Speaker: Xinfu Chen, University of Pittsburgh, USA
Title: Introduction to Free Boundary Problems
Abstract: This
mini-course introduces some basic theory on free boundary problems and
its associated variational problems. As an example, an American put
problem will be thoroughly investigated. More specifically the course
will cover the following topics:
1) Obstacle Problem and Variational Inequality
2) Stefan Problem
3) American Put from Mathematical Finance
4) Convexity of free boundary for Stefan problem
Speaker: Mikhail Feldman, University of Wisconsin-Madison, USA
Title: Shock Reflection, von Neumann Conjectures, and Free Boundary Problems
Abstract: In this course, we discuss shock reflection problem for compressible gas dynamics, and von Neumann conjectures on transition between regular and Mach reflections. We will describe recent results on existence and properties of regular reflection solutions for potential flow equation. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear elliptic equation, with ellipticity degenerating near a part of the fixed boundary. We will discuss the techniques andmethods used in the study of such free boundary problems.
Speaker: Monica Torres, Purdue University, USA
Title: Divergence Measure Fields, Conservation Laws, and Free Boundary Problems
Abstract: In this series of lectures we will present the theory of divergence-measure fields, focusing on the existence of traces on boundaries of sets of finite perimeter and the corresponding Gauss-Green formulas. The necessary background on measure theory and the fine properties of BV functions and sets of finite perimeter will be introduced first. Applications of the theory of divergence-measure fields to nonlinear hyperbolic conservation laws will be explained, including the development of the formulation of Cauchy fluxes that allow the inclusion of shock waves, and the derivation of systems of balance laws.
In the second part of the mini-course, further analytical properties of the divergence operator will be presented. The understanding of this operator is fundamental as many important equations are in divergence form. In particular, several results concerning the solvability of the equation div F = T in various spaces of functions will be explained. The case when T is a measure will be of particular interest. Remarkably, the solvability of this equation is connected to the analysis of BV*, the dual of the spaces of functions of bounded variation. This space is widely used in image processing to model the noise of an image.
In the last part of the course, the question concerning the structure of entropy solutions and the regularity of the shock waves will be discussed, as well as the connections with divergence-measure fields. The shock waves are free boundaries for hyperbolic equations, and thus un understanding of their structure and regularity is required.