Global solutions to involutive systems

Bergamasco, A.; Kirilov, A.; Nunes, W.; Zani, S.

doi:10.1090/proc/12633

Global solutions to involutive systems
HTML articles powered by AMS MathViewer

by A. P. Bergamasco, A. Kirilov, W. V. L. Nunes and S. L. Zani PDF

Proc. Amer. Math. Soc. 143 (2015), 4851-4862 Request permission

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 $a$.

References

Adalberto P. Bergamasco, Remarks about global analytic hypoellipticity, Trans. Amer. Math. Soc. 351 (1999), no. 10, 4113–4126. MR 1603878, DOI 10.1090/S0002-9947-99-02299-0
Adalberto P. Bergamasco, Cleber de Medeira, and Sérgio Luís Zani, Globally solvable systems of complex vector fields, J. Differential Equations 252 (2012), no. 8, 4598–4623. MR 2881049, DOI 10.1016/j.jde.2012.01.007
Adalberto P. Bergamasco and Alexandre Kirilov, Global solvability for a class of overdetermined systems, J. Funct. Anal. 252 (2007), no. 2, 603–629. MR 2360930, DOI 10.1016/j.jfa.2007.03.013
Adalberto P. Bergamasco, Alexandre Kirilov, Wagner Vieira Leite Nunes, and Sérgio Luís Zani, On the global solvability for overdetermined systems, Trans. Amer. Math. Soc. 364 (2012), no. 9, 4533–4549. MR 2922600, DOI 10.1090/S0002-9947-2012-05414-6
Adalberto P. Bergamasco, Wagner V. L. Nunes, and Sérgio Luís Zani, Global analytic hypoellipticity and pseudoperiodic functions, Mat. Contemp. 18 (2000), 43–57 (English, with English and Portuguese summaries). VI Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 1999). MR 1812862
Adalberto P. Bergamasco, Wagner V. L. Nunes, and Sérgio Luís Zani, Global properties of a class of overdetermined systems, J. Funct. Anal. 200 (2003), no. 1, 31–64. MR 1974087, DOI 10.1016/S0022-1236(02)00055-1
Adalberto P. Bergamasco and Gerson Petronilho, Global solvability of a class of involutive systems, J. Math. Anal. Appl. 233 (1999), no. 1, 314–327. MR 1684389, DOI 10.1006/jmaa.1999.6310
Shiferaw Berhanu, Paulo D. Cordaro, and Jorge Hounie, An introduction to involutive structures, New Mathematical Monographs, vol. 6, Cambridge University Press, Cambridge, 2008. MR 2397326, DOI 10.1017/CBO9780511543067
Fernando Cardoso and Jorge Hounie, Global solvability of an abstract complex, Proc. Amer. Math. Soc. 65 (1977), no. 1, 117–124. MR 463721, DOI 10.1090/S0002-9939-1977-0463721-8
François Trèves, Hypo-analytic structures, Princeton Mathematical Series, vol. 40, Princeton University Press, Princeton, NJ, 1992. Local theory. MR 1200459