On the global solvability for overdetermined systems
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- by Adalberto P. Bergamasco, Alexandre Kirilov, Wagner Vieira Leite Nunes and Sérgio Luís Zani PDF
- Trans. Amer. Math. Soc. 364 (2012), 4533-4549 Request permission
Abstract:
We consider a class of systems of two smooth vector fields on the 3-torus associated to a closed 1-form. We prove that the global solvability is completely determined by the connectedness of the sublevel and superlevel sets of a primitive of this 1-form in the minimal covering.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, Paulo D. Cordaro, and Pedro A. Malagutti, Globally hypoelliptic systems of vector fields, J. Funct. Anal. 114 (1993), no. 2, 267–285. MR 1223704, DOI 10.1006/jfan.1993.1068
- Adalberto P. Bergamasco, Paulo D. Cordaro, and Gerson Petronilho, Global solvability for certain classes of underdetermined systems of vector fields, Math. Z. 223 (1996), no. 2, 261–274. MR 1417431, DOI 10.1007/PL00004558
- Adalberto P. Bergamasco and Paulo L. Dattori da Silva, Global solvability for a special class of vector fields on the torus, Recent progress on some problems in several complex variables and partial differential equations, Contemp. Math., vol. 400, Amer. Math. Soc., Providence, RI, 2006, pp. 11–20. MR 2222462, DOI 10.1090/conm/400/07527
- A. P. Bergamasco and P. L. Dattori da Silva, Solvability in the large for a class of vector fields on the torus, J. Math. Pures Appl. (9) 86 (2006), no. 5, 427–447 (English, with English and French summaries). MR 2271625, DOI 10.1016/j.matpur.2006.08.001
- 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 and Abdelhamid Meziani, Solvability near the characteristic set for a class of planar vector fields of infinite type, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 1, 77–112 (English, with English and French summaries). MR 2141289
- 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 Sérgio Luís Zani, Prescribing analytic singularities for solutions of a class of vector fields on the torus, Trans. Amer. Math. Soc. 357 (2005), no. 10, 4159–4174. MR 2159704, DOI 10.1090/S0002-9947-05-03905-X
- Adalberto P. Bergamasco and Sérgio Luís Zani, Globally analytic hypoelliptic vector fields on compact surfaces, Proc. Amer. Math. Soc. 136 (2008), no. 4, 1305–1310. MR 2367104, DOI 10.1090/S0002-9939-07-09097-1
- Adalberto P. Bergamasco and Sérgio Luís Zani, Global analytic regularity for structures of co-rank one, Comm. Partial Differential Equations 33 (2008), no. 4-6, 933–941. MR 2424383, DOI 10.1080/03605300701833565
- 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
- Jorge Hounie, Globally hypoelliptic and globally solvable first-order evolution equations, Trans. Amer. Math. Soc. 252 (1979), 233–248. MR 534120, DOI 10.1090/S0002-9947-1979-0534120-1
- Abdelhamid Meziani, Hypoellipticity of nonsingular closed 1-forms on compact manifolds, Comm. Partial Differential Equations 27 (2002), no. 7-8, 1255–1269. MR 1924466, DOI 10.1081/PDE-120005837
- François Treves, Study of a model in the theory of complexes of pseudodifferential operators, Ann. of Math. (2) 104 (1976), no. 2, 269–324. MR 426068, DOI 10.2307/1971048
- François Trèves, Hypo-analytic structures, Princeton Mathematical Series, vol. 40, Princeton University Press, Princeton, NJ, 1992. Local theory. MR 1200459
Additional Information
- Adalberto P. Bergamasco
- Affiliation: Departamento de Matemática, ICMC-USP, Caixa Postal 668, 13560-970, São Carlos, SP, Brazil
- Email: apbergam@icmc.usp.br
- Alexandre Kirilov
- Affiliation: Departamento de Matemática, UFPR, Caixa Postal 19081, Curitiba, PR, 81531-990, Brazil
- Email: akirilov@ufpr.br
- Wagner Vieira Leite Nunes
- Affiliation: Departamento de Matemática, ICMC-USP, Caixa Postal 668, 13560-970, São Carlos, SP, Brazil
- Email: wvlnunes@icmc.usp.br
- Sérgio Luís Zani
- Affiliation: Departamento de Matemática, ICMC-USP, Caixa Postal 668, 13560-970, São Carlos, SP, Brazil
- Email: szani@icmc.usp.br
- Received by editor(s): January 14, 2010
- Received by editor(s) in revised form: July 8, 2010
- Published electronically: March 21, 2012
- Additional Notes: The first and the fourth authors were partially supported by CNPq and FAPESP; the second author was partially supported by CNPq; the third author was partially supported by FAPESP
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 4533-4549
- MSC (2010): Primary 35A01, 35N10, 58J10
- DOI: https://doi.org/10.1090/S0002-9947-2012-05414-6
- MathSciNet review: 2922600