# zbMATH — the first resource for mathematics

Free and moving boundary problems. (English) Zbl 0547.35001
Oxford Science Publications. Oxford: Clarendon Press. X, 425 p. £45.00 (1984).
This book is welcome by the growing population of scientists interested in free boundary problems for partial differential equations. It is well known that a common feature of such problems is the fact that part of the boundary of the domain in which the p.d.e. is to be solved is unknown and has to be determined by means of additional conditions imposed on it. In evolutionary problems the unknown boundary generally moves and for such cases the author reserves the name of ”moving boundary problem” (MBP), while steady state problems are referred to as ”free boundary problems” (FBP). A list of the problems dealt with in the first two chapters gives an idea of the variety of physical phenomena falling in this domain of investigation.
(A) MBP: melting (Stefan problem), melting with ablation or convection, multiphase problems, alloys solidification, displacement of immiscible fluids in a porous medium (Muskat problem), oxygen diffusion-consumption in tissues, change of phase with concentrated thermal capacities. (B) FBP: filtration of water through a dam, other filtration problems (coastal aquifers, canals and ditches, etc.), unsaturated flows, the Hele-Shaw cell, electrochemical machining, lubrication of a journal bearing.
All the problems above are clearly motivated, formulated and commented. Many references are given to further extensions and developments. Chapters 3-6 are concerned with MBP. Analytical solutions are expressed in chapter 3 with references to the Stefan problem. In the same chapter various integral formulations are described and methods to obtain approximate solutions are illustrated.
Chapter 4 is devoted to front-tracking numerical methods, including different types of finite-difference and finite-elements methods. The method of lines is also treated at length. Front-fixing numerical methods (chapter 5) are based on coordinate transformations, mapping the original moving domain in a fixed one (the variable originally related to the unknown boundary now appears in the differential equation). One of the most important method in this class is the isotherm migration method, mainly developed by the author. This chapter is closed by a comparative analysis among the methods presented.
Next chapter describes methods labelled as fixed-domain methods. The enthalpy method and the alternating-phase truncation method are treated starting from their historical background and reporting all subsequent applications. Also variational inequalities formulations and corresponding discretized versions are included in this chapter, whose concluding section is a dense summary of recent literature. Chapters 7-8 are devoted to analytical and numerical solutions of free (i.e. stationary unknown) boundary problems. Seepage problems are discussed in chapter 7 under various aspects with particular reference to classical methods providing approximate solutions. The concluding chapter is a review of algorithms applicable to a large variety of FBP’s.
This book accomplishes the difficult task of presenting an up-to-date account of an extremely complex subject, maintaining a pleasant and fully understandable style. With its rich choice of topics and its remarkably extensive bibliography it is a compulsory reading for everybody engaged in this research field.
Reviewer: A.Fasano

##### MSC:
 35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations 35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations 35R35 Free boundary problems for PDEs 49J40 Variational inequalities 65Z05 Applications to the sciences