A generalization of an integrability theorem of Darboux

Abstract: In his monograph "Le\c{c}ons sur les syst`emes orthogonaux et les coordonn\'ees curvilignes. Principes de g\'eom\'etrie analytique", 1910, Darboux stated three theorems providing local existence and uniqueness of solutions to first order systems of the type [\partial_{x_i} u_\alpha(x)=f^\alpha_i(x,u(x)),\quad i\in I_\alpha\subseteq{1,\dots,n}.] For a given point $\bar x\in \mathbb{R}<sup>n$ it is assumed that the values of the unknown $u_\alpha$ are given locally near $\bar x$ along $${x\,|\, x_i=\bar x_i \, \text{for each}\, i\in I_\alpha}$$. The more general of the theorems, Th\'eor`eme III, was proved by Darboux only for the cases $n=2$ and $3$. In this work we formulate and prove a generalization of Darboux's Th\'eor`eme III which applies to systems of the form [{\mathbf r}i(u\alpha)\big|x = f_i^\alpha (x, u(x)), \quad i\in I\alpha\subseteq{1,\dots,n}] where $\mathcal R={{\mathbf r}<em>i}</em>{i=1}<sup>n$ is a fixed local frame of vector fields near $\bar x$. The data for $u_\alpha$ are prescribed along a manifold $\Xi_\alpha$ containing $\bar x$ and transverse to the vector fields ${{\mathbf r}<em>i\,|\, i\in I</em>\alpha}$. We identify a certain Stable Configuration Condition (SCC). This is a geometric condition that depends on both the frame $\mathcal R$ and on the manifolds $\Xi_\alpha$; it is automatically met in the case considered by Darboux. Assuming the SCC and the relevant integrability conditions are satisfied, we establish local existence and uniqueness of a $C<sup>1$-solution via Picard iteration for any number of independent variables $n$.