×

On the global solvability for overdetermined systems. (English) Zbl 1275.35004

Summary: 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.

MSC:

35A01 Existence problems for PDEs: global existence, local existence, non-existence
35N10 Overdetermined systems of PDEs with variable coefficients
58J10 Differential complexes
35B10 Periodic solutions to PDEs
35F05 Linear first-order PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adalberto P. Bergamasco, Remarks about global analytic hypoellipticity, Trans. Amer. Math. Soc. 351 (1999), no. 10, 4113 – 4126. · Zbl 0932.35046
[2] 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. · Zbl 0777.58041 · doi:10.1006/jfan.1993.1068
[3] 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. · Zbl 0863.58062 · doi:10.1007/PL00004558
[4] 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. · Zbl 1108.35026 · doi:10.1090/conm/400/07527
[5] 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). · Zbl 1157.35304 · doi:10.1016/j.matpur.2006.08.001
[6] Adalberto P. Bergamasco and Alexandre Kirilov, Global solvability for a class of overdetermined systems, J. Funct. Anal. 252 (2007), no. 2, 603 – 629. · Zbl 1158.58011 · doi:10.1016/j.jfa.2007.03.013
[7] 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). · Zbl 1063.35051
[8] 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). · Zbl 0979.35036
[9] 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. · Zbl 1034.32024 · doi:10.1016/S0022-1236(02)00055-1
[10] 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. · Zbl 1077.35004
[11] 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. · Zbl 1139.35038
[12] 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. · Zbl 1153.35006 · doi:10.1080/03605300701833565
[13] Shiferaw Berhanu, Paulo D. Cordaro, and Jorge Hounie, An introduction to involutive structures, New Mathematical Monographs, vol. 6, Cambridge University Press, Cambridge, 2008. · Zbl 1151.35011
[14] Fernando Cardoso and Jorge Hounie, Global solvability of an abstract complex, Proc. Amer. Math. Soc. 65 (1977), no. 1, 117 – 124. · Zbl 0335.58015
[15] Jorge Hounie, Globally hypoelliptic and globally solvable first-order evolution equations, Trans. Amer. Math. Soc. 252 (1979), 233 – 248. · Zbl 0424.35030
[16] Abdelhamid Meziani, Hypoellipticity of nonsingular closed 1-forms on compact manifolds, Comm. Partial Differential Equations 27 (2002), no. 7-8, 1255 – 1269. · Zbl 1017.58014 · doi:10.1081/PDE-120005837
[17] François Treves, Study of a model in the theory of complexes of pseudodifferential operators, Ann. of Math. (2) 104 (1976), no. 2, 269 – 324. · Zbl 0354.35067 · doi:10.2307/1971048
[18] François Trèves, Hypo-analytic structures, Princeton Mathematical Series, vol. 40, Princeton University Press, Princeton, NJ, 1992. Local theory. · Zbl 0565.35079
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.