News Release

Mathematics Addresses Problems of
Science and Technology

For further information, contact:
Avner Friedman
Director, Minnesota Center for Industrial Mathematics
University of Minnesota
Telephone: 612-624-4025, 612-625-3377
Email: friedman@math.umn.edu

August 3, 2000

PROVIDENCE, RI---Free boundary problems are mathematical constructs that provide powerful means for understanding various phenomena in science and technology. In recent years, important new examples of free boundary problems have arisen, demanding the development of new mathematical ideas and techniques.

Free boundary problems are used to represent phenomena that change, either with time or with the variation of another parameter. The equations describing the nature of the change are valid only within a certain region. Sometimes a part of the boundary of this region is not prescribed in advance, and this part is called the "free boundary" and is determined in tandem with the solution to the equations.

One example comes from the manufacturing of photographic film. In this process, sheets of fluid are dispersed in layers onto the film base, which is moving horizontally on a conveyor belt. The fluid falls under the influence of gravity and is often guided by control of air pressure. An important problem is to understand exactly how the fluid falls onto the film. In particular, for good quality coating, the angle of contact between the fluid and the film must be kept constant. Experiments have been done to measure this angle, but there is no theory giving a precise relationship between the speed of the moving film and the contact angle.

The motion of the fluid can be described by partial differential equations that govern fluid flow. In such a description, the boundary between the fluid and the air forms a free boundary. New mathematical techniques developed specifically to address this free boundary problem have helped industrial researchers to understand better the shape the fluid takes as it coats the film.

The article, "Free Boundary Problems in Science and Technology" by Avner Friedman, describes a number of examples in which progress in mathematical techniques for addressing free boundary problems has led to new understanding of important problems in science and technology, such as semiconductor manufacturing and tumor growth. The article will appear in the September 2000 issue of Notices of the AMS. A PDF file containing the article may be downloaded at http://www.ams.org/notices/200008/fea-friedman.pdf.