Workshop

Many problems in physics, industry, finance, biology, and other areas can be described by partial differential equations that exhibit apriori unknown sets, such as interfaces, moving boundaries or shocks for example. The study of such sets, also known as free boundaries, often plays a central role in the understanding of such problems. The aim of this workshop is to introduce several free boundary problems arising in completely different areas. Schedule 09:30 - 10:30   Toti Daskalopoulos (Columbia University): Degenerate Geometric Flows and related Free-Boundary Problems Abstract: We will discuss the evolution of hyper-surfaces in R^n+1 by functions of their principal curvatures. Examples include the evolution by powers of the Gaussian curvature and the Harmonic mean curvature flow. Such flows result to fully-nonlinear parabolic PDE which become degenerate at points or interfaces where one or more of the principal curvatures vanish. The question of the optimal regularity of solutions is often connected to the study of related free-boundary problems. The topics presented will include: An overview of the geometric flows arising by the evolution of hyper-surfaces by functions of their principal curvatures. Short time existence of solutions with the optimal regularity and related degenerate linear problems. Sharp a priori estimates. Long time existence of regular solutions. Singularities. Regularity questions and open problems. 10:30 - 11:00   Tea   11:00 - 12:00   Mikhail Feldmann (University of Wisconsin): Free Boundary Problems in Shock Analysis Abstract: Shocks in gas or compressible fluid arise in various physical situations, and often exhibit complex structures. One example is reflection of shock by a wedge, first described by Ernst Mach in 1878. In later works, experimental and computational studies and asymptotic analysis have shown that various patterns of reflected shocks may occur, including regular and Mach reflection. However, many fundamental issues related to shock reflection are not understood, including transition between different reflection patterns. For this reason it is important to establish mathematical theory of shock reflection. Some results in this direction were obtained recently. In this course we start by discussing the basic equations of gas dynamics: steady and self-similar compressible Euler system and potential flow equation. These equations are of mixed elliptic-hyperbolic type. Shocks correspond to certain discontinuities in the solutions. In many cases, study of solutions with shocks can be reduced to solving a free boundary problem for the elliptic part of the solution. We discuss existence, stability and regularity of steady and self-similar solutions with shocks in several cases, including shock reflection problem.   12:00 - 14:00   Lunch   14:00 - 15:00   Inwon Kim (University of California, Los Angeles): Homogenization for free boundary problems Abstract: I will discuss recent results on the homogenization of several free boundary problems, with oscillatory boundary conditions.   15:00 - 15:30   Tea   15:30 - 16:30   Arshak Petrosyan (Purdue University): Monotonicity formulas and obstacle type problems (Lecture Notes) Abstract: We will discuss several types of monotonicity formulas (Alt-Caffarelli-Friedman, Almgren, Weiss, Monneau) and their generalizations to study different aspects of obstacle type problems, such as the optimal regularity, classification of free boundary points, the study of blowups, the regularity of the free boundary, and the structure of the singular set. On Tuesday, a reception will be held for all participants after the last talk. Bibliography (PDF) Accommodation: A block of rooms has been reserved at the Hotel Durant. Please mention MSRI and the workshop name when making reservations via phone, fax or e-mail. If you are making your reservations on line, please go to Hotel Durant website, choose your dates of stay and enter the "MSRIWORKSHOP" promo code in the box. The cut-off date for reservations is December 17, 2010. The rate is $110 per night plus tax. A block of rooms has been reserved at the Rose Garden Inn. Reservations may be made by calling 1-800-992-9005 OR directly on their website. Click on Corporate at the bottom of the screen and when prompted enter code MATH (this code is not case sensitive). By using this code a new calendar will appear and will show MSRI rate on all room types available. The cut-off date for reservations is December 27, 2010. New, completely renovated Hotel Shattuck Plaza has rooms available for you! MSRI’s preferred rate is $131. Guests can either call the hotel’s main line, 510-845-7300, and ask for the MSRI rate or go to http://www.hotelshattuckplaza.com, click on "Corporate Rates" in the reservation screen and type the code msri10.