TY - JOUR

T1 - A Generalization of an Integrability Theorem of Darboux

AU - Benfield, Michael

AU - Jenssen, Helge Kristian

AU - Kogan, Irina A.

N1 - Funding Information:
This work was supported in part by the NSF Grants DMS-1311353 (PI: Jenssen) and DMS-1311743 (PI: Kogan).
Publisher Copyright:
© 2018, Mathematica Josephina, Inc.

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 -