zbMATH — the first resource for mathematics

Global solvability for a class of overdetermined systems. (English) Zbl 1158.58011
The paper continues the study [J. Funct. Anal. 200, No. 1, 31–64 (2003; Zbl 1034.32024)] of a particular locally integrable structure on \(\mathbb T^3\) associated with a smooth, real, closed 1-form \(b\) on \(\mathbb T^2\). When \(b\) is exact, then a necessary and sufficient condition of global solvability is connectedness of all the sublevel and superlevel sets of the global primitive of \(b\). This follows from a general result of F. Cardoso and J. Hounie [Proc. Am. Math. Soc. 65 117–124 (1977; Zbl 0335.58015)].
Now, the authors extend the result to the “incommensurable” case, when \(b\) is not exact. The modified necessary and sufficient condition refers to the primitive of the pull-back of \(b\) on the universal covering \(\mathbb R^2 \to \mathbb T^2\).

58J10 Differential complexes
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
Full Text: DOI
[1] Arnold, V.I., Topological and ergodic properties of closed 1-forms with incommensurable periods, Funct. anal. appl., 25, 2, 81-90, (1991) · Zbl 0732.58001
[2] Bergamasco, A., Remarks about global analytic hypoellipticity, Trans. amer. math. soc., 351, 4113-4126, (1999) · Zbl 0932.35046
[3] Bergamasco, A.; da Silva, P. Dattori, Global solvability for a special class of vector fields on the torus, (), 11-20 · Zbl 1108.35026
[4] Bergamasco, A.; da Silva, P. Dattori, Solvability in the large for a class of vector fields on the torus, J. math. pures appl., 86, 427-447, (2006) · Zbl 1157.35304
[5] Bergamasco, A.; Meziani, A., Solvability near the characteristic set for a class of planar vector fields of infinite type, Ann. inst. Fourier (Grenoble), 55, 77-112, (2005) · Zbl 1063.35051
[6] Bergamasco, A.; Zani, S., Prescribing analytic singularities for solutions of a class of vector fields on the torus, Trans. amer. math. soc., 357, 4159-4174, (2005) · Zbl 1077.35004
[7] Bergamasco, A.; Cordaro, P.; Malagutti, P., Globally hypoelliptic systems of vector fields, J. funct. anal., 114, 267-285, (1993) · Zbl 0777.58041
[8] Bergamasco, A.; Cordaro, P.; Petronilho, G., Global solvability for certain classes of undetermined of vector fields, Math. Z., 223, 223-274, (1996) · Zbl 0863.58062
[9] Bergamasco, A.; Nunes, W.; Zani, S., Global analytic hypoellipticity and pseudoperiodic functions, Mat. contemp., 18, 43-57, (2000) · Zbl 0979.35036
[10] Bergamasco, A.; Nunes, W.; Zani, S., Global analytic hypoellipticity for a class of overdetermined systems, J. funct. anal., 31-64, (2003) · Zbl 1034.32024
[11] Bergamasco, A.; Cordaro, P.; Petronilho, G., Global solvability for a class of complex vector fields on the two-torus, Comm. partial differential equations, 29, 785-819, (2004) · Zbl 1065.35088
[12] Cardoso, F.; Hounie, J., Global solvability of an abstract complex, Proc. amer. math. soc., 65, 117-124, (1977) · Zbl 0335.58015
[13] Epstein, D.B.A., Curves on 2-manifolds and isotopies, Acta math., 115, 83-107, (1966) · Zbl 0136.44605
[14] Hirsch, M.W., Differential topology, Grad. texts in math., vol. 33, (1976), Springer-Verlag New York · Zbl 0121.18004
[15] Hounie, J., Global hypoelliptic and globally solvable first order evolution equations, Trans. amer. math. soc., 252, 233-248, (1979) · Zbl 0424.35030
[16] Meziani, A., Hypoellipticity of nonsingular closed 1-forms on compact manifolds, Comm. partial differential equations, 27, 1255-1269, (2002) · Zbl 1017.58014
[17] Treves, F., Solvability of a model in the theory of complexes of pseudodifferential operators, Ann. of math. (2), 104, 269-324, (1976) · Zbl 0354.35067
[18] Treves, F., Hypoanalytic structures (local theory), (1992), Princeton Univ. Press Princeton, NJ · Zbl 0787.35003
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.