Existence Theorems for Analytic Linear Partial Differential Equations

Abstract

The theory of analytic systems of partial differential equations was first systematically investigated by Riquier and Elie Cartan around 1900. The existence of local solutions involves an algebraic problem, finding formal power series solutions, and an analytic problem, proving the convergence of formal power series solutions. Cartan defined the notion of an involutive system of partial differential equations and, using his theory of exterior differential systems, was able to show the existence of formal power series solutions for involutive partial differential equations of first order and to prove the convergence by the Cauchy-Kowalewski theorem. His result was extended by Kihler to systems of partial differential equations of higher order and is known today as the Cartan-Kihler theorem. Adjoining to a system of partial differential equations of order k the equations obtained by differentiating the original equations gives rise to a system of partial differential equations of order k +1, the prolongation of the system, which has the same solutions as the original equations. Cartan conjectured that, by prolonging a system a sufficient number of times, one would obtain an involutive system; this result was proved by Kuranishi in 1957 within the framework of Cartan's theory of exterior differential systems, and is now referred to as the Cartan-Kuranishi prolongation theorem. In 1961, Spencer introduced, in his fundamental paper [6] on the deformation of pseudogroup structures, certain cohomology groups Hkj associated to a partial differential equation (see ? 3), which are dual to homology groups of a Koszul complex; and so the cohomology groups Hki vanish for all sufficiently large k (see Lemma 3.1). The vanishing of these cohomology groups was shown by Serre to be equivalent to Cartan's notion of involutiveness (see V. W. Guillemin and S. Sternberg [3]). It then became possible to analyse the role played by involutiveness in Cartan's theory of partial differential equations. In this paper, we prove the Cartan-Kahler theorem for systems of linear partial differential equations formulated in terms of Ehresmann's theory of