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3. Internet Seminar 1999/2000 - Semigroups generated by elliptic operators
In the academic year 1999/00, the TULKA group from Ulm (Chief: Prof. Dr. W. Arendt) has organized an Internet Seminar on Semigroups generated by elliptic operators. It was a great success with over 35 participants from Czech Republic, Germany, Great Britain, Finland, China, Hungary, Italy, Japan, Morocco, Spain and USA.
After an electronic learning course the organization
team from Ulm organized a final workshop
at the Heinrich-Fabri Institut in Blaubeuren.
Mathematical content:
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 Rn. We start with L2-theory, but will extend the results to Lp and also to C-spaces. Some elementary results of potential theory will be developed in order to understand the Laplacian on C0(Ω).
In a second part, variational methods will be studied systematically. We will see how criteria by Beurling-Deny allow one to pass from L2(Ω) to Lp(Ω) 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. |
Workshop program - 19.6.-23.6.2000
| Monday, June 19 | |
| 7.00 | Early Morning Jogging (voluntary!) |
| 8.00-9.00 | Breakfast |
| 9.00-9.10 | Opening |
| Chair : Prof. Wolfgang Arendt (Universität Ulm) | |
| 9.10-9.50 | Prof. Giorgio Metafune (Università di Lecce) |
| Introduction to Feller Semigroups | |
| 10.00-10.20 | Pedro Miana (University of Zaragoza) |
| Semigroups and Laplace Transforms | |
| 10.30-11.00 | Coffee break |
| 11.00-11.20 | Neil Dew (University of Oxford) |
| Approximation and Real Characterisation | |
| 11.30-11.50 | Julien Bourgeois (Université Marne la Vallé / Universität Ulm) |
| The Trotter Product Formula | |
| 12.30 | Lunch |
| Prof. Yuri Latushkin (Missouri/Columbia) | |
| 14.00-14.20 | László Székelyhidi (University of Oxford) |
| Strong Continuity and Interpolation | |
| 14.30-14.50 | Andreas Schumacher (Universität Tübingen) |
| Compactness, Spectrum and Extrapolation | |
| 15.00-15.30 | Coffee break |
| 15.30-15.50 | Alex Us (University of Bristol) |
| Dissipative Operators in Hilbert Spaces | |
| 16.00-16.20 | Vincenzo Manco (Universtà di Lecce) |
| Regularly Dissipative Operators | |
| 16.30-16.50 | Fouzi Zaouch (Universität Rostock) |
| Elliptic Operators | |
| 18.30 | Dinner |
| Tuesday, June 20 | |
| 7.00 | Early Morning Jogging (voluntary!) |
| 8.00-9.00 | Breakfast |
| Chair: Prof. Diego Pallara (Università di Lecce) | |
| 9.10-9.50 | Prof. Stig-Olof Londen (University of Helsinki) |
| On Fractional Evolution Equations | |
| 10.00-10.20 | Jari Hätönen (University of Helsinki) |
| Introduction to Control Theory I | |
| 10.30-11.00 | Coffee break |
| 11.00-11.20 | Boris Thomaschewski (TU Ilmenau) |
| Introduction to Control Theory II | |
| 11.30-11.50 | Genni Fragnelli (Università di Lecce) |
| Stabilizability of the Wave Equation | |
| 12.30 | Lunch |
| Chair:Prof. Jaroslav Milota (Charles University, Prague) | |
| 14.00-14.20 | Cornelia Kaiser (Universität Karlsruhe) |
| Resolvent Positive Operators and Integrated Semigroups | |
| 14.30-14.50 | Petr Honzik (Charles University, Prague) |
| Dirichlet Regularity and the Dirichlet Laplacian | |
| 15.00-15.30 | Coffee break |
| 15.30-18.00 | Hiking around Blaubeuren |
| 18.30 | Dinner |
| Wednesday, Juny 21 | |
| 7.00 | Early Morning Jogging (voluntary!) |
| 8.00-9.00 | Breakfast |
| Chair:Ralph Chill (Universität Ulm) | |
| 9.10-9.50 | Prof. Eva Fasangova (Charles University, Prague) |
| Volterra Equations and Quadratic Forms in the Study of a Physical Problem | |
| 10.00-10.20 | Carolina Cofano (Universität T"ubingen) |
| Semigroups for Delay Differential Equations | |
| 10.30-11.00 | Coffee break |
| 11.00-11.20 | Thomas Stricker (Universität T"ubingen) |
| Abstract Wave Equation I | |
| 11.30-11.50 | Viking Högnäs (University Helsinki) |
| Abstract Wave Equation II | |
| 12.30 | Lunch |
| Chair: Prof. Rainer Nagel (Universität Tübingen) | |
| 14.00-14.20 | Jan Vybiral (Charles University, Prague) |
| Holomorphic Semigroups - an Overview (Part 1) | |
| 14.30-14.50 | Balint Farkas (Eotvos Lorand University, Budapest) |
| Holomorphic Semigroups - an Overview (Part 2) | |
| 15.00-15.30 | Coffee break |
| 15.30-15.50 | Michal Benes (Charles University, Prague) |
| Maximal Regularity I | |
| 16.00-16.20 | Markus Duelli (Universität Ulm) |
| Maximal Regularity II | |
| 16.30-18.00 | Presentation of the Internet Seminar to the Rector of the University of Ulm and local newspapers. |
| 18.30 | Dinner |
| 20.00 | Concert at the Blaubeuren Monastery |
| Thursday, June 22 | |
| 7.00 | Early Morning Jogging (voluntary!) |
| 8.00-9.00 | Breakfast |
| Chair: Ti-Jun Xiao (Hefei) | |
| 9.10-9.50 | Prof. Rainer Nagel (Universität Tübingen) |
| Why do Populations Need Semigroups? | |
| 10.00-10.20 | Horst Heck (TU Darmstadt) |
| Gaussian Estimates - L^p-Holomorphy | |
| 10.30-11.00 | Coffee break |
| 11.00-11.20 | Tomas Barta (Charles University, Prague) |
| Gaussian Estimates - Second Order Operators and Dirichlet Boundary | |
| 11.30 | Plenum Discussion |
| 12.30 | Lunch |
| Chair: Prof. Silvia Romanelli (Università di Bari) | |
| 14.00-14.20 | Simona Fornaro (Università di Lecce) |
| L^{\infty} -Contractivity of Semigroups Generated by Sectorial Forms | |
| 14.30-14.50 | Tomomi Yokota (Science University of Tokyo) |
| Invariance of Closed Convex Sets under Semigroups of Nonlinear Operators | |
| 15.00-15.30 | Coffee break |
| 15.30-15.50 | Markus Biegert (Universität Ulm) |
| The Neumann Laplacian on Spaces of Continuous Functions - Entire Space R^n and Half-Space R^n_+ | |
| 16.00-16.20 | David Opela (Charles University, Prague) |
| The Neumann Laplacian on Spaces of Continuous Functions - General Bounded Smooth Domains | |
| 19.00 | Conference Dinner |
| 21.00 | Last Evening - Fare Well Party |
| Friday, June 23 | |
| 7.00 | Early Morning Jogging (voluntary!) |
| 8.00-9.00 | Breakfast |
| Chair: Prof. Jin Liang (Hefei) | |
| 9.10-9.50 | Prof. Yuri Latushkin (University of Missouri-Columbia) |
| A Spectral Mapping Theorems and Invariant Manifolds for Nonlinear Schroedinger Equations | |
| 10.00-10.20 | Ingrid Carbone (Università di Bari) |
| Differential Operators, Semigroups and Positive Linear Operators on Unbounded Intervals | |
| 10.30-11.00 | Coffee break |
| 11.00-11.20 | Delio Mugnolo (Università di Bari) |
| A Semigroup Approach to a Parabolic Problem Involving Nonlinear Operators and Nonlinear Boundary Conditions | |
| 11.30-11.50 | Hadd Said (Cadi Ayyad University, Marrakech) |
| Critical Spectrum and Regularity | |
| 12.00 | Closing ceremony |
| 12.30 | Lunch |
| Saturday, June 19 | |
| 8.00-9.00 | Breakfast |
| Departure | |
Tübingen: R.Nagel, F. Kühnemund
Ulm: W.Arendt
Karlsruhe: L.Weis
Office: S. Thomaschewski