A Generalization of an Integrability Theorem of Darboux

Abstract

In his monograph “Systèmes Orthogonaux” (Darboux, in Leçons sur les systèmes orthogonaux et les coordonnées curvilignes, Gauthier-Villars, Paris, 1910) Darboux stated three theorems providing local existence and uniqueness of solutions to first-order systems of the type

$$\begin{aligned} \partial _{x_i} u_\alpha (x)=f^\alpha _i(x,u(x)), \quad i\in I_\alpha \subseteq \{1,\dots ,n\}. \end{aligned}$$

For a given point ${\bar{x}}\in \mathbb {R}^n$ 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, Théorème 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 Théorème III which applies to systems of the form

$$\begin{aligned} {{\mathbf {r}}}_i(u_\alpha )\big |_x = f_i^\alpha (x, u(x)), \quad i\in I_\alpha \subseteq \{1,\dots ,n\} \end{aligned}$$

where ${\mathcal {R}}=\{{{\mathbf {r}}}_i\}_{i=1}^n$ 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 $\{{{\mathbf {r}}}_i\,|\, i\in I_\alpha \}$. 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 $C^1$-solution via Picard iteration for any number of independent variables n.