Contemporary Mathematicians. Boston-Basel-Stuttgart: Birkhäuser. 648 p., 758 p. DM 498.00 (1985).
Fritz John was born in Berlin in 1910, obtained his Ph. D. from the University of Göttingen in the ominous year 1933; he left Germany in 1934 for Great Britain and later the USA. The papers collected in these two volumes cover the period 1934-1984 and they have been arranged by subject. There are the following sections: I) Random Transfer, II) Differential Equations; III) Ill-posed problems; IV) Formation of Singularities, V) Numerical Methods, VI) Elasticity Theory; VII) Water Waves, VIII) Geometric Inequalities and Convexity; IX) Functions of Bounded Mean Oscillations; X) Miscellaneous.
The editor, J. Moser, starts his preface with the following appreciation: "The mathematical works of Fritz John whose deep and original ideas have had a great influence on the development of various fields in mathematical analysis are made available with these volumes. His works are certainly well known to the experts, but knowledge of his contributions may not have spread as widely as it should have. For example, the concept of functions of bounded mean oscillations plays a central role in harmonic analysis today, but it is perhaps less known that this class of functions was introduced by John as early as 1961, motivated by his work in elasticity theory. With the publication of this collection, a wider circle of mathematicians will become familiar with, and appreciate, the fertile ideas of Fritz John".
The editing has been done with great care. There is a personal foreword by Gårding, a list of publications of John, a biographical sketch; moreover most sections end with remarks by the author and comments by a number of distinguished mathematicians. We mention some examples.
In section II we find a paper on algebraic conditions for hyperbolicity of systems of partial differential equations, which according to P. D. Lax, represents ’a very original contribution to algebraic geometry over . To the disappointment of many applied mathematicians, this has been a subject seriously neglected by algebraists. In section IV we have papers on singularities and blow-up of solutions of nonlinear wave equations. S. Klainerman summarizes some of the results as follows: ’he (John) showed, in dimension , that if the initial data has finite support and sufficiently small amplitude of order , a classical solution of the corresponding initial value problem, in the whole space, exists for at least an interval of time of length while the sharp similar bound in dimension one is only . It should be mentioned that the estimate for is not sharp, but altogether such dimension-dependent estimates are very intrigueing. Finally in section VI-(elasticity) there are comments by W. T. Koiter who also discusses open problems as the interaction of boundary layers and interior solutions in nonlinear elasticity.
A final word about collected papers like this. This century has shown very high productivity in mathematics, also there has been a corresponding proliferation of papers, journals and science literature. It is very difficult to find one’s way in all this material and even to have daily access to everything one needs. Collected papers by important mathematicians like F. John are very helpful in research and all science libraries should acquire these volumes.