||This book is based on a course given at Carnegie Mellon and on a lecture series given at the Summer School on Mathematical Physics in Ravello, Italy; the Istituto Mauro Picone in Rome; and the Institute for Mathematics and its Applications, University of Minnesota.
The book is not meant to be comprehensive; it presents topics that have interested me over the past few years, and it represents a point of view different from that prevalent in the physical literature. Being an engineer by training, but a mathematician by choice, I have tried to make the book comprehensible to both mathematicians and physical scientists. Throughout I emphasize issues that are foundational in nature, as I am more interested in the interplay between mathematics and physics than in the solution of specific problems. I present what I hope are rational derivations of those free-boundary problems that form the basis of the subject. These problems should be of great interest to analysts. Unfortunately, technical mathematical issues such as existence, uniqueness, and regularity of solutions are discussed only superficially; these are better left to mathematicians more competent in this area than I.
Throughout the development of this material I have profited greatly from discussions with Sigurd Angenent, Paolo Podio-Guidugli, and David Kinderlehrer. I also acknowledge valuable discussions with Perry Leo, Jose Matias, William Mullins, Robert Sekerka, Mete Soner, Allan Struthers, Peter Voorhees, and William Williams.
My research as presented here was supported by the Army Research Office and the National Science Foundation; this support is gratefully acknowledged.