Nonlinear Analysis (2005)

Organizers: Henrik Shahgholian (Stockholm), Peter Markowich (WPI c/o Univ. Vienna)

(Co-funded with ESF program Global)

This Thematic Program aims at the study of properties of solutions of nonlinear partial differential equations (PDE) and free boundary problems (FBP) from the viewpoints of theory and applications.

The investigation of global and geometric properties of solutions of partial differential equations is a key research area in mathematics that has many practical applications. The focus of this program will be on problems where not only geometric aspects will play a central role but also the fact that a-priori the domain under consideration is unknown (whence the name FBP).

Regularity questions of unknown sets have been in focus for several decades. Many times, modeling phenomena in nature by partial differential equations give rise to unknown boundaries. It is of great importance to analyze these boundaries from a rigorous mathematical point of view. Recent years have also seen many new directions as well as new techniques entering into this very technical field. Several areas such as potential theory, micromagnetics, image processing, computer vision, geometric measure theory, free boundary problems in physics/mechanics, and many other subjects have gained or regained from the new techniques.

The local study of solutions of PDEs is often carried out through a "zooming" technique or blow-up analysis. This procedure reduces the problem to the classification of global solutions (solutions in the whole space). It is often employed, for instance, for minimal surfaces, free boundary problems and for semilinear equations.

Similar global symmetry questions are now under study in relation with problems where the reaction only takes place on the boundary, and in the interior of the material there is only diffusion. Phase transition type (or layer) solutions are for first time under analysis. Here, as in the famous (and partially solved) conjecture of De Giorgi, the combination of PDE and variational tools provides very strong techniques.

A further theme in focus is the theory of optimal stopping, central in economics. For example,

1. the problem of finding the optimal time of starting (or stopping) an economic activity under uncertainty can often be formulated as an optimal stopping problem.
2. Second, it has been proved that the price of an American option in finance is given by the solution of an optimal stopping problem. Today it is well known that an optimal stopping problem for a diffusion is equivalent to a free boundary problem, and such problems are again equivalent to certain variational inequalities involving partial differential operators (the generators of the underlying diffusions). We need to solve the corresponding PDEs in the so-called continuation region and combine this with high contact (smoothness) conditions. This gives an explicit link between PDEs and optimal stopping and hence with finance.

Events

Workshop on Free Boundaries and PDEs in Biology
Location: WPI Nordbergstr. 15 / Time: 21. Mar 2006 (Tue) - 22. Mar 2006 (Wed); Opening: 9:00
Organiser(s) Danielle Hilhorst (Univ. Paris sud) Christian Schmeiser (WPI and Univ. of Vienna) Henrik Shagholian (KTH Stockholm)
Remark: Program




Workshop "Symmetries of solutions of nonlinear elliptic PDEs
Location: Nordbergstrasse 15/ Time: 23. Jan 2006 (Mon) - 27. Jan 2006 (Fri); Opening: 10:00
Organiser(s) Henrik Shahgholian (henriksh@kth.se) Wolfgang Reichel (wolfgang.reichel@math.unizh.ch) Alberto Farina (alberto.farina@u-picardie.fr)




New developments in Hele-Shaw flow (or Laplacian growth)
Location: WPI Nordbergstraße, Seminar Room Time: 13. Dec 2005 (Tue) - 13. Dec 2005 (Tue); Opening: 9:00
Organiser(s) Björn Gustafsson (KTH Stockholm) Henrik Shahgholian (KTH Stockholm)




Conference on nonlinear PDEs in honor of Luis Caffarelli
Location: Nordbergstraße 15, UZA 4, Rooms 206 and 207 Time: 24. Oct 2005 (Mon) - 25. Oct 2005 (Tue); Opening: 9:00
Organisation(s) WPI CNRS Organiser(s) Alberto Farina (Inst. CNRS Pauli) Peter Markowich (WPI c/o Univ. Wien) Contact: wittgenstein.mathematik@univie.ac.at
Remark: Confirmed speakers: L. Caffarelli, I. Gamba, T. Souganidis, H. Shahgholian, J.L.Vazquez, A.Farina Program




NLPDE 05 : Summer School and Workshop on "Non Linear PDEs" (external website )
Location: Seminar Room C207, Fak.f. Mathematik, Nordbergstr. 15, UZA4 Time: 19. Jul 2005 (Tue) - 27. Jul 2005 (Wed); Opening: 14:00
Topics: Joint event of the EU network HYKE with the WPI Programm on Nonlinear PDEs and Applications and the ESF programme on "Global and Geometric Aspects of Nonlinear PDEs" of H. Shahgholian and P. Markowich . A 4 days school will be followed by a 3 days workshop
Organisation(s) WPI Organiser(s) Yann Brenier , CNRS Nice Peter Markowich , WPI c/o Vienna University Norbert J. Mauser , WPI c/o Vienna University Henrik Shahgholian , Royal Institute of Technology (KTH), Stockholm




Workshop "PDEs in Industry and Engineering" (external website )
Location: Vienna/Strobl Time: 26. Jun 2005 (Sun) - 30. Jun 2005 (Thu); Opening: 14:00
Topics: "Semiconductor modeling using quantum mechanical, kinetic and fluid dynamical models" "Free boundary problems in flame propagation" "Semiclassical analysis of Schroedinger equations with applications to quantum mechanical systems, eg Bose-Einstein condensation"
Organisation(s) WPI ALFA Project "PDEs in Industry and Engineering Wittgenstein Award - Peter Markowich Organiser(s) Peter A. Markowich , University of Vienna, WPI Jorge P. Zubelli , IMPA, Rio de Janeiro, Brazil
Remark: Contact/Information: Renate Feikes, wittgenstein.mathematik@univie.ac.at




Talks in the framework of this thematic program... (by date) , (by name)

Pauli Fellows