This book is amiably dedicated to the “Free Boundary Fraternity”. Not by chance: the huge theory of the Stefan problem and of the many free boundary problems closely related to it was developed (mainly during the second half of the last century) by a “fraternity” of many persons, clearly recognizable going through the various chapters, who in most cases have been linked to each other by a sincere friendship. It is a nice trait of this book that this feeling is made explicit.
The author undertook a formidable and frightening task because the material he had to deal with was not only very large but also very diversified. Phase change alone (which makes the core of the Stefan problem) presents many different physical situations and requires an impressive variety of mathematical techniques.
The book is well organized, so that, in spite of its complexity, the exposition is seemingly effortless and reading is greatly facilitated by a very judicious mixing of phenomenology and mathematics.
Very appropriately the book opens with a large number of examples of free boundary problems arising in different applications. Thus the reader is given an idea of the great practical relevance of the theory that will be presented, anticipating some basic notions and results. The second chapter illustrates the thermodynamics of phase change, including special topics like supercooling, the effect of the interface curvature, etc. Various extensions are then analyzed along with other problems, like the oxygen diffusion-consumption problem, which differ from the Stefan problem in the fact that the interface conditions give no explicit information on the velocity of the interface.
A group of chapters follows in which the author goes more deeply in the analysis of the mathematical structure of phase change problems with supercooling or superheating in their various formulations (phase-field models, kinetic undercooling, curvature effects, weak formulations, mushy regions in the presence of heat sources, blow-up phenomena). Another chapter deals with steady state problems and quasi-steady problem, with specific reference to Hele-Shaw flows.
The variational techniques needed to treat both the elliptic and the parabolic Stefan problem are then illustrated.
The Stefan problem with a hyperbolic heat conduction model is also treated extensively. Moreover, the book includes a large chapter on the inverse Stefan problem, which is analyzed in great detail.
Perhaps the chapters most adherent to the title of the book are the last two. There existence, uniqueness and regularity of solutions are investigated for Stefan problems in one or two phases and in one or more space dimension, illustrating a large portion of the literature.
The book is written in a plain style, sistematically avoiding the most technical details, but focussing on the essential ideas and on the main results.
A book like this, bridging physics and mathematics in a concrete and readable way, was very much needed. Prof. Gupta’s book fulfills that task nicely. It could be objected that it is not complete. Problems not treated are for instance the Stefan problem with concentrated capacities, the problems connected with the growth of large single crystals in a crucible and classes of problems in the flow of fluids through porous media (besides the porous medium equation and the dam problem shortly dealt with in the book). Many other examples of practical interest, like for instance the so-called electromachining problem, are not included. However the field of free boundary problem is so large that the target of completeness would only be appropriate to an encyclopaedia, rather than to a book of reasonable size. Correspondingly, the bibliography is also the result of a selection. Somebody in the “fraternity” may not find in it some of his favourite papers, but it must also be said that the total number of papers on free boundary problems in the time period spanned by the book is probably exceeding 5000 and that a reasonable bibliography could only contain less than \(10\%\) of them.
In conclusion I believe that the author should be praised for his commendable effort and that the book will be useful for the free boundary people and for those who wish to join the fraternity.