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Workshop

Introductory Workshop: Free Boundary Problems, Theory and Applications January 18, 2011 - January 21, 2011
 Registration Deadline: January 18, 2011 over 2 years ago November 07, 2010 over 2 years ago
Parent Program: Free Boundary Problems, Theory and Applications
Organizers Tatiana Toro* (University of Washington)
Speaker(s) Panagiota Daskalopoulos (Columbia University), Craig Evans (University of California, Berkeley), Mikhail Feldman (University of Wisconsin), Inwon Kim (University of California), Arshak Petrosyan, Georg Sebastian Weiss
Description

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:
1. An overview of the geometric flows arising by the evolution of hyper-surfaces by functions of their principal curvatures.
2. Short time existence of solutions with the optimal regularity and related degenerate linear problems. Sharp a priori estimates.
3. Long time existence of regular solutions.
4. Singularities.
5. 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.

Funding & Logistics

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To apply for funding, you must register by the funding application deadline displayed above.

Students, recent Ph.D.'s, women, and members of underrepresented minorities are particularly encouraged to apply. Funding awards are typically made 6 weeks before the workshop begins. Requests received after the funding deadline are considered only if additional funds become available.

Show Lodging

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.

A block of rooms has been reserved at the Hotel Durant. Reservations may be made by calling 1-800-238-7268. When making reservations, guests must request the MSRI preferred rate. If you are making your reservations on line, please go to this link and enter the promo/corporate code MSRI123. Our preferred rate is \$129 per night for a Deluxe Queen/King, based on availability.

Additional lodging options (short term housing page - Short-term-housing

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Schedule
Jan 18, 2011
 09:25 AM - 09:40 AM Welcome 09:40 AM - 10:40 AM Degenerate Geometric Flows and related Free-Boundary Problems 11:10 AM - 12:10 PM Free Boundary Problems in Shock Analysis Mikhail Feldman (University of Wisconsin) 02:00 PM - 03:00 PM Homogenization for free boundary problems Inwon Kim (University of California) 03:30 PM - 04:30 PM Monotonicity formulas and obstacle type problems Arshak Petrosyan
Jan 19, 2011
 09:30 AM - 10:30 AM Degenerate Geometric Flows and related Free-Boundary Problems 11:00 AM - 12:00 PM Free Boundary Problems in Shock Analysis Mikhail Feldman (University of Wisconsin) 02:00 PM - 03:00 PM Homogenization for free boundary problems Inwon Kim (University of California) 03:30 PM - 04:30 PM Monotonicity formulas and obstacle type problems Arshak Petrosyan
Jan 20, 2011
 09:30 AM - 10:30 AM Degenerate Geometric Flows and related Free-Boundary Problems 11:00 AM - 12:00 PM Free Boundary Problems in Shock Analysis Mikhail Feldman (University of Wisconsin) 02:00 PM - 03:00 PM Homogenization for free boundary problems Inwon Kim (University of California) 03:30 PM - 04:30 PM Monotonicity formulas and obstacle type problems Arshak Petrosyan
Jan 21, 2011
 09:30 AM - 10:30 AM Degenerate Geometric Flows and related Free-Boundary Problems 11:00 AM - 12:00 PM Free Boundary Problems in Shock Analysis Mikhail Feldman (University of Wisconsin) 02:00 PM - 03:00 PM Homogenization for free boundary problems Inwon Kim (University of California) 03:30 PM - 04:30 PM Monotonicity formulas and obstacle type problems Arshak Petrosyan