A generalization of an integrability theorem of Darboux

Abstract

In his monograph "Leçons sur les systèmes orthogonaux et les coordonnées curvilignes. Principes de géométrie analytique", 1910, Darboux stated three theorems providing local existence and uniqueness of solutions to first order systems of the type \[\partial_{x_i} u_α(x)=f^α_i(x,u(x)),\quad i\in I_α\subseteq\{1,\dots,n\}.\] 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 \[{\mathbf r}_i(u_α)\big|_x = f_i^α(x, u(x)), \quad i\in I_α\subseteq\{1,\dots,n\}\] 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 $n$.

21 pages, 2 figures