A Generalization of an Integrability Theorem of Darboux

In Chapter I, Livre III of his monograph “Systèmes Orthogonaux” Darboux stated three integrability theorems. They provide local existence and uniqueness of solutions to systems of first order Partial Differential Equations of the type: ∂xiuα(x) = f α i (x, u(x)), i ∈ Iα ⊆ {1, . . . , n}. Here x = (x1, . . . , xn) , u = (u1, . . . , um), and f α i (x, u(x)) are given functions. For each dependent variable uα, the system prescribes partial derivatives in the coordinate directions given by the index set Iα. The data are given locally near a fixed point x̄ ∈ R, and prescribe each unknown uα along the affine subspace spanned by the coordinate vectors complimentary to the coordinate vectors defined by indices in Iα. Darboux’s first theorem applies to determined systems, in which case |Iα| = 1 for all α, while his second theorem is Frobenius’ Theorem for complete systems, in which case |Iα| = n for all α. The third theorem addresses the general situation where Iα are arbitrary subsets of {1,. . . ,n} varying with α. Under the appropriate integrability conditions, Darboux proved his third theorem in the cases n = 2 and n = 3. However, his argument does not appear to generalize in any easy manner to cases with more than three independent variables. In the present work, we formulate and prove a theorem that generalizes Darboux’s third theorem to systems of the form ri(uα) ∣∣ x = f i (x, u(x)), i ∈ Iα ⊆ {1, . . . , n} where {ri}i=1 is an arbitrary local frame of vector-fields near x̄. Furthermore, the data for uα can be prescribed along an arbitrary submanifold through x̄ transversal to the subset of vector-fields {ri | i ∈ Iα}. Our proof applies to any number of independent variables and uses a nonstandard application of Picard iteration. The approach requires only C smoothness of the f i and the initial data; for analytic systems with analytic data, the result is a consequence of the Cartan-Kähler theorem.