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Global solutions to involutive systems


Authors: A. P. Bergamasco, A. Kirilov, W. V. L. Nunes and S. L. Zani
Journal: Proc. Amer. Math. Soc. 143 (2015), 4851-4862
MSC (2010): Primary 35A05, 35N10, 58J10
DOI: https://doi.org/10.1090/proc/12633
Published electronically: May 7, 2015
MathSciNet review: 3391043
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Abstract | References | Similar Articles | Additional Information

Abstract: In this work we consider a class of systems of two vector fields on the 3-torus associated to a closed smooth complex 1-form $ c=a+ib$ with $ b$ exact. Necessary conditions and sufficient conditions for this system to be globally solvable are provided in terms of the position of the global extrema and the connectedness of sublevel and superlevel sets of the primitives of $ b$, together with arithmetical properties of the periods of $ a$.


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Additional Information

A. P. Bergamasco
Affiliation: Departamento de Matemática, ICMC-USP, Caixa Postal 668, 13560-970, São Carlos, SP, Brazil
Email: apbergam@icmc.usp.br

A. Kirilov
Affiliation: Departamento de Matemática, UFPR, Caixa Postal 19081, 1, 81531-980, Curitiba, PR, Brasil
Email: akirilov@ufpr.br

W. V. L. Nunes
Affiliation: Departamento de Matemática, ICMC-USP, Caixa Postal 668, 13560-970, São Carlos, SP, Brazil
Email: wvlnunes@icmc.usp.br

S. L. Zani
Affiliation: Departamento de Matemática, ICMC-USP, Caixa Postal 668, 13560-970, São Carlos, SP, Brazil
Email: szani@icmc.usp.br

DOI: https://doi.org/10.1090/proc/12633
Keywords: Global solvability, complex 1-form, periodic solutions
Received by editor(s): February 28, 2014
Received by editor(s) in revised form: September 11, 2014
Published electronically: May 7, 2015
Additional Notes: The first author was supported in part by CNPq and FAPESP
The third author was supported in part by FAPESP
The fourth author was supported in part by CNPq and FAPESP
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2015 American Mathematical Society

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