## 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 ∂xiuα(x)=fiα(x,u(x)),i∈Iα⊆{1,⋯,n}.For a given point x¯ ∈ R^{n} it is assumed that the values of the unknown u_{α} are given locally near x¯ along {x|xi=x¯ifor eachi∈Iα}. 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 ri(uα)|x=fiα(x,u(x)),i∈Iα⊆{1,⋯,n}where R={ri}i=1n is a fixed local frame of vector fields near x¯. The data for u_{α} are prescribed along a manifold Ξ _{α} containing x¯ and transverse to the vector fields {ri|i∈Iα}. We identify a certain Stable Configuration Condition (SCC). This is a geometric condition that depends on both the frame R and on the manifolds Ξ _{α}; 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.

Original language | English (US) |
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Pages (from-to) | 3470-3493 |

Number of pages | 24 |

Journal | Journal of Geometric Analysis |

Volume | 29 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 2019 |

## All Science Journal Classification (ASJC) codes

- Geometry and Topology