Free Boundary Problems in Science and Technology, Volume 47, Number 8

854 NOTICES OF THE AMS VOLUME 47, NUMBER 8 F ree boundary problems deal with solving partial differential equations (PDEs) in a domain, a part of whose boundary is unknown in advance; that portion of the boundary is called a free boundary. In addition to the standard boundary conditions that are needed in order to solve the PDEs, an additional condition must be imposed at the free boundary. One then seeks to determine both the free boundary and the solution of the differential equations. The theory of free boundaries has seen great progress in the last thirty years; for the state of the field up to 1982 we refer to [3] and [10]. Two types of problems stimulated this progress. The first one is the “obstacle problem”, which, in the time-independent case, can be stated as a variational problem. The second one is the Stefan problem, describing the process of melting and solidification. These two problems will be briefly reviewed in the first two sections. The main objective of the present article, however, is to describe several more recent free boundary problems, selected from an increasingly large number of such problems that arise in science and technology and that require the development of new mathematical ideas and analytical techniques. Since the aim of this article is not to provide an exhaustive review of the subject of free boundary problems, but rather to brief the general mathematical public on a few of the current research directions, we had to drastically curtail the list of references and the full historical allocation of credit.