Free Boundary Problems

 

ABSTRACT:

In these lectures I shall introduce several mathematical problems that appear in various applied sciences, and have the common denominator of leading to the so-called Free Boundary (Value) Problems (FBP). The term Free Boundary Problem refers to a problem in which one or several unknown variables must be determined in different domains of the space, or space-time.
If the domains were known, the problem reduces to solving the equations, usually partial differential equations (PDEs). The novelty of a FBP lies in the fact that the domains are a priori unknown and have to be determined as a part of the problem. The process is usually controlled by several underlying mathematical conditions that are derived from certain physical laws or other constraints governing the phase transition.In these lectures I shall explain mainly the basic theory of FBP of obstacle type. The first lecture is intended for a broader audience. The remaining lectures shall be available for non-experts, including advanced undergraduate students, who are aware of PDE tools, and would like to learn standard and occasionally advanced tools in FBP.

Lecture 1.  AN INTRODUCTION TO FREE BOUNDARY PROBLEMS (FBP)
(Tuesday May 2, 16:00-17:30, CDT Lecture Room 3)

In the first lecture we  will be dealing with the development of the topic in the last few decades. I shall give a heuristic tour on various models in applications that leads to FB of obstacle type. This encompasses: constraint minimization, control theory for PDEs, optimal stopping time and dynamic programing, option pricing, optimal switching, inverse problems in potential theory. I shall also touch upon the topic of internal diffusion limited aggregation, and connect to FB. At the end of the lecture, I shall specify a few questions that have been subject of study in the last few decades.

Lecture 2.  MATHEMATICAL ANALYSIS OF FREE BOUNDARY PROBLEMS
(Thursday May 4, 14:00-15:30, CDT Lecture Room 3)

Second lecture will concern general approaches to analysis of the FBP,  from a regularity point of view. As a prototype I shall consider a specific model and its mathematical treatment. Here I shall give some detailed account of mathematical preliminaries and develop standard tools that are central in the theory. These include optimal regularity of solutions, non-degeneracy of solutions, Lebesgue, and Hausdorff measure of the FB. For the purpose of the task we shall introduce various tools such as: Monotonicity formulas, Scaling and Blow-up techniques, and others.

Lecture 3. REGULARITY THEORY
(Tuesday May 9, 15:00-16:30, CDT Lecture Room 3) 

The objective with this lecture is to give a general account of how one can show that a FB is regular, given certain a priori conditions. We shall prove that if  a priori the FB is  not too  “bad” (say has no cusps), it has good chances of being regular, up to C^1.  Two main tools for such an analysis are: Classification of global solutions, and cones of Directional monotonicity argument. The latter means the the solution function can be shown to be locally monotone in a cone of directions around a FB point, and hence the FB should be Lipschitz.

Lecture 4. CURRENT STATUS AND FUTURE DEVELOPMENT
(Thursday May 11, 15:00-16:30, CDT Lecture Room 3)

The fourth lecture will concern the developments in the last few years, and the connection to other areas, as well as possible future directions of the theory. Here I shall mention in bypassing, without much deepening into topics, the following cases: FB for systems, quasilinear and semil-inear PDEs with FB, nonlocal problems, and other related developments.

 

References:

1) Petrosyan, Arshak; Shahgholian, Henrik; Uraltseva, Nina Regularity of free boundaries in obstacle-type problems. Graduate Studies in Mathematics, 136. American Mathematical Society, Providence, RI, 2012. x+221 pp. ISBN: 978-0-8218-8794-3
2) There also several articles, that I shall refer to during the lectures.