Vol. 20, No. 3, 1967

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“The δ-Poincaré estimate”

William John Sweeney

Vol. 20 (1967), No. 3, 559–570

The main application of the theorem proved here is to establish the local solvability of a system of linear partial differential equations, in the analytic case, by a homological procedure based on the associated Spencer resolution and δ-cohomology. The theorem states that the δ-cohomology associated with an involutive system of partial differential equations vanishes in a normed sense. From this one can show that the Spencer resolution associated with an involutive system is exact for analytic data, and thus by a result of D. G. Quillen the corresponding inhomogeneous system has local solutions, provided the inhomogeneous term is analytic and satisfies the appropriate compatibility conditions in the overdetermined case. It is well known that if an arbitrary system is prolonged a sufficient number of times, the resulting system will have vanishing δ-cohomology. According to a result of J. P. Serre this is equivalent to the resulting system being involutive. Thus the question of local solvability reduces to the involutive case, and we obtain the classical existence theorem of Cartan-Kähler.

Mathematical Subject Classification
Primary: 35.30
Secondary: 55.00
Received: 22 June 1965
Published: 1 March 1967
William John Sweeney