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On local solvability of underdetermined systems of vector fields. (English) Zbl 0708.58025
The authors study the solvability of the equation \[ (I)\quad \delta_{n-I}\tilde u=\tilde f, \] where \(\tilde f\) is an element of the space \({\mathcal C}^{\infty}(A,\Lambda^ n(C\otimes T^*\Omega /{\mathcal L}^{\perp}))\) of germs of the sections of \(\Lambda^ n(C\otimes T^*\Omega)/{\mathcal L}^{\perp})\) at A and \(\tilde u\in {\mathcal C}^{\infty}(A,\Lambda^{n-I}(C\otimes T^*\Omega /{\mathcal L}^{\perp}))\) is a formal integrable structure over the smooth paracompact manifold \(\Omega\), A is a fixed arbitrary point of \(\Omega\), n is a fiber dimension of \({\mathcal L}.\)
The authors introduce a solvability condition denoted \({\mathcal P}n-I\) and prove that it is necessary for the local solvability of (I) even for locally integrable smooth structures.
Reviewer: S.Bajzaev

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