is usually not the case for
hyperbolic equations when one has to stick to the standard spaces in the
C0 theory. The deeper reasons for this will hopefully
become more clear during the lecture.
Despite complex analysis is a major tool, participants are not assumed to
be familiar with this topic: A crash-course on complex analysis introducing
all required concepts will be given in some of the first lectures.
Depending on the interests of the audience, it might be possible that
some results are treated less deeply and that the remaining time is used
instead to give an introduction to various other topics on dynamical systems
Possible such topics include the theory and application of compact and Fredholm
operators or topological methods (in particular degree theory) and their
application to dynamical systems and PDEs.
General literature concerning (analytic and C0) semigroups,
in decreasing order of relevance for the lecture:
The last reference also contains a lot about parabolic PDEs.
Of course, also a lot of standard references about PDEs will be useful for
the lecture, e.g.
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer, New York, Berlin, Heidelberg, 1992
- D. Henry, Geometric theory of semilinear parabolic equations, Lect. Notes Math. 840, Springer, Berlin, New York, 1981
- H. Amann, Linear and quasilinear parabolic problems I , Birkhäuser, Basel, Boston, Berlin, 1995
- A. Lunardi, Analytic semigoups and optimal regularity in parabolic problems, Birkhäuser, Basel, Boston, Berlin, 1994
- E. Hille and R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Coll. Publ., Providence, R. I., 1957
- A. Belleni-Morante and A. C. McBride, Applied nonlinear semigroups, John Wiley & Sons, Chichester, New York, Weinheim, 1998
- I. I. Vrabie, C0-semigroups and applications, 2nd ed., Elsevier, Amsterdam, 2003
- E. M. Ouhabaz, Analysis of heat equations on domains, Princeton Univ. Press, Princeton, Oxford, 2005
If you wish to refresh your general background on functional analysis or
integration theory, I recommend (again in decreasng order of relevance):
- L. C. Evans, Partial differential equations, 2nd ed., Amer. Math. Soc., Providence, R. I., 2010
Most of the above mentioned books are in the library in the "Handapparat".
- H. Brezis, Functional analysis, Sobolev spaces, and partial differential equations, Springer, New York, Dordrecht, Heidelberg, London, 2011
- H. W. Alt, Lineare Funktionalanalysis, 2nd ed., Springer, Berlin, Heidelberg, New York, 1992
- L. V. Kantorovich and G. P. Akilov, Functional analysis, 2nd ed., Pergamon Press, Oxford, 1982
- J. Appell and M. Väth, Elemente der Funktionalanalysis, Vieweg & Sohn, Braunschweig, Wiesbaden, 2005
- M. Väth, Integration theory. A second course, World Scientific Publ., Singapore, New Jersey, London, Hong Kong, 2002
- T. Kato, Perturbation theory for linear operators, Springer, New York, 1966
- D. Werner, Funktionalanalysis, 4th ed., Springer, Berlin, Heidelberg, New York, 2002
- N. Dunford and J. T. Schwartz, Linear operators I, 3rd ed., Int. Publ., New York, 1966
- H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland, Amsterdam, New York, Oxford, 1978
Dates for the Exam (Klausur)
Parts of the exercise hours will be used for lectures.
This will be announced from lecture to lecture.
Do not get scared about the following first exercise too much:
It is about a somewhat exceptional topic which will be explained in more detail