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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 \(\mathbb R^2 \to \mathbb T^2\).

MSC:
58J10 Differential complexes
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
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