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
Click the links below for the lecture notes of this Mini Course.
Syllubus Part One Part Two Part Three
Click the links below for the problems and solutions of this Mini Course.
Problems and Solutions
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.
Click the link below for the slide Prof. Feldman used for this Mini Course.
Slide
Click the links below for the problems and solutions of this Mini Course.
Problems and Solutions
Speaker: Monica Torres, Purdue University, USA
Title: Divergence Measure Fields, Conservation Laws, and Free Boundary Problems
Abstract: The divergence operator is of fundamental importance as many important equations are in divergence form. In the first part of the mini-course we will present several results concerning the solvability of the equation div F = T in various spaces of functions. The case when T is a Radon measure m will be of particular interest. A vector field F satisfying div F = m is called a divergence-measure field. Remarkably, the solvability of this equation is connected to the analysis of BV*, the dual of the space of functions of bounded variation. This space is widely used in image processing to model the noise of an image.
In the second part of the mini-course we will continue the analysis of divergence-measure fields, focusing on the existence of normal traces and the corresponding Gauss-Green formulas. 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. Since the shock waves are free boundaries for hyperbolic equations, the analysis of their structure and regularity is an important question that will be discussed in this mini-course, as well as the connections with divergence-measure fields.
In the third part of the course we will discuss a free boundary problem arising in the modeling of populations. This problem involves fully-nonlinear elliptic operators. We will show the existence of solutions, non-degeneracy properties at regular points and the analysis of the blow-up of the solution. Finally, we will show that we can apply the regularity theory developed by Caffarelli to obtain regularity of the interfaces for this problem.
Click the links below for the slides of this Mini Course.
Slides - Part One Slides - Part Two
Click the links below for the lecture notes of this Mini Course.
Part One Part Two Part Three
Click the links below for the problems and solutions of this Mini Course.
Problems Solutions