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Complexes of differential operators. Rev. a. upd. transl. from the Russian by P. M. Gauthier. (English) Zbl 0852.58076
Mathematics and its Applications (Dordrecht). 340. Dordrecht: Kluwer Academic Publishers. xviii, 396 p. (1995).
The best known example of an overdetermined system of linear partial differential operators is perhaps the one associated with the “gradient” of a function. The compatibility conditions in the case of 3 dimensions are given by the operator “curl”, which in turn has the “divergence” operator as a compatibility condition. In \(n\) dimensions, this corresponds of course to the de Rham system, which, when we consider everything on a manifold, is deeply related to the geometry of the manifold under consideration. Indeed, for many years, overdetermined systems of partial differential equations were mainly considered in the context of geometry (starting with S. Lie and E. Cartan). When an overdetermined system is given, the right hand sides of the equations will in general have to satisfy compatibility conditions. In favorable situations, one might be able to write the set of compatibility conditions in the form of an overdetermined system of partial differential equations, and to study this new system one will have to consider the set of compatibility conditions for the new system, etc.: see the case of the “gradient” above, or the Cauchy-Riemann system in \(n\) complex variables. The natural object of study is therefore seen to be that of complexes of partial differential operators. It is again clear that there will be many formal difficulties in the study of overdetermined systems and of the associated complexes. Many aspects of the formal theory (e.g. formal integrability) of overdetermined systems have been studied in the sixties by D. Spencer, D. Quillen, H. Goldschmidt, R. Sweeney and M. Kuranishi, but still many problems remain open. Roughly speaking, at the same time, a quite satisfactory theory of general systems and complexes of linear partial differential operators with constant coefficients was developed by Ehrenpreis, Malgrange and Palamodov. (For results on the formal theory, cf. the book of J. F. Pommaret: ‘Systems of partial differential equations and Lie pseudogroups’, Gordon and Breach (1978; Zbl 0401.58006), for constant coefficient operators, the books of L. Ehrenpreis, ‘Fourier analysis in several complex variables’, Interscience Publ. (1970; Zbl 0195.10401), and V. P. Palamodov, ‘Linear partial differential operators with constant coefficients’. Mir Publ., Moscow (1967), and Springer Verlag, Grundlehren Series, vol. 168 (1970; Zbl 0191.43401).) At present, the main emphasis in the study of overdetermined systems lies perhaps on the study of systems of vector fields and in particular on what are called CR structures (problems of hypoellipticity, extendability and solvability).
The book under review is an extended and revised edition of a book previously published for Nauka, Novosibirsk, in Russian (1990; Zbl 0758.58002) and is thought as an introduction to the general theory of overdetermined systems and complexes of linear partial differential operators. The main arguments treated in the book are: a short introduction to the formal theory, a number of results on the theory of complexes of constant coefficient operators, tensor products of differential complexes, parametrices and fundamental solutions for complexes, Green-type operators and homotopy formulas on manifolds with boundary, Plemelj-Sohotskii formulas for general elliptic complexes, Cauchy problems and boundary problems for complexes, (Grothendieck-) duality theory for elliptic complexes and the Atiyah-Bott-Lefschetz fixed point theorem for elliptic complexes. (Results on CR-systems and results related to what goes nowadays under the name “algebraic analysis” are not presented in the book.) The book contains a number of examples, historical comments and exercises. Many results presented in the book are due to the author himself. Unfortunately, due of course to space limitations, a number of results are stated without proofs, and many page numbers in the index are wrong (since in the final stage of the publishing process the “source” file was processed once more, after the index had been made by hand, as I understand.) This is outweighed by the amount of material presented and I think that mathematicians interested in the theory will find a lot of material in the book that they cannot find in book-form elsewhere.

58J10 Differential complexes
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
35N05 Overdetermined systems of PDEs with constant coefficients