### 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) |
---|---|

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 |

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### All Science Journal Classification (ASJC) codes

- Geometry and Topology

### Cite this

*Journal of Geometric Analysis*,

*29*(4), 3470-3493. https://doi.org/10.1007/s12220-018-00119-6

}

*Journal of Geometric Analysis*, vol. 29, no. 4, pp. 3470-3493. https://doi.org/10.1007/s12220-018-00119-6

**A Generalization of an Integrability Theorem of Darboux.** / Benfield, Michael; Jenssen, Helge Kristian; Kogan, Irina A.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A Generalization of an Integrability Theorem of Darboux

AU - Benfield, Michael

AU - Jenssen, Helge Kristian

AU - Kogan, Irina A.

PY - 2019/12/1

Y1 - 2019/12/1

N2 - 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¯ ∈ Rn it is assumed that the values of the unknown uα 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 uα 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 C1-solution via Picard iteration for any number of independent variables n.

AB - 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¯ ∈ Rn it is assumed that the values of the unknown uα 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 uα 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 C1-solution via Picard iteration for any number of independent variables n.

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U2 - 10.1007/s12220-018-00119-6

DO - 10.1007/s12220-018-00119-6

M3 - Article

AN - SCOPUS:85057594271

VL - 29

SP - 3470

EP - 3493

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

SN - 1050-6926

IS - 4

ER -