Summer 2017

Differentialgleichungen III

PD Dr. Martin Väth

Schedule, Summer 2017

Lecture:

Monday, 16.15-17.45, SR E.31, Arnimallee 7

Exercise:

Thursday 12.15-13.45, 017, Arnimallee 6 (Pi building)

Changes of the schedule:

Tuesday (May 16) 12.15-13.45 additional lecture in Room SR 210, Arnimalle 3

Tuesday (June 6) 12.15-13.45 additional lecture in Room SR 140, Arnimalle 7

Registration

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 organizing things.

Instructions to do this:

1. Go to the KVV homepage.
2. 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.)
3. Go to Membership, choose Joinable Sites, choose the lecture (Differentialgleichungen III) and press the join button.

Content

The lecture is a continuation of part 1 and 2. As such, it is assumed that participants are familiar with

1. Basic concepts of ODEs and dynamical systems (e.g. notions and criteria for stability)
2. 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)
3. 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)
4. 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 and PDEs. 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.

Literature

1. A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer, New York, Berlin, Heidelberg, 1992
2. D. Henry, Geometric theory of semilinear parabolic equations, Lect. Notes Math. 840, Springer, Berlin, New York, 1981
3. H. Amann, Linear and quasilinear parabolic problems I , Birkhäuser, Basel, Boston, Berlin, 1995
4. A. Lunardi, Analytic semigoups and optimal regularity in parabolic problems, Birkhäuser, Basel, Boston, Berlin, 1994
5. E. Hille and R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Coll. Publ., Providence, R. I., 1957
6. A. Belleni-Morante and A. C. McBride, Applied nonlinear semigroups, John Wiley & Sons, Chichester, New York, Weinheim, 1998
7. I. I. Vrabie, C₀-semigroups and applications, 2nd ed., Elsevier, Amsterdam, 2003
8. E. M. Ouhabaz, Analysis of heat equations on domains, Princeton Univ. Press, Princeton, Oxford, 2005

1. L. C. Evans, Partial differential equations, 2nd ed., Amer. Math. Soc., Providence, R. I., 2010

1. H. Brezis, Functional analysis, Sobolev spaces, and partial differential equations, Springer, New York, Dordrecht, Heidelberg, London, 2011
2. H. W. Alt, Lineare Funktionalanalysis, 2nd ed., Springer, Berlin, Heidelberg, New York, 1992
3. L. V. Kantorovich and G. P. Akilov, Functional analysis, 2nd ed., Pergamon Press, Oxford, 1982
4. J. Appell and M. Väth, Elemente der Funktionalanalysis, Vieweg & Sohn, Braunschweig, Wiesbaden, 2005
5. M. Väth, Integration theory. A second course, World Scientific Publ., Singapore, New Jersey, London, Hong Kong, 2002
6. T. Kato, Perturbation theory for linear operators, Springer, New York, 1966
7. D. Werner, Funktionalanalysis, 4th ed., Springer, Berlin, Heidelberg, New York, 2002
8. N. Dunford and J. T. Schwartz, Linear operators I, 3rd ed., Int. Publ., New York, 1966
9. H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland, Amsterdam, New York, Oxford, 1978

Dates for the Exam (Klausur)

July 13 during lecture.

Exercises

Possible Solutions to the Exercises

Manuscript