Free boundary problems arise naturally in the mathematical formulation of a great variety of real phenomena in technology and applied sciences. The mathematical literature has increased enormously in recent years, not only on mathematical journals but also in proceedings of meetings and in books. A rather comprehensive bibliographical account containing more than 2500 titles up to 1987 has been compiled by D. Tarzia [1], covering only moving and free boundary problems for the heat diffusion equation, specially Stefan-type problems.
Although the first work on this topic seems due to Lamè and Clapeyron, who in 1831 considered a simple model for the solidification of a liquid sphere [2], the prototype of a free boundary problem for the heat equation, the ice-water problem, is now usually called the Stefan problem, after the austrian mathematical physicist Joseph Stefan, who wrote a series of papers on this problem one century ago [3]. Even if the applied interest of this problem is limited, its mathematical relevance is quite large as a typical model where new theoretical and numerical methods have been developed and tested as a starting point for the analysis of more complicated problems.
Over the past two decades, the development of free boundary problems has enlarged the scope of this interdisciplinary research topic up to a point where now is almost impossible to cover all the literature. A rather good idea of the field can be found on the contents of the Conference Series on "Free Boundary Problems: Theory and Applications", contained in the FBP-News #1 (Montecatini—1981, Maubuisson—1984, Irsee—1987 and Montreal—1990) and in the FBP-News #2 (Toledo—1993). However these larger conferences do not exhaust the matter and a significant number of other workshops and meetings took also place in the last five years like, for instance, Óbidos—1988 [4], Oberwolfach—1989 [5], Jyväskylä—1990 [6], Southampton—1991 [7], Novosibirsk—1991 [8], Milano—1993 [9], besides a NATO Advanced Study Institute held in Montreal—1990 [10] (see Appendix for their contents).
An important event for the development of these mathematical studies was the whole academic year 1990—91 programme on "Phase Transitions and Free Boundaries", organized by the "Institute of Mathematics and its Applications" (IMA) at the University of Minnesota, with short and long term visitors, a lecture series and a number of specialized workshops, whose proceedings appeared as IMA volumes in its collection (see [11–15]).
The IMA research workshops covered a wide range of subjects, such as: On the Evolution of Phase Boundaries [11], involving ideas and methods from nonlinear partial differential equations, asymptotic analysis, numerical computation and experiment, to describe phase transitions and interface dynamics; Degenerate Diffusions [12], reporting both mathematical models and techniques in nonlinear diffusion involving free boundaries or sharp interfaces; Shock Induced Transitions and Phase Structures in General Media [13], presenting experimental, physical and mathematical questions in molecular dynamics, shear induced dynamic phase transitions, systems allowing change of type, shock stability and theoretical implications in their constitutive structure; Variational Problems [14], presenting mathematical methods applied to a variety of models arising, for instance, in elastic/plastic contact problems, Hele—Shaw cells, crystal growth, computer vision, magneto-hydrodynamics, bubble growth and in stochastic control and economics; and Free Boundaries in Viscous Flows [15], focusing a broad range of subjects on interfacial phenomena, whose free boundaries are interfaces of the fluid with either second immiscible fluids or else deformable solid boundaries due to mechanical displacement or to phase transformation.
Additionally, one should mention also that in the mathematical modelling of current industrial processes the emergence of free boundary problems is a constant feature and an inexhaustible source of new and challenging mathematical questions as can be found, for instance, in the series of books [16], based on seminars involving industrial scientists engaged in research and development of new or improved products, or in most recent proceedings of meetings, such as [17] for example.
Of course, nothing has been said about publications in scientific journals, either of mathematical, physical or engineering nature. Just to give an example, in the already mentioned work of Tarzia [1], up to 1987, about five hundred titles (28% of the 1798 quoted titles) of papers published in twenty mathematical journals (abbreviated title) are quoted, namely:
Recently new journals, such as European J. Appl. Math. or Adv. Math. Sci. Appl., have appeared and have already published mathematical papers relevant to the FBP field. However the number of works on free boundary problems has increased and it is then natural to discuss the question if there should exists or not a new mathematical journal devoted exclusively to this interdisciplinary topic. The answer to this question, which has been already raised by several people, is not a minor point to the development of the mathematical research in free boundary problems and should be considered very seriously.
REFERENCES
[1] G. LAMÉ & B.P. CLAFEYRON — Mêmoire sur le solidification par refroidissement d'un globe solid, Annales Chemie Physique, 47 (1831), 250–156.
[2] J. STEFAN — Über linige Probleme der Theorie der Wärmeleitung, Zit. Akad. Wissenchaften (Wien), Math. cl. 98 (1989), pg. 473–484; 614–634; 965–983; 1418–1442.
[3] D.A. TARZIA — A bibliography on Moving-Free Boundary Problems for the Heat-Diffusion Equation. The Stefan Problem, Prog. Nazionale M.P.I., Firenze, 1988.
[4] J.F. RODRIGUES (Ed.) — Mathematical Models for Phase Change Problems, Int. Ser. Num. Math., 88, Birkhäuser, Basel, 1989.
[5] K.-H. HOFFMANN & J. SPREKELS (Eds.) — Free Boundary Value Problems, Int. Ser. Num Math, 95, Birkhäuser, Basel, 1990.
[6] P. NEITTAANMÄKI (Ed.) — Numerical Methods for Free Boundary Problems, Int. Ser. Num. Math., 99, Birkhäuser, Basel, 1991.
[7] L.C. WROBEL & C.A. BREBBIA (Eds.) — Computational Modelling of Free and Moving Boundary Problems, Computational Mechanics Publications, Southampton, 1991.
[8] S.N. ANTONTSEV, K.-H. HOFFMANN & A.M. KHLUDNEV (Eds.) — Free Boundary Problems in Continuum Mechanics, Int. Ser. Num. Math., 106, Birkhäuser, Basel, 1992.
[9] L.C. WROBEL & C.A. BREBBIA (Eds.) — Computational Modelling of Free and Moving Boundary Problems II, Computational Mechanics Publications, Southampton, 1993.
[10] M.C. DELFOUR & G. SABIDUSSI (Eds.) — Shape Optimization and Free Boundaries, Kluwer, Dordrecht, 1992.
[11] M.E. GURTIN & G. MCFADDEN (Eds.) — On the Evolution of Phase Boundaries, IMA series, Vol. 43, Springer, New York, 1991.
[12] W.-M. NI, L.A. PELETIER & J.L. VAZQUEZ (Eds.) — Degenerate Diffusion, IMA series, Vol. 47, Springer, New York, 1992.
[13] R. FOSDICK, E. DUNN & M. SLEMROD (Eds.) — Shock Induced Transitions and Phase Structures in General Media, IMA series, Vol. 52, Springer, New York, 1993.
[14] A. FRIEDMAN & J. SPRUCK (Eds.) — Variational Problems, IMA series, Vol. 53, Springer, New York, 1993.
[15] R.A. BROWN & S.H. DAVIS (Eds.) — Free Boundaries in Viscous Flows, IMA series, Vol. 61, Springer, New York, 1993.
[16] A. FRIEDMAN — Mathematics in Industrial Problems, Parts 1–6, IMA Series, Vols. 16, 24, 31, 38, 49 and 57 (1993), Springer, New York.
[17] H. KAWARADA, N. KENMOCHI & N. YANAGIHARA (Eds.) — Nonlinear Mathematical Problems in Industry, Gakuto Int. Series Math. Sci. and Applications, Tokyo, 1993.
| (article in FBP News #3, Nov./Dec. 93 p.12-13) |